Oscillation Theory for Functional Differential Equations

,., . . . . . . . . . .. . . . .... , . . . . . " . t OSCILLATION THEORY FOR FUNCTIONAL DIFFERENTIAL EQUATIONS L. H. E...

Author: Lynn Erbe | Q. Kong | B.G. Zhang

128 downloads 1488 Views 15MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

,., . . . . . . . . . .. . . . .... , . . . . . " . t

OSCILLATION THEORY FOR FUNCTIONAL DIFFERENTIAL EQUATIONS

L. H. Erbe Qingkai Kong B. G. Zhang


PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed

University of Delaware Newark, Delaware

CHAIRMEN OF THE EDITORIAL BOARD S. Kobayashi

Edwin Hewitt

University of California, Berkeley Berkeley, California

University of Washington Seanle, Washington

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology Marvin Marcus University of California, Santa Barbara W: S. Massey Yale University

Ani! Nerode Cornell University

Donald Passman University of Wrsconsin-Madison Fred S. Roberts Rutgers University Gian-Carlo Rota Massachruetts Institute of Technology David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universitllt Siegen Mark Teply University of Wrsconsin-Milwaukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1 • K. Yeno, Integral Formulas in Riemannian Geometry ( 1970) 2. S. Kobeyeshi, Hyperbolic Manifolds and Holomorphic Mappings (1970) 3. V. S. Vledimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. Littlewood, trans.) (1970) 4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation ed.; K. Makowski, trans.) (1971) 5. L. Nerici et el., Functional Analysis and Valuation Theory (1971) 6. S. S. Psssmen, Infinite Group Rings (1971) 7. L. Dornhoff, Group Representation Theory. Part A: Ordinary Representation Theory. Part B: Modular Representation Theory ( 1971, 197 2) 8. W. Boothby end G. L. Weiss, eds., Symmetric Spaces (1972) 9. Y. Metsushime, Differentiable Manifolds (E. T. Kobayashi, trans.) (1972) 10. L. E. Werd, Jr., Topology (1972) 11. A. Bebekhenien, Cohomological Methods in Group Theory (1972) 12. R. Gilmer, Multiplicative Ideal Theory (1972) 13. J. Yeh, Stochastic Processes and the Wiener Integral (1973) 14. J. Berros-Neto, Introduction to the Theory of Distributions (1973) 15. R. Lersen, Functional Analysis (1973) 16. K. Yeno endS. lshihere, Tangent and Cotangent Bundles (1973) 17. C. Procesi, Rings with Polynomial Identities (1973) 18. R. Hermenn, Geometry, Physics, and Systems (1973) 19. N. R. Wellech, Harmonic Analysis on Homogeneous Spaces (1973) 20. J. Dieudonn6, Introduction to the Theory of Formal Groups (1973) 21. I. Veismen, Cohomology and Differential Forms (1973) 22. B.-Y. Chen, Geometry of Submanifolds (1973) 23. M. Mercus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975) 24. R. Lersen, Banach Algebras (1973) 25. R. 0. Kujele end A. L. Vitter, eds., Value Distribution Theory: Part A; Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1973) 26. K. B. Stolersky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Megid, The Separable Galois Theory of Commutative Rings (1974) 28. B. R. McDoneld, Finite Rings with Identity (1974) 29. J. Seteke, Linear Algebra (S. Koh et al., trans.) (1975) 30. J. S. Golen, Localization of Noncommutative Rings (1975) 31. G. Klembeuer, Mathematical Analysis (1975) 32. M. K. Agoston, Algebraic Topology (1976) 33. K. R. Goodeerl, Ring Theory (1976) 34. L. E. Mensfield, Linear Algebra with Geometric Applications (1976) 35. N.J. Pullmen, Matrix Theory and Its Applications (1976) 36. B. R. McDoneld, Geometric Algebra Over Local Rings (1976) 37. C. W. Groetsch, Generalized Inverses of Linear Operators (1977) 38. J. E. Kuczkowski end J. L. Gersting, Abstract Algebra (1977) 39. C. 0. Christenson end W. L. Voxmen, Aspects of Topology (1977) 40. M. Negate, Field Theory (1977) 41 . R. L. Long, Algebraic Number Theory ( 1977) 42. W. F. Pfeffer, Integrals and Measures (1977) 43. R. L. Wheeden end A. Zygmund, Measure and Integral (19771 44. J. H. Curtiss, Introduction to Functions of a Complex Variable (1978) 45. K. Hrbecek end T. Jech, Introduction to Set Theory (1978) 46. W. S. Massey, Homology and Cohomology Theory (1978) 47. M. Marcus, Introduction to Modern Algebra ( 1978) 48. E. C. Young, Vector and Tensor Analysis (1978) 49. S. B. Nadler, Jr., Hyperspaces of Sets (1978) 50. S. K. Segel, Topics in Group Kings (1978) 51. A. C. M. ven Rooij, Non-Archimedean Functional Analysis (1978) 52. L. Corwin end R. Szczerbe, Calculus in Vector Spaces (1979)

C. Sadosky, Interpolation of Operators and Singular Integrals (1979) J. Cronin, Differential Equations ( 1980) C. W. GroBtsch, Elements of Applicable Functional Analysis (1980) /. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980) H. I. Fraadan, Deterministic Mathematical Models in Population Ecology ( 1980) S. 8. Chae, Lebesgue Integration (1980) C. S. Rees eta/., Theory and Applications of Fourier Analysis (1981) L. Nachbin, Introduction to Functional Analysis (R. M. Aron, trans.) (1981) G. Orzech and M. Orzech, Plane Algebraic Curves ( 1981 l R. Johnsonbaugh and W. E. Pfaffanbargar, Foundations of Mathematical Analysis (1981) 63. W. L. Voxman and R. H. Goatschal, Advanced Calculus (1981) 64. L. J. Corwin and R. H. Szczerba, Multivariable Calculus (1982) 65. V. /. lstrilfascu, Introduction to Linear Operator Theory (1981) 66. R. D. Jarvinen, Finite and Infinite Dimensional Linear Spaces (1981) 67. J. K. Seem and P. E. Ehrlich, Global Lorentzian Geometry (1981) 68. D. L. Armacost, The Structure of Locally Compact Abelian Groups (1981) 69. J. W. Brewer and M. K. Smith, ads., Emily Noether: A Tribute (1981) 70. K. H. Kim, Boolean Matrix Theory and Applications ( 1982) 71. T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982) 72. D. B.Gau/d, Differential Topology (1982) 73. R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983) 74. M. Carma/i, Statistical Theory and Random Matrices (1983) 75. J. H. Carruth at a/., The Theory of Topological Semigroups (1983) 76. R. L. Faber, Differential Geometry and Relativity Theory (1983) 77. S. Barnett, Polynomials and Linear Control Systems (1983) 78. G. Karpilovsky, Commutative Group Algebras (1983) 79. F. Van Oystaayan and A. Verschoran, Relative Invariants of Rings (1983) 80. I. Vaisman, A First Course in Differential Geometry (1984) 81. G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984) 82. T. Patrie and J. D. Randall, Transformation Groups on Manifolds (1984) 83. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) 84. T. Albu and C. NastiJsascu, Relative Finiteness in Module Theory (1984) 85. K. Hrbacak and T. Jach, Introduction to Set Theory: Second Edition ( 1984) 86. F. Van Oystaeyen and A. Varschoren, Relative Invariants of Rings (1984) 87. 8. R. McDonald, Linear Algebra Over Commutative Rings (1984) 88. M. Namba, Geometry of Projective Algebraic Curves ( 1984) 89. G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) 90. M. R. Bremner eta/., Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985) 91. A. E. Fekete, Real Linear Algebra (1 985) 92. S. 8. Chaa, Holomorphy and Calculus in Normed Spaces (1985) 93. A. J. Jerri, Introduction to Integral Equations with Applications (1985) 94. G. Karpi/ovsky, Projective Representations of Finite Groups (1985) 95. L. Narici and E. Beckenstain, Topological Vector Spaces (1985) 96. J. Weeks, The Shape of Space (1985) 97. P. R. Gribik and K. 0. Kortanek, Extremal Methods of Operations Research ( 1985) 98. J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis (1986) 99. G. D. Crown at a/., Abstract Algebra (1986) 100. J. H. Carruth eta/., The Theory of Topological Semigroups, Volume 2 (1986) 101. R. S. Doran and V. A. Belfi, Characterizations of C*-Aigebras (1986) 102. M. W. Jeter, Mathematical Programming (1986) 1 03. M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications (1986) 104. A. Verschoren, Relative Invariants of Sheaves (1987) 105. R. A. Usmani, Applied Linear Algebra (1987) 106. P. Blass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p > 0 (1987) 107. J. A. Reneke eta/., Structured Hereditary Systems (1987) 53. 54. 55. 56. 57. 58. 59. 60. 61 . 62.

108. H. Busflmann and B. B. Phsdkfl, Spaces with Distinguished Geodesics (1987) 109. R. Hartfl, lnvertibility and Singularity for Bounded Linear Operators ( 1988) 110. G. S. Lsdds fit al., Oscillation Theory of Differential Equations with Deviating Arguments (1987) 111 • L. Dudkin fit a/., Iterative Aggregation Theory ( 1987) 1 12. T. Okubo, Differential Geometry (1 987) 1 13. D. L. Stanaland M. L. Stancl, Real Analysis with Point-Set Topology (1 987) 1 14. T. C. Gard, Introduction to Stochastic Differential Equations (1 988) 1 15. S. S. Abhyankar, Enumerative Combinatoric& of Young Tableaux (1 988) 1 16. H. Stradsand R. FBrnstflinflr, Modular Lie Algebras and Their Representations (1988) 117. J. A. Huckaba, Commutative Rings with Zero Divisors (1988) 1 18. W. D. Wallis, Combinatorial Designs ( 1988) 119. W. Witsfaw, Topological Fields (1988) 120. G. Karpilovsky, Field Theory (1988) 121. S. Caflnflpflfll and F. Van Oystaflyfln, Brauer Groups and the Cohomology of Graded Rings (1989) 122. W. Kozlowski, Modular Function Spaces (1988) 123. E. Lowfln-Colsbundsrs, Function Classes of Cauchy Continuous Maps (1989) 124. M. Pavfll, Fundamentals of Pattern Recognition (1 989) 125. V. Lakshmikantham st a/., Stability Analysis of Nonlinear Systems (1 989) 1 26. R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1 989) 1 27. N. A. Watson, Parabolic Equations on an Infinite Strip ( 1989) 128. K. J. Hsstings, Introduction to the Mathematics of Operations Research (1 989) 129. B. Rnfl, Algebraic Theory of the Bianchi Groups (1989) 130. D. N. Dikranjan fit a/., Topological Groups (1989) 131. J. C. Morgan II, Point Set Theory (1990) 132. P. Siler and A. Witkowski, Problems in Mathematical Analysis (1 990) 133. H. J. Sussmann, Nonlinear Controllability and Optimal Control (1 990) 134. J.-P. Rorflns fit a/., Elements of Bayesian Statistics (1990) 135. N. Shflll, Topological Fields and Near Valuations (1 990) 136. B. F. Doolin and C. F. Martin, Introduction to Differential Geometry for Engineers (1990) 137. S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method (1990) 138. J. Okninski, Semigroup Algebras (1990) 139. K. Zhu, Operator Theory in Function Spaces (1990) 140. G. B. Pricfl, An Introduction to Multicomplex Spaces and Functions (1991 l 141. R. B. Darst, Introduction to Linear Programming (1991) 142. P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991 l 143. T. Husain, Orthogonal Schauder Bases (1991) 144. J. Foran, Fundamentals of Real Analysis (1991 l 145. W. C. Brown, Matrices and Vector Spaces (1991) 146. M. M. Rao and Z. D. Ran, Theory of Orlicz Spaces (1991) 147. J. S. Golan and T. Head, Modules and the Structures of Rings (1991) 148. C. Small, Arithmetic of Finite Fields ( 1 991) 149. K. Yang, Complex Algebraic Geometry ( 1991 l 150. D. G. Hoffman fit a/., Coding Theory (1991 l 151. M. 0. Gonzalez, Classical Complex Analysis (1992) 152. M. 0. Gonzalez, Complex Analysis (1992) 153. L. W. Baggett, Functional Analysis (1992) 154. M. Sniedovich, Dynamic Programming (1992) 155. R. P. Agarwal, Difference Equations and Inequalities (1992) 156. C. Brezinski, Biorthogonality and Its Applications to Numerical Analysis (1992) 157. C. Swartz, An Introduction to Functional Analysis (1992) 158. S. B. Nadler, Jr., Continuum Theory (1992) 159. M. A. AI-Gwaiz, Theory of Distributions (1992) 160. E. Perry, Geometry: Axiomatic Developments with Problem Solving (1992) 161. E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering (1992) 162. A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis (1992) 163. A. Charlier eta/., Tensors and the Clifford Algebra (1992)

P. Bilflf' and T. Nadzieja, Problema end Examples in Differential Equations (1992) E. Hansen, Global Optimization Using Interval Analysis (1992) S. Guerre-DelabriiJre, Classical Sequences in Banach Spaces (1992) Y. C. Wong, Introductory Theory of Topological Vector Spaces (1992) S. H. Kulkarni and B. V. Umaye, Real Function Algebras (1 992) W. C. Brown, Matrices Over Commutative Rings (1993) J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (1993) W. \1. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (1993) 172. E. C. Young, Vector and Tensor Analysis: Second Edition (1993) 173. T. A. Sick, Elementary Boundary Value Problema (1993) 174. M. Pavel, Fundamentals of Pattern Recognition: Second Edition (1 993) 175. S. A. Albeverio et al., Noncommutative Distributions (1993) 178. W. Fulks, Complex Variables 11993) 177. M. M. Rao, Conditional Measures and Applications (1993) 178. A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (1 994) 179. P. Neittaanmiki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1994) 180. J. Cronin, Differential Equations: Introduction and Qualitative Theory, Second Edition (1994) 181. S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) 182. X. Mao, Exponential Stability of Stochastic Differential Equations ( 1994) 183. B.S. Thomson, Symmetric Properties of Real Functions (1994) 184. J. E. Rubio, Optimization and Nonstandard Analysis (1994) 185. J. L. Sueso, P. Jars, and A. Varschoren, Compatibility, Stability, and Sheaves (1995) 188. A. N. Michel and K. Wang, Qualitative Theory of Dynamical Systems ( 1995) 187. M. R. Darnel, Theory of Lattice-Ordered Groups (1995) 188. Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (1995) 189. L. J. Corwin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (1995) 190. L. H. Erbe, Q. Kong, B. G. Zhang, Oscillation Theory for Functional Differential Equations ( 1995)

184. 185. 188. 187. 188. 189. 170. 171.

Additional Volumes in Preparation


L. H. Erbe University of Alberta Edmonton, Alberta, Canada

Qingkai Kong Northeastern Illinois University DeKalb, Illinois

B. G. Zhang . Ocean University of Oingdao Oingdao, Shandong, People's Republic of China

Marcel Dekker, Inc.

New York• Basel• Hong Kong

Library of Congress Cataloging-in-Publication Data Erbe, L. H. Oscillation theory for functional differential equations I L. H. Erbe, Qingkai Kong, B. G. Zhang. p. em. -(Monographs and textbooks in pure and applied mathematics ; 190) Includes bibliographical references and index. ISBN 0-8247-9598-9 (acid-free) 1. Functional differential equations-Numerical solutions. 2. Oscillations. I. Kong, Qingkai. II. Zhang, B. G. III. Title. IV. Series. QA372.E63 1994 515' .352-dc20 94-38757 CIP

The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the address below. This book is printed on acid-free paper.

Copyright© 1995 by MARCEL DEKKER, INC. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. MARCEL DEKKER, INC. 270 Madison Avenue, New York, New York 10016 Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

Preface

Various models of functional differential equations (FDEs) have been posed in different sciences, strongly motivating research in the qualitative theory of FDEs. As a part of this approach, oscillation theory of FDEs has developed very rapidly during the last decade. It has concerned itself largely with the oscillatory and nonoscillatory properties of solutions. Specifically, the topics dealt with include the following: 1. 2. 3. 4. 5. 6.

Finding conditions for oscillation of all solutions; Finding conditions for the existence of one nonoscillatory solution; Finding conditions for the existence of one oscillatory solution; Finding conditions for the nonexistence of oscillatory solutions; Estimating the distance between zeros of oscillatory solutions; Finding an asymptotic classification of nonoscillatory solutions and finding conditions for existence of solutions with designated asymptotic properties.

The first monograph using this approach was that of G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, and it has received much attention. This book covers the work up to 1984 and concerns mainly the effect of the deviating argument on the oscillation of solutions. However, it does not touch upon FDEs of neutral type. In the past decade research on the oscillation theory of FDEs of neutral type has been very active and fruitful, and has attracted the attention of many mathematicians worldwide. The recent interesting book by I. Gyori and G. Ladas iii

Preface

iv

summarizes some important work in this area, especially the contributions of the group centered at the University of Rhode Island. For instance, the relation between the distribution of the roots of characteristic equations and the oscillation of all solutions, and linearized oscillation theory have been developed largely by this group. However, their book does not deal with some other important problems, such as topics 3-6 mentioned above, although much progress has been made in these areas. Also, their book does not consider equations with nonlinear neutral terms of the form

(x(t)

+ cx"'(t- r))' + p(t)xP(t- r) =

0,

0 0, and max

OO

(1.2.4)

has a nonoscillatory solution if and only if pe

~

(1.2.5)

1.

In Chapter 3, we shall prove that every solution of (x(t)- cx(t- 1))'

+ px(t -1) = 0,

t "2. to

(1.2.6)

is oscillatory if pe>1-c,

cE(0,1).

(1.2.7)

Assume that 1- c < pe ~ 1, c E (0, 1), then the oscillations of (1.2.6) are caused by the neutral term. If pe > 1, then every solution of (1.2.4) oscillates. In Chapter 3, we shall show that if c < 0, pe > 1, then Eq. (1.2.6) has a nonoscillatory solution. This nonoscillation is caused by the neutral term. Therefore the presence of the neutral term can cause or destroy the oscillation of the solutions. In this book we will investigate systematically the effect of deviating arguments and neutral terms on the oscillation of systems in the sense that their presence causes or destroys the oscillation of the solutions. We shall present many of the recent developments in the following problems ( 1) (2) (3) (4) (5)

Criteria for all solutions to be oscillatory. The classification and existence of nonoscillatory solutions. The existence of oscillatory solutions. The distance between adjacent zeros of the oscillatory solutions. Some applications to ecology models.

1.3.

Formulation of Boundary Value Problems for Functional Differential Equations

The main sources of boundary value problems for ordinary differential equations without deviating arguments are boundary value problems for partial differential equations.

7

Preliminaries

For functional differential equations the situation is different. The usual sources of boundary value problems for functional differential equations arise from variational problems for these equations. For example, we consider the problem of an extremum of the functional

V(x(t)) =

i

t,

F(t,x(t),x(t- r),x(t),:i:(t- r))dt

(1.3.1)

to

with fixed or movable boundary values and initial functions. These variational problems lead to the following simple boundary value problem statements. 1. Find the continuous and smooth solutions of the equation

x"(t) = f(t,x(t),x(t- r1(t)), ... ,x(t- Tm(t))), x'(t),x'(t- r1(t)), ... ,x'(t- rm(t)), x"(t- r1(t)), ... , x"(t- Tm(t))

(1.3.2)

fortE (to, tl) having a given initial function 0 there exists an N = N(c) such that llxn- xmll < c for all n,m ~ N(c). Clearly a convergent sequence is a Cauchy sequence but the converse may not be true. A space X where every Cauchy sequence of elements of X converges to an element of X is called a complete space. A complete normed vector space is said to be a Banach space. Let M be a subset of a Banach space X. A point x E X is said to be a limit point of M if there exists a sequence of vectors in M which converges to x. We say a subset M is closed if M contains all of its limit points. The union of M and its limit points is called the closure of M and will be denoted by M. Let N, M be normed spaces, and X be a subset of N. An operator T: X --t M is continuous at a point x E X if and only if for any c > 0 there is a 6 > 0 such that IITx- Tyll < c for ally EX such that llx- Yll < 6. Tis continuous on X, or simply continuous, if it is continuous at all points of X. The following result is well-known. Theorem 1.4.1. Every continuous mapping of a closed bounded convex set in Rn into itself has a fixed point.

A subset S of a Banach space X is compact, if and only if every infinite sequence of elements of S has a subsequence which converges to an element of S. We say M is relatively compact if every infinite sequence in S contains a subsequence which converges to an element in X. That is, M is relatively compact, if M is compact. A subset Sin C([a, b], R) with norm

11!11

= sup lf(x)l :rE[a,b]

is relatively compact if and only if it is uniformly bounded and equicontinuous on [a, b] (Arzela-Ascoli Theorem). A family S in C([a, b], R) is called uniformly bounded if there exists a positive number M such that

lf(t)l :5 M

for all

t E [a, b],

all

f

E S.

Sis called equicontinuous iffor every c > 0 there exists a 6 = 6(c) > 0 such that lf(tt)- j(t2)l < c for all t~, t2 E [a, b] with lt1 - t2l < 6 and for all f E S.

Chapter 1

10

Theorem 1.4.2. (Schauder's Fixed Point Theorem). Let S be a closed, convex and nonempty subset of a Banach space X. LetT : S --1 S be a continuous mapping such that TS is a relatively compact subset of X. Then T has at least one fixed point in S. That is, there exists an x E S such that Tx = x. Remark 1.4.1. In oscillation theory we usually want to prove that the family of functions is uniformly bounded and cquicontinuous on [to, +oo ). According to Levitan's result, the family S is equicontinuous on [to, oo) if for any given c; > 0, the interval [to, oo) can be decomposed into a finite number of subintervals in such a way that on each subinterval all functions of the family S have oscillations less than c. A topology T on a linear space E is called locally convex if and only if every neighborhood of the element 0 includes a convex neighborhood of 0. A real valued function p( x) defined on a linear space X is called a semi norm on X iff the following conditions are satisfied:

p(x

+ y) ~ p(x) + p(y)

p(ax)

=

jajp(x),

a

is scalar.

From this definition, we can prove that a seminorm p(x) satisfies p(O) = 0, p(x 1 - x 2 ) ~ jp(x 1 ) - p(x 2 )j. In particular, p(x) ~ 0. However, it may happen that p(x) = 0 for x f 0. A family P of seminorms on X is said to be separating iff to each x =/= 0 there corresponds at least one p E P with p( x) =/= 0. For a separating seminorm family P, if p( x) = 0 for every p E P, then x = 0. A locally convex topology T on a linear space is determined by a family of seminorms {Pa- : a E I}, I being the index set. Let E be a locally convex space, x E E, {xn} E E. Then and only if Pa-(Xn- x) --1 0 as n --1 oo, for every a E I.

Xn --1

x in E if

A set S C E is bounded if and only if the set of numbers {Pa(x), xES} is bounded for every a E I. A complete metrizable locally convex space is called a Frechet space.

Theorem 1.4.3. (Schauder-Tychonov Theorem). Let X be a locally convex linear space, let S be a compact convex subset of X, and let T : S - t S be a continuous mapping with T(S) compact. Then T has a fixed point in S.

11

Preliminaries

For example C([t 0 , oo ), R) is a locally convex space consisting of the set of all continuous functions. The topology of C is the topology of uniform convergence on every compact interval of [to, oo ). The seminorm of the space

C([t 0 , oo ), R) is defined by p0

(x)

=

max Jx(t)J,

:r:E[to,]

x E C,

a E (to,oo).

Let X be any set. A metric in X is a function d : X x X following properties for all x, y, z EX :

-+

R having the

1. d(x,y) ~ 0 and d(x,y) = 0 if and only if x = y 2. d(y,x) = d(x,y) 3. d(x,z) :S d(x,y) + d(y,z) A metric space is a set X together with a given metric in X. A complete metric space is a metric space X in which every Cauchy sequence converges to a point in X. Let (X, d) be a metric space and let T : X -+ X. If there exists a number r E [0, 1) such that d(Tx,Ty) :S rd(x,y)

for every

x,y EX,

then we say T is a contraction mapping on X.

Theorem 1.4.4. (The Banach Contraction Mapping Principle). A contraction mapping on a complete metric space bas exactly one fixed point. Theorem 1.4.5. Banach space, let maps of Q into X contraction and B

(Krasnoselskii's Fixed Point Theorem).

Let X be a

Q be a bounded closed convex subset of X and let A, B be

such that Ax + By E Q for every pair x, y E Q. If A is a is completely continuous, then the equation Ax +Ex =x

bas a solution in

n.

A nonempty and closed subset ]{ of a Banach space X is called a cone if it possesses the following properties:

Chapter 1

12 (i) if a E R+ and x E K, then ax E K (ii) if x, y E K then x + y E K (iii) if x E K- {0}, then -x fl. K.

We say that a Banach space X is partially ordered if X contains a cone K with a nonempty interior. The ordering :5 in X is then defined as follows: x :::; y

if and only if

y- x E K.

Let M be a subset of a partially ordered Banach space X. Set

M == {x E X : y $ x for every

y E M}.

We say that the point x 0 E X is the infimum of M if xo E M and for every x EM, x 0 :5 x. The supremum of M is defined in a similar way. Theorem 1.4.6. (Knaster's Fixed Point Theorem). Let X be a partially ordered Banach space with ordering :5 . Let M be a subset of X with the following properties: The infimum of M belongs to M and every nonempty subset of M has a supremum which belongs to M. Let T : M ~ M be an increasing mapping, i.e., x :5 y implies Tx :::; Ty. Then T has a fixed point in M. Let X be a Banach space let K be a cone in X, and let

:5 be the order in

X induced by K, i.e, x :::; y if and only if y - x is an element of K. Let D be a subset of K, and T: D ~ K a mapping. We denote by (x, y) the closed order interval between x andy, i.e., (x,y) == {x E Z: x $ z :5y}. We assume that the cone K is normal in X, which implies that order intervals are norm bounded. The cones of nonnegative functions are normal in the space of continuous functions with supremum norm and in the space Lp. Theorem 1.4. 7. Let X be a Banach space, K a normal cone in X, D a subset of K such that if x, y are elements of D, x $ y, then ( x, y) is contained in D, and let T : D ~ K be a continuous decreasing mapping which is compact on any closed order interval contained in D. Suppose that there exists x 0 E D such

Preliminaries

13

that T 2 x 0 is defined (where T 2 xo = T(Tx 0 )) and further more Tx 0 , T 2 x 0 are (order) comparable to xo. Then T has a fixed point in D provided that either:

(I) Txo $ x 0 and T 2 xo $ xo or Txo ;::: xo and T 2 xo ;::: xo, or (II) The complete sequence of iterates {Tnxo}~=O is defined and there exists Yo E D such that Tyo E D and Yo $ Tnxo for all n.

Theorem 1.4.8. Let X be a Banach space, A : X -+ X be a completely continuous mapping such that I - A is one to one, and let n be a bounded set such that 0 E (I- A)(!l). Then the completely continuous mappingS: n-+ X has a fixed point in 0 if for any .X E (0, 1), the equation

x = .XSx + (1 - .X)Ax has no solution

X

on the boundary

an of il.

The following theorem is a very useful version of the Topological Transversality Theorem of Granas. Theorem 1.4.9. Suppose that X is a normed space (possibly incomplete). Let S be a bounded, closed, convex subset of X, containing the origin 0 in its interior D. Let H : [0, 1] x S-+ X be a homotopy of compact transformations such that H(O, as) c sand H(t, X) =F X (0 $ t $ 1, X E as). Then tp = H(1,.) has a fixed point in S. Theorem 1.4.10. (Leray-Schauder Alternative). Let C be a convex subset of a normed linear space E and assume 0 E C. Let F : C -+ C be a completely continuous operator and let

£(F)= {x E C: x = .XFx

for some

0

Then either £(F) is unbounded or F has a fixed point.

< .X < 1}.

2 Oscillations of First Order Delay Differential Equations

2.0. Introduction In this chapter, we will describe some of the recent developments in the oscillation theory of first order delay differential equations. This theory is interesting from the theoretical as well as the practical point of view. It is well known that homogeneous ordinary differential equations (ODEs) of first order do not possess oscillatory solutions. But the presence of deviating arguments can cause the oscillation of solutions. In this chapter, we will see these phenomena and we will show various techniques used in the oscillation and nonoscillation theory of differential equations with deviating arguments. We will present some criteria for oscillation, for the existence of positive solutions and results in the distribution of zeros of oscillatory solutions of DDEs of first order. In the last section we will discuss the oscillation and nonoscillation behavior of some basic ecological delay equations.

14

15

Oscillations of First Order Delay Equations

2.1. Stable Type Equations with a Single Delay 2.1.1.

Oscillation results

We consider the linear delay differential inequalities and equations of the form

x'(t)

where p, T E Set

+ p(t)x(r(t))

::=; 0

(2.1.1)

x'(t) + p(t)x(r(t)) 2:: 0

(2.1.2)

x'(t) + p(t)x(r(t)) = 0

(2.1.3)

C([to,oo),~),

r(t) ::=; t and lim r(t) = oo.

R+ = [O,oo),

1--+oo

m = liminf [' p(s)ds 1--+oo

(2.1.4)

1T(I)

and

M =lim sup [' p(s)ds.

(2.1.5)

1T(I)

1--+oo

The following lemmas will be used to prove the main results in this section. All the inequalities in this section and in the latter parts hold eventually if it is not mentioned specifically. Lemma 2.1.1.

Set

6(t)

= max {T(s);

s E [t 0 ,tl}

(2.1.6)

and m > 0. Then we have lim inf [' p( s )ds = lim inf [' p( s )ds = m. l--+oo

16(1)

t--+oo

1 T(l)

Proof: Clearly, 6(t) ;::: r(t) and so

[' p(s)ds ::=; 16(t)

1 1

T(l)

p(s)ds.

(2.1. 7)

16

Chapter 2

Hence liminf [' p(s)ds::; liminf [' p(s)ds. t-+oo

}6(t)

t-+oo

If (2.1.7) does not hold, then there exist m' tn - oo as n --t oo and

lim n-+oo

l tn

6(tn)

}r(t)

> 0 and a sequence {tn} such that

p(s)ds::; m' < m.

By definition, li( tn) = max ( r( s) : s E (to, tnl), and hence there exists t~ E [to, tn] such that li(tn) = r(t~). Hence

l tn

p(s)ds =

ltn

6(tn)

p(s)ds >

lt~

r(t:,)

p(s)ds.

r(t:,)

It follows that {J:rt:.)p(s)ds}~=l is a bounded sequence having a convergent subseque nce, say

l

t~.

p(s)ds- c::; m',

as

k--. oo,

r(t:..>

which implies that liminf [' p(s)ds::; m', t-+oo

lr(t)

contradic ting (2.1.4). Lemma 2.1.2. then

D

Let x(t) be an eventually positive solution of (2.1.1). Hm > ~ ,

lim x(r(t)) = oo. t-+oo

x(t)

(2.1.8)

H m ::; ~' then liminf x(r(t)) >A t-+oo

x(t)

-

(2.1.9)

17


where A is the smaller positive root of the equation (2.1.10)

Proof: Let t 1 be a sufficiently large number so that x(r(t)) Hence

> 0 fort !:: t1.

x'(t) < -p(t) x(r(t)) < -p(t). x(t) x(t) -

(2.1.11)

Integrating it from r(t) tot we have that eventually

r

x(r(t)) ;:::: exp( p(s)ds). x(t) lr(t) Then for any e > 0, there exists Te such that x(r(t))

~!::e

m

-e,

t!::Te.

Substituting (2.1.12) into (2.1.11) we have ~N? ~ -(em - e)p(t), hence

. . x(r(t))

limmf t-+oo

X

(t )

(2.1.12)

t !:: Te, and

!:: exp(m expm).

Set Ao = 1, A1 = exp(m>.o), ... ,>.n = exp(mAn-1), .... For a sequence {en} with en > 0 and en --+ 0 as n --+ oo, there exists a. sequence {tn} such that tn --+ oo as n--+ oo and (2.1.13) If m > 1, then n-..oo lim An= oo, and (2.1.8) holds. If m = 1, then lim An= e; and e e n-+oo if m < ~. then An tends to the smaller root of Eq. (2.1.10). 0

Remark 2.1.1. From Theorem 2.1.1 we will see that (2.1.1) has no eventually positive solutions if m > ~ .

Chapter 2

18

Lemma 2.1.3. In addition to the hypotheses of Lemma 2.1.1, assume T is nondecreasing, 0 :5 m :5 ~ , and x(t) is an eventually positive solution of (2.1.1). Set

. 'nf r = 11m1 t.-oo

x(t) t

(2.1.14)

- (( )) . X T

Then 2

A(m) := _1_-_m_-_v __ 1_-_2_m_-_m_

2

< r < 1. -

-

(2.1.15)

Proof: Assume that x(t) > 0 fort > T1 ;::: t 0 , and there exists a sequence {Tn} such that T1 < Tz < T3 < ... and r(t) > Tn fort> Tn+1, n = 1, 2, .... Hence x(r(t)) > 0 fort> T2 • In view of (2.1.1), x'(t) :50 on (T2 , +oo). Clearly, (2.1.15) holds form= 0. If 0 < m :5 ~ , for any c E (0, m), there exists Ne such that

t

p(s)ds>m-c,

for

t>Ne.

(2.1.16)

JT(t)

For a fixed e we will show that for each t > Ne there exists a At such that r(At) < t < At and {At

lt In fact, for a given t,

(2.1.17)

p(s)ds = m- c.

f(A) := ftA p(s)ds is continuous and lim /(A)> m-e> A-+oo

0 = f(t). Hence there exists At > t such that j(>.t) = m- c, i.e., (2.1.17) holds. From (2.1.16) we have

1

At

p(s)ds >m-e=

T(At)

!At

p(s)ds,

f

therefore r(At) < t. Integrating (2.1.1) from t (> max {T4 ,Ne}) to At we have At

x(t)- x(At);:::

1 t

p(y)x(r(y))dy.

We see that r(t) ~ r(y) ~ r(At) < t fort~ y ~At.

(2.1.18)

19


Integrating (2.1.1) from r(y) tot we have that fort::;; y::;; At

x(r(y))- x(t) 2::

lt

p(u)x(r(u))du

r(y)

2:: x(r(t))

lt (lt

p(u)du

r(y)

= x(r(t))

p(u)du

r(y)

-fy p(u)du) t

> x (r( t)) [(m - c) -

J.Y p( u )dul·

(2.1.19)

From (2.1.18) and (2.1.19) we have

x(t) x(.At) + x(t)(m- e)+

t (m- e)

2

x(r(t)).

{2.1.21)

Hence

x(t) (m-e? x(r(t)) > 2(1- m +e)

d :=

(2.1.22)

b

and then

Substituting this into (2.1.21) we obtain

x(t) > x(t)(m + d1- e)+

t {m- e) 2 x(r(t)),

and hence

x(t) {m-e) 2 x(r(t)) > 2{1- m- d1 +e)

d :=

2.

In general we have

x(t) {m-e) 2 x(r(t)) > 2(1-m-dn+e) :=dn+I>

n= 1' 2 ' " ' '

It is not difficult to see that if e is small enough, 1 ~ dn Hence lim dn = d exists and satisfies

> dn-l,

n--+oo

-2d2 + 2d{1- m +e)= (m- e) 2 ,

n = 2, 3, ....


21

i.e., d= 1-m+e± y'1-2(m-e)-(m-e) 2 2

Therefore, for all large t

x(t) x(r(t))

-----> Letting e

-+

1-m+ e- y'l- 2(m- e)- (m- e) 2 ----~----~--~----~~--~2

0, we obtain that

x(t) 1-m-v'l-2m-m2 x(r(t)) ;::: 2 = A(m). D

This shows that (2.1.15) holds. Assume that 0 < M $ 1, eventually positive solution of (2.1.1). Set

Lemma 2.1.4.

r is nondecreasing. Let x(t) be an

liminf x(r(t)) =f.. t-+oo

x( t)

Then (2.1.23)

Proof: For a given e E (O,M), there exists a sequence {tn} such that tn-+ oo as n-+ oo and

1

tn

p(s)ds >M-e,

tn

> T,

n = 1, 2, ....

r(tn)

Set ()~ = 1- y'1- (M-e) . It is easy to see that 0 < e. Hence there exists Pn} such that r(tn) < An < tn and

n = 1,2, ....

()E

<M-e for small

22

Chapter 2 Integrating (2.1.11) from An to tn we obtain

Similarly, we have

x(r(tn))- x(An) ~

!.

>. ..

p(s)x(r(s))ds

r(t,.)

~ x(r(An))

!.

>. ..

p(s)ds

r(t,.)

= x(r(An)) [

['" p(s)ds- ['" p(s)ds]

Jr(t,.)

J>.,.

> x(r(An))[(M- c)- 8,.]. From the above inequalities we get

x(An) > 8,.x(r(tn))

> 8,.[x(An) + x(r(An))(M- c- 8,.)] and then

x(r(An)) 1-8,. x(An) < 8,.(M- c- 8,.) '

n

= 1,2, ...

which implies that

e
e- T ~ t*- 3r ~ tl + T. On the other hand, x(t) is decreasing fort~ t 1 + r, so x(e- r) > x(t*- 2r), which contradicts (2.2.11). Therefore x(t) lies above the lineR. on [t*- 2r, t*- r]. We see that

1

t•-r

( X S

)d

S

x(t•- r)

>

+ z(t•- 2r)

T.

2

t•-2r

Using (2.2.10) we have

J

t•-r

t•-2r

x(s)ds >

2x(t•- r) + rx(t•- 2r) r. 2

(2.2.12)

By Lemma 2.2.1 no+ 3 (1 + ln r) } x(t*- r). x(t*- 2r) > { 1 + r-

(2.2.13)

Combining (2.2.12) and (2.2.13) we obtain

l

t•-r

x(s)ds>

2+r(1+

no+ 3 (1+lnr))

;

r·x(t*-r).

(2.2.14)

t•-2r

Integrating (2.2.1) from t•- r tot* and using (2.2.14) we obtain { 1- 2 + r( 1 + ; ( 1 + ln r)) r }x(t*- r)- x(t*) > 0 and hence 3 (1+lnr)) } { 1- 2-r(1+ ""; T >0

which contradicts the definition of n 0 as given in (2.2.9). From Theorem 2.2.1 and Lemma 2.2.1 we have the following corollary.

0


35

Corollary 2.2.1. Assume that T > ~, x(t) is a solution of (2.2.1) and positive and decreasing on [tl> t 1 + T], then x(t) can not always be positive on [t 1 + T, t1 +(no -1)T]. Then we consider

x'(t) + p(t)x(t- T) = 0.

(2.2.15)

By the method of Theorem 2.1.3 and 2.1.5 the results of (2.2.1) can be extended to Eq. (2.2.15) Theorem 2.2.1. Assume that (i) p E C(t 0 ,oo),(O,oo)), p(t) ¢. 0 on any subinterval of[t 0 ,oo), liminfjt p(s)ds=k>!.

(ii)

t-+oo

(2.2.16)

e

t-r

Let x(t) be a solution of (2.2.15) on [T.,, +oo), T., ;::: t 0 • Let c: > 0 be a small number such that k - c: > ~ . Then there exists t~ ;::: T., such that the distances between adjacent zeros of x(t) is less than (1 + ne)T on [t~, +oo ), where

n~

=max{4,

3+ [ 2 - 2(k-c:)-(k-c:)2]}· (k- c:)(1 + ln(k- c:))

Proof: In view of the assumptions there exists

i

t

p(s)ds>k-c:

for

t~ ;::: t 0

(2.2.17)

such that

t;:::te.

(2.2.18)

t-r

Set u = a(t) = ft: p(s )ds and z( u) = x(u- 1 (u)). As in the proof of Theorem 2.1.4, (2.2.15) reduces to

( 1

cr-t(u)

z'(u)z u-

)

p(s)ds

= 0.

(2.2.19)

cr- 1 (u)-r

We shall show that x(t) has zeros on any interval [t1, t 1 + (1 + n.)T], t 1 ;::: t •. If not, x(t) > 0 on [t1, t1 + (1 + ne)T], then x'(t) < 0 on [t 1 + T, t 1 + (1 + n~)T]. Set

Chapter 2

36

Then z(u) > 0 for u ~

Ut

and z'(u)

1

< 0 for u ~ u2 and

.,.-t(u)

u-

p(s)ds

~

u 2 for

u

~

u

~ Ut.

u3 .

.,.-•(u)-r

Since

1

.,.-•(u)

p(s)ds > k- e:,

for

.,.- 1 (u)-r

we have

z'(u) + z(u- (k- e:)) < 0,

u ~ u3.

for

z(u) is positive and decreasing for u ~ u3. By Corollary 2.2.1, z(u) has zeros on [u3, U3 +(n., -1)(k -e:)]. Hence there exists u E [u3, u3 + (n., -1)(k- e:)] such that z(u) = 0. Set t = u- 1 (U), then x(i) = 0. On the other hand, from (2.2.18) we have

i

t 1 +(n.+t)r

p(s)ds > (n., -1)(k- e:).

•+2r

Hence

i

t, +(n.+l)r

to

p(s)ds >

1t

p(s)ds >

1t'

to

+2r

p(s)ds,

t0

which derives that

This implies that x(t)

> 0, contradicting that x(t) = 0.

Remark 2.2.1. Consider the autonomous equation

x'(t)+px(t-r)=O,

p>O,

r>O,

(2.2.20)

where pr > ~ . Then according to Theorem 2.2.2 the distance between adjacent zeros of solutions of (2.2.20) is less than

2- 2pr- (pr) 2 ]} r max { 5,4 + [ pr(l + ln(pr)) .

(2.2.21)

Oscillation.s of First Order Delay Equations

37

2.3. Unstable Type Equations We now consider the first order linear delay differential equation with unstable type of the form

x'(t) = p(t)x(r(t)) where p, r E C([t 0 ,oo),~), Theorem 2.3.1.

r(t)

~

(2.3.1)

t and limt.....oor(t) = oo.

Assume that 00 {

lto

p(t)dt

= 00.

(2.3.2)

Then Eq. (2.3.1) always has an unbounded positive solution, and every bounded solution of (2.3.1) is oscillatory. Proof: The proof of the first part can be seen from Theorem 3.5.1. Now we will prove that every bounded solution of (2.3.1) is oscillatory. Let x(t) be a bounded positive solution of (2.3.1), then x(t) is nondecreasing and bounded. Hence lim x(t) =I.> 0 exists. From (2.3.1), we obtain t-+oo

x'(t)

~ ~ p(t)

for all large t.

(2.3.3)

(2.3.1) and (2.3.2) imply that lim x(t) = oo, contradicting the boundedness of 0 x(t).

Example 2.3.1. Consider the equation

r.::a; 2 ).

y'(t) = 2ty 0 fort E [r"+ 1 (bn), bn]· By Lemma 2.4.1 x(t) > o:;x(r(t)), t E [r"- 2 (bn),ri+ 1 (bn)] and x'(t) ~ 0, t E [r"(bn), bn]· Integrating (2.4.4) from r(t) tot fort E [r"- 2 (bn), ri+ 1 (bn)] we obtain

-x(t) + x(r(t))

~ f' p(s)x(r(s))ds ~ x(r(t)) f' p(s)ds. J..(t) J..(t)

Hence

l

t ( ) x(t) psds~1- ( ()), r(t) X T t

te [T A:-2( bn),r i+l( bn)·]

In view of that x(t) > o:;x(r(t)) fortE [r"- 2 (bn),ri+ 1 (bn)], we obtain (2.4.18) which contradicts (2.4.17).

D

Similarly, we have the following lemma. Lemma 2.4.3.

Assume the assumptions of Lemma. 2.4.2 hold. Then the

Chapter 2

44

differential inequality x'(t) + p(t)x(r(t)) ~ 0

(2.4.19)

has no solution x(t) with x(t) < 0 on [r"'+ 1 (bn),bn]· Combining Lemma 2.4.2 and Lemma 2.4.3 we obtain the following oscillation criterion.

Under the assumptions of Lemma. 2.4.2 every solution of Theorem 2.4.1. Eq. (2.4.1) has a.t least one zero on [r"'+ 1 (bn),bn]· Consequently, every solution of (2.4.1) is oscillatory. Remark 2.4.1. The first part of Theorem 2.1.2 can follow from Theorem 2.4.1. In fact, the assumptions of Lemma 2.4.2 are satisfied in this case. Example 2.4.1. Consider the equation

x'(t) + 0.8x(t- sint) = 0

(2.4.20)

where r(t) = t - sin t is nondecreasing. Let b be the real root of the equation

b=

f + sinb

which lies on the interval ( f, 1r). It is easy to see that bE ( f, ~1r). Let bn = 2n1r+ b, n = 0, 1, 2, .... Then r(bn) = 211" + f, r 2 (bn) = 2n?r + f - 1, r 3 (bn) = 2n?r + f- 1- cos 1 > 2n?r. Clearly r 3 (bn) < bn and r(t) < t fortE (r(bn), bn)· Let k = 3, then the condition (i) of Lemma 2.4.1 is satisfied and [' 0.8ds lr(t)

= 0.8sin t 2:: (0.4) v'2 = m3,

t E [r(bn), bn]·

On the other hand, we see that

1

T(bn)

T 2(bn)

O.Sds

= 0.8 > 0.63161 2:: 1- ao.

That is, all assumptions of Lemma 2.4.2 are satisfied. Therefore, by Theorem 2.5.1, every solution of (2.4.20) is oscillatory.


45


x'(t) +a sin~ x(t- 2;) = 0

(2.4.21)

e-li, t).

where a E Set k = 3, and

bn = 2(2n- 1)11". Then the assumptions of Lemma 2.5.1 hold

a :::;;

l

t

t-.!!:

s a sin 2 ds,

2

and

,~-·

~

s

Jbn-~ asin2ds=2a>1- 2( 1 -a), i.e., all assumptions of Lemma 2.4.2 are satisfied. Therefore by Theorem 2.4.1 every solution of (2.4.21) is oscillatory.

2.5. Equations with Positive and Negative Coefficients We consider the linear delay differential equation with nonpositive and nonnegative coefficients of the form

x'(t) + p(t)x(t- r)- Q(t)x(t- u) = 0 where p; Q E C{[to,oo),.R;.), r,u E {O,oo). The following is an oscillation criterion for (2.5.1). Theorem 2.5.1.

Assume that

T ~

u

~

0 and

p(t)=p(t)-Q(t+u-r)~O,

lim inf t-+oo

and for every A> 0

it

t-T

{¢0),

p( s )ds > 0,

t~t1 ~to,

{2.5.1)

Chapter 2

46 Then every solution of (2.5.1) is oscillatory.

In the next chapter we will prove a more general result, so we omit the proof here. From (2.5.2) we can derive some explicit conditions easily.

H the condition (2.5.2) is replaced by the explicit condition

Corollary 2.5.1.

li~inf [el' p(s)ds + t

t-1'

00

lt-u

Q(s + u- T)ds] > 1,

(2.5.3)

t-1'

then the conclusion of Theorem 2.5.1 is true. The following is a result for the existence of positive solutions of (2.5.1). Theorem 2.5.2. Assume that i) T > u ~ 0 and

p(t) = p(t)- Q(t + u- T)

~

0 (¢ 0),

t

~ t1 ~ t0 ,

I,';'

p(t)dt = oo, ii) iii) there exist T ~ t 1

and~·

sup{;.

exp(~·

t~T

> 0 such that

1'

t-,.

p(u)du)

(2.5.4) Then Eq. (2.5.1) has a positive solution which tends to zero as t-+ oo. To prove Theorem 2.5.2, we first prove the following lemma. Lemma 2.5.1. integral inequality

Assume that condition i) in Theorem 2.5.2 holds. H the

(2.5.5)

47


has a continuous positive solution z E C[t1- r, oo) with lim z( t) = 0. Then there t-+00 exists a positive solution x : [t 1 - r, oo) -+ (0, oo) of the integral equation

i~~.r Q(s + u)z(s)ds +

1:

p(s + r)z(s)ds

= z(t),

Proof: Choose T;?: t 1 so large that z(t) > z(T), for t 1 Define a set of functions as follows:

n ={wE C([t 1 and a mapping T on

(Tw)(t) =

-

t;?: t1.

r :::; t < T.

0:::; w(t):::; z(t),

r,oo),R+):

(2.5.6)

t;?: t1 - r}

n.

{ J/~; Q(s + u)w(s)ds + ft~rp(s + r)w(s)ds, (Tw)(T) + z(t)- z(T),

t;?: T (2.5.7)

t1- T:::; t < T.

Clearly, Tfl c n. Define a sequence of functions as follows:

zo=z,

Zn=Tzn-b

n=1,2, ....

It is easy to see that

. 0:::; Zn(t):::; Zn-l(t):::; · · ·:::; z(t), t;?: t1- T. Hence J~~ zn(t) = x(t) exists and x(t) is continuous and nonnegative for t ;:::

t1-

T.

By Lebesgue's dominated convergence theorem x(t) satisfies (2.5.6). Since x(t) > 0 fort E [tt - r, r), it follows from (2.5.6) that x(t) > 0 for all t ;?: t 1 - r. The proof is complete.

Proof of Theorem 2.5.2:

Set

z(t) = exp(-

.x·i:+r p(u)du).

(2.5.8)

Chapter 2

48

We shall show that (2.5.8) satisfies integral inequality (2.5.5). In fact, (2.5.4) implies that ;. exp (A +

•l~r p( u )du)

l~~cr Q(s + u --r)exp(;\* 1' p(u)du )ds 5, 1

for

t

?_ T.

(2.5.9)

It is not difficult to see the relations ;. exp(x· [' p(u)du) J1-r

=

roo

p(s + -r)exp(>.·

J1-2r

r'

p(u)du)ds

(2.5.10)

J5+r

and

l~~cr Q(s + u- r)exp( ).* 1' p(u)du )ds =

r-r-CT Q(s + u)exp(.x·j!5+r p(u)du)ds.

h-2r

(2.5.11)

Substituting (2.5.10) and (2.5.11) into (2.5.9) we obtain

roo

p( S

+ T )exp ().

lt-2r

+

*1

1

s+r

['-r-cr Q(s

p( U )du) ds +u)exp(.x• [' p(u)du)ds 5, 1, Js+r

lt-2r

or

p( s + T )exp (_x * ['' p( U )du) ds

roo

~+r

h-2r

+

1!-r-a

Q( s + 0" )exp (.x.

f''

Js+r

t-2r

p( u )du) ds 5, exp ( -

x·jt~ p( u)du)'

i.e.,

j oo

t-2r

p(s + r)z(s)ds +

jt-r-cr Q(s t-2r

+ u)z(s)ds 5, z(t- r),

t?. T.

49


The hypothesis of Lemma 2.5.1 is satisfied. Therefore the integral equation (2.5.6) has a positive solution x(t) with x(t) :5 z(t). By differentiation we obtain

x'(t) = Q(t)x(t- u)- p(t)x(t- r). 0

The proof is complete.

From Theorem 2.5.1 and Theorem 2.5.2 we obtain the following corollary. Corollary 2.5.2. If p > Q 2::: 0 are constants of Eq. (2.5.1) oscillates if and only if

->.p + pe>..;;r where

p=

- Qe>..;;a

> 0,

T

> u 2::: 0, then every solution

for all

)..

>0

(2.5.12)

p- Q.

Remark 2.5.1. H Q(t)

= 0, (2.5.4) becomes that there exists ).. • > 0 such that

sup { ;. exp ().. I~T

•f'

p( u )du) } :5 1.

(2.5.13)

1-r

In particular, we take)..* = e, then (2.5.4) leads to

1 1

1

p(u)du :5 -, e

1-r

for

t 2::: T.

(2.5.14)


x'(t) + z(t; 2) x(t- 2)- !x(t) = 0 where p(t)

z(~2 2 )

-

= z(~; 2 >, Q(t) = h T

(2.5.15)

t

t

= 2, u

= 0.

Thus p(t)

= p(t)- Q(t + u- r) =

1 ~ 2 > 0 fort 2::: 10 and f 17: p(t)dt = oo. It is easy to see that there exists

T 2::: 10 such that

l

lne p(u)du :5 - , 2e 1-r t

l

1

t

Q(s- r)ds :5 -, 2e 1-r

for

t 2::: T.

Choose )..* = 2e, then (2.5.4) holds. Hence Eq. (2.5.15) has a positive solution which tends to zero as t -+ oo. In fact, x(t) = is such a solution of (2.5.15).

t

Chapter 2

50

2.6. Equations with Several Delays 2.6.3. Equations with constant parameters

First we discuss the oscillation problem of the following equation n

x'(t) + LP;x(t- r;)

=0

(2.6.1)

i=l

where p; E R,

r; ;:::

0 are constants. A known result for it is

Theorem 2.6.1. equation

Eq. (2.6.1) is oscillatory if and only if its characteristic n

f(>.) = >. + LP;e-~r, = 0

(2.6.2)

i=l

has no real roots. This theorem is of theoretical interest. But this condition is not easy to verify. In the following we give a new necessary and sufficient condition for oscillation of (2.6.1) which leads to some sharp explicit conditions for oscillation. Let n

D = {i = {i1, ... ,ln)li;;:?: 0,

i = 1, .. . ,n,El; = 1}. i=l

For any i E D define

>.· L ....!.ln(ep;r;fl;) n

f(l) =

(2.6.3)

i=l r;

where i; = 0 implies that the corresponding term vanishes. (i) f(l) has a maximum at a point i 0 = (i~, ... ,i~) on D Theorem 2.6.2. which is uniquely determined by the condition that

>.;

1 = -ln(ep;r;/l;), T;

have the same value if l; =f; 0.

i = 1, ... , n

(2.6.4)

51

Ouillations of First Order Delay Equations

(ii) Eq. (2.6.1) is oscillatory if and only if f(£ 0 )

> 0.

Another version of Theorem 2.6.2 is Theorem 2.6.3. Eq. (2.6.1) is oscillatory if and only if there exists an f.* = (ii, ... ,i~) such that >.; =f Aj for some i,j E In = {1, ... , n} and f(i*);:::: 0 where>.;, i E In is defined by (2.6.4).

Corollary 2.6.1.

Assume that n

PITt

>

~

for

n = 1,

~p·r·

>1 L....J••-e

and

for

n

>_ 2.

i=l

Then every solution of Eq. (2.6.1) is oscillatory.

Proof: We only prove it for the case that n ;:::: 2. Choose 0 < a ~ 1 such that E7= 1 aep;r; = 1. Let li = aep;r;, i = 1, ... , n, and f.* = (li, ... , £~). Then i* E D, and >.; are different fori = 1, ... , n. Furthermore, 1

f(i*)

n

= aln- l:Pi;:::: 0. a i=l

According to Theorem 2.6.3, Eq. (2.6.1) is oscillatory.

0

From Theorems 2.6.2 and 2.6.3 we can derive some other explicit conditions for oscillation of (2.6.1). In the next chapter we will prove more general results than Theorems 2.6.2 and 2.6.3. Therefore we omit the proofs here. Remark 2.6.1. The following still is an open problem. Find an explicit necessary and sufficient condition for the oscillation of (2.6.1) which does not depend on solving an equation.

52

Chapter 2

2.6.4. Equations with variable parameters We consider the first order delay differential equations of the form n

x'(t) + LPi(t)x(t- ri(t)) = 0

(2.6.5)

i=l

where pi, Let

t-ri(t) --too ast --too,

TiE C([to,oo),R+),

f

i E In= {l, ... ,n}.

E C(R, R) and define

type

f

=lim sup tIn lf(t)l.

(2.6.6)

t-+oo

Here f(t) is allowed to vanish and we denote ln 0 = -oo. Clearly, a function f with type -oo tends to zero faster than any exponential functions as t --too. We start discussion with some basic lemmas.

Lemma 2.6.1.

If liminf { inf _!, ~ Pi(t)e.\ri(t)} t-+oo .\>0 ).. L.J

>1

(2.6.7)

i=l

and Eq. (2.6.5) has nonoscillatory solutions. Then every nonoscillatory solution of (2.6.5) has type -oo. Proof: Without loss of generality assume Eq. (2.6.5) has a solution x(t), ~ 0. From (2.6. 7) we may assume that to is so large that

t 2:: t 0

(2.6.8)

n

and hence EPi(t) > 0 fort 2:: t 0 • We define a sequence {tk},

k = 0, 1, 2, ... by

i=l

tk = sup{ t: minl~i~n(t- r;(t)) ~ tk-b i E In}, k 2:: 1, then tk 2:: tk-h k ~ 1. It is easy to see that {tk} is unbounded. Otherwise, there exists a constant A such that limk-+oo tk =A and hence Ti(Ai) = 0, i E In. But then (2.6.8) and hence (2.6.7) is not true. Therefore limk-+oo tk = oo. x'(t) From (2.6.5) we have x'(t) < 0 fort~ t 1 . Define u(t) = - X(t) > 0, t ~ t1.

-----------~-------

---- - - -

53


Then u(t)

= tPi(t)expjt i=l

u(s)ds,

t

~ t1.

(2.6.9)

t-r,(t)

We shall show that lim u(t) = oo. To this end, define .A 1 = inf u(t), and t~t 1

t-oo

n

Ak+l =

inf

t~tA:+l

LP;(t)exp(.Akr;(t)). i=l

(2.6.10)

Clearly, each Ak is well defined. It is easy to see that

= tPi(t)expit

u(t)

i=l

u(s)ds

t-r,(t)

n

~ LPi(t)exp(.AI Ti(t)) ~ A2

for

t ~ t2.

i=l By induction, we have u(t) 2:: Ak

t

for

~ tk,

k

= 1, 2, ....

Now we shall prove that lim Ak = oo. k-oo

We first claim that A1 > 0. Otherwise there is an arbitrarily large t such that u(s) 2:: u(t) > 0 for all s E [to,t]. Then from (2.6.9)

u(t) = tp;(t)exp i=l

it

u(s)ds

> to

~ tPi(t)exp(u(t)r;(t)) i=l

t-Ti(t)

which contradicts (2.6.8). From (2.6.8) and (2.6.10) there is an o: > 1 such that Ak+l 2:: o:Ak for k = 1, 2, ... , and hence Ak --+ oo as k --+ oo. Since x(t) exp(- fe: u(s )ds) and hence x(t) has type -oo. The proof is complete. 0

Lemma 2.6.2. Let f : R+ --+ (0, oo) be continuous. If there exist b, B > 0 such that f( t) / f( t - b) 2:: B for t ~ b, then there exists M > 0 such that

f(t)

~ M exp

(f;

lnB),

for t ~b.

(2.6.11)

Chapter 2

54 The proof is easy so we omit it here.

Lemma 2.6.3. Let p(t) ~ 0 and r(t) be continuous functions fort ~to ~ 0 and lett- r(t) be nondecreasing such that t- r(t) ~ oo as t ~ oo. Assume

lim inf t-+oo

it

p( s )ds > 0

(2.6.12)

t-r(t)

and lim inf r( t) t--+00

> 0.

(2.6.13)

Then the differential inequality x(t)[x'(t)

+ p(t)x(t- r(t))] 50

(2.6.14)

does not have any nonoscillatory solutions with type -oo. Proof: Let x(t) be a nonoscillatory solution of (2.6.14). Without loss of generality, we assume x(t) > 0 fort ~ t 0 • From (2.6.12) and (2.6.13) there are b, B > 0 such that for some t 1 ~ t 0

i

t t-r(t)

p(s)ds

~ B,

r(t)

~ b,

t

~ t1

(2.6.15)

which implies that

(2.6.16) (see Lemma 2.1.3). Since r(t)

~band

x(t) is nonincreasing we have

x(t)fx(t- b)~ By Lemma 2.6.2, type x

> -oo.

B2

4' t

~ t1.

(2.6.17)

0

Remark 2.6.1. The following example shows that the conditions in Lemma 2.6.3 are sharp.

55


Example 2.6.1. The equation

(2.6.18) has a solution x(t) (2.6.12) holds.

= exp(-

Theorem 2.6.4.

Assume that (2.6. 7) holds and

!t 2 ) with type -oo. We see that (2.6.13) fails and

(2.6.19)

Then every solution of (2.6.5) is oscillatory.

Proof: Let x(t) be an eventually positive solution of (2.6.5). Then x(t) > 0 and x'(t) :::; 0 for all large t. We first show that (2.6.20) If not, there exists a sequence {t k} such that lim t k = oo and Pi (t k) Ti (t k) ~ 0 k-->oo

as k ~ oo, i E In. If Pi(tk)ri(tk) = 0 for some tk and all i E In, then 1

n

XLPi(tk) exp [ri(tk)]:::; npo/'A--+ 0,

as

>..--+ oo.

i=l

Hence we may assume that max {Pi(tk)ri(tk), i E In} For each k, set aik = oo if Pi(tk)ri(tk) = 0 and aik = Pi(tk)ri(tk) > 0. From (2.6.19) we have

> 0 for all k = 1, 2, ....

T}t,.)

ln (2po/Pi(tk)) if

In view of the fact that Pi(tk)r,(tk) ~ 0 as k--+ oo, it is not difficult to see that aik ~ oo ask~ oo for each i E In. Now let A.(tk) =min {aibi = 1,2, ... ,n}. Then we have 0 < A.(tk) < oo and A.(tk) ~ oo ask~ oo. Thus 1

n

A.(t ) LPi(tk)exp(A.(tk), r,(tk)) k

i=l

Chapter£

56

which contradicts (2.6.7) and hence (2.6.20) holds. Next, if we let p(t) = p;(t), r(t) = r;(t) be such that Pj(t)rj(t) = max{p;( t)r;( t), i E In} for each t, then 0 ::; p(t) ::; Po, 0 :5 r(t) ::; ro. From (2.6.20), there is a constant c > 0 such that p(t)r(t) ;?: c for all large t and so p(t) ;?: c/ro and r(t);?: cfpo. From (2.6.5) we have n

0 = x'(t)

+ LP;(t)x(t- r;(t)) i=l

;?: x'(t)

+ p(t)x(t- r(t))

;?: x'(t) + .!:.x(t-.!:. ), ro Po

and so, by Lemma 2.6.3, type x > -oo. On the other hand, by Lemma 2.6.1, under (2.6.1) the nonoscillatory solution x has type -oo. This contradiction completes the proof of the theorem. 0 Considering that the functions p;(t) and r;(t) may be unbounded, we present the following oscillation criterion. Theorem 2.6.5.

Suppose that there exists a nonempty subset I of the set

In= {1, ... ,n} such that liminf u(t) t-oo

>0

(2.6.21)

and liminfft t-oo

LP;(s)ds > 0

(2.6.22)

t-u(t) iEJ

whereu(t) = min{r;(t), i E I}. H(2.6.7) holds, then every solution ofEq. (2.6.5)

is oscillatory.

57


Proof: Assume that (2.6.5) has a solution x(t) > 0 for t ~ to ~ 0, by Lemma 2.6.1, it is sufficient to show that type x > -oo. From (2.6.7) we may assume that t 0 is sufficiently large so that (2.6.8) holds. Then, from (2.6.5), x'(t) < 0 fort~ t1 ~to, where t 1 =sup {t : min(t- r;(t)) :5 to, i E In}· Since n

0 = x'(t) + LP;(t)x(t- r;(t)) i=l

t ~ t1•

~ x'(t) + x(t- u(t)) Ep;(t), iEI

By virtue of (2.6.2) and (2.6.22) and Lemma 2.6.3, we have type x > -oo. The 0 proof is complete. The following example shows that condition (2.6.19) or conditions (2.6.21) and (2.6.22) are important for Theorem 2.6.4 and Theorem 2.6.5 to hold. Example 2.6.2. Consider the equation

x'(t) + Pl(t)x(t) + P2(t)x(t -1) = 0. Choose a, b > 0 so that a= b/(1- e- 6) > 1. Let t 0 be such that to> // ln(1- e- 6 ) > 0. Pl(t) = a(g(t) -1), P2(t) =a exp(-g(t)), where g(t) = (1- exp( -b))exp(bt), t ~ t 0 • For any ,.\ > 0,

(2.6.23)

t ~ t0 ,

'nf g(t)- 1 + exp(..\- g(t))] ,[ . ,.\ =A- 1 +a1 t2:to

The function

/(..\) = g(t)- 1 + e;p(..\- g(t))

,

,.\ > 0

minimized at ..\ = g( t) and the minimum is 1. Hence n

-,.\ + inf 2:Pk(t)e.\r.(t) ~ ..\( -1 +a) > 0 t2:to k=l

Chapter 2

58

for all .\ > 0, and so condition (2.6.7) holds. On the other hand, it is easy to check that Eq. (2.6.23) has a positive solution

x(t) = exp(- exp(bt)),

t ~ t0 •

Furthermore, neither (2.6.19) nor (2.6.21) holds. Therefore nonoscillatory solutions with type -oo may exist (see Lemma 2.6.1). Corollary 2.6.2. Assume that i) (2.6.19) holds or both (2.6.21) and (2.6.22) bold, n

ii)

liminf

L: p;(t)r;(t) > ~.

(2.6.24)

t-oo i=l

Then every solution of (2.6.5) is oscillatory. In order to present a nonoscillation criterion we need the following lemma.

Suppose that there exists a nonempty set I

Lemma 2.6.4.

i) r;(t) > 0

fort~

to,

~

{In} such that

i E I, and

ii) l:p;( t) > 0 fort ~ to. iEJ

If y is an eventually positive solution of the differential inequality n

y'(t) + LPk(t)y(t- Tk(t)) $0,

(2.6.25)

k=l

then there exists an eventually positive solution x(t) of (2.6.5) with

x(t) $ y(t)

for all large t,

and lim x(t) = 0. t-oo

Proof: Since y is an eventually positive solution of (2.6.25) we have that eventually

y'(t) $- LP;(t)y(t- r;(t)) $- LP;(t)y(t) iEJ

iEl

59


and hence, because of i), we have the y'(t) < 0 eventually. Choose T sufficiently large. For any t;::: t;::: T integrating (2.6.5) from t to have t we

+

y(t);::: y(t)

i t

n

:l::>A:(s)y(s- TA:(s))ds A:=l

i A:=l t

>

t

n

~:::>A:(s)y(s- TA:(s))ds.

t

Because tis arbitrary we get

Define a set fl = {:r: E C([T,oo),J4): 0:5 :r:(t) :5 y(t),

t;::: T}.

For any :r: E fl, we define

t;::: T x(t) = { :r:(t), :r:(T) + (y(t)- y(T)), T :5 t :5 T. Then define a mapping T on ('T:r:)(t) =

n as follows:

ioo A:=l

tPA:(t)x(s- TA:(s))ds,

t;::: T.

t

Clearly, Tis well-defined, and ('T:r:)(t) :5 y(t) on t;::: T and hence 'Tfl Defineasequence{:r: n(t)}, n=0,1,2, ... asfollows: :r:o = y, Xn+l='T:r:n,

t;::: T, t;:::T,

n=0,1,2, ....

By induction, we see that 0 :5 Xn(t) :5 Xn-l(t) :5 · · · :5 y(t),

t;::: T,

c n.

Chapter 2

60

and hence x( t) = lim xn(t) exists on [T, oo ). Then we can apply the Lebesgue's n ..... oo dominated convergence theorem to show that x = T x, i.e.,

x(t) =

l oo

n

LPk(s)x(s- T~;(s))ds,

t

~

T.

A:=l

Obviously, lim x(t) t ..... oo

= 0. Also, we have n

x'(t) =- LPA:(t)x(t- Tk(t)),

t ~ T,

k=l

which means that x or x is a solution on [T, oo) of Eq. (2.6.5). Moreover, it is easy to see that x(t) $ y(t) fort~ T. It remains to prove that x is positive on [T, oo ). Clearly, x > 0 fort E [T, T) and x ~ 0 for t ~ T. If there is a t ~ T such that x(t) = 0, set

t* then x(t*)

= inf{f I x(f) = o,

= 0, x(t) > 0 fort x(t*) =

E [T, t*) and T;(t)

roo

lt

t•

t

~ T},

> 0,

i E I, t ~to. Hence

00

LP~;(s)x(s- Tk(s))ds > 0 k=l

contradicting that x(t*) = 0. Theorem 2.6.6. Suppose that there exists a nonempty set I ~ In so that i) and ii) in Lemma 2.6.4 are satisfied. Moreover, assume that for a sufficiently large T 0 ~to n

->.+sup LPk(t)exp(>.Tk(t)) $0,

>. > 0.

for some

(2.6.26)

t~To k=l

Then there exists a positive solution x of Eq. (2.6.5) satisfying

x(t) $ exp (->.t)

for alllarge t

Proof: The point To is sufficiently large so that t -

T~;(t) ~

(2.6.27)

t 0 , for t

~

T0 ,

61

Oscillations of First Order Delay Equations k E In. Set y(t)

= exp( -,\t), t;;:: to. Then n

n

y'(t)

+ LPk(t)y(t- Tk(t))

= -,\y(t) + LPk(t)exp(- -\(t- Tk(t)))

k=l

k=l

This means that y(t) is an eventually positive solution of (2.6.25). By Lemma 2.6.4, Eq. (2.6.5) has a solution x(t) satisfying x(t) $ y(t) = exp( -.\t) D eventually.

Corollary 2.6.3.

Consider the differential equation n

x'(t) + LPkx(t- Tk) = 0

(2.6.28)

k=l where Pk, Tk are positive constants, k E In. Then evezy solution of (2.6.28) is oscillatory if and only if n

-.\ + LPkexp (ATk) > 0

for all ,\ > 0.

(2.6.29)

k=l

Corollary 2.6.4. Assume that the conditions of Lemma 2.6.4 hold. Moreover, assume that there exist T ;;:: to and .\ > 0 such that

(2.6.30)

for a To ;;:: t 0

and .\

> 0.

Then there exists a positive solution x of Eq. (2.6.5) satisfying lim x(t) t-+oo

= 0 and

Chapter 2

62 eventually

Proof: Choose T0 so large that t- Tk(t) ~to fort ~To,

k E In. Set

It is easy to see that y is a positive solution of (2.6.25) So the conclusion of the corollary follows from Lemma 2.6.4. n

Suppose that

Remark 2.6.2.

l: Pk(t) >

0, for t ~ to, and for a sufficiently

k=l

large To~ to sup

t~To

{

max

k=1,2, ... ,n

l

t

t-rk(t)

(

n ) 1 LP;(s) ds } ~-.

i=l

e

(2.6.32)

Then (2.6.30) holds. Consequently, the conclusion of Corollary 2.6.4 holds. Corollary 2.6.5. Assume that the assumptions i) and ii) in Lemma 2.6.4 hold. Then every solution of (2.6.5) is oscillatory if and only if differential inequality (2.6.25) has no eventually positive solutions.

From this corollary we can derive some comparison results. With Eq. (2.6.5) we associate the equation n

x'(t) +

L q;(t)x(t- u;(t)) = 0.

(2.6.33)

i=l

Theorem 2.6.7. and

Assume that the assumptions i) and ii) in Lemma 2.6.4 hold,

(2.6.34)

63


eventually. Then every solution of Eq. (2.6.5) is oscillatory implies the same for (2.6.33). Proof:

If not, let x(t) be an eventually positive solution of (2.6.33). Then

eventually n

x'(t) =- L

q;(t)x(t- u;(t))

i=l n

:::; - LP;(t)x(t- u;(t)) i=l n

$- LP;(t)x(t -r;(t)). i=l

By Corollary 2.6.5, Eq. (2.6.5) has an eventually positive solution contradicting the assumption. D

Corollary 2.6.6.

If Eq. (2.6.5) has an eventually positive solution and (2.6.35)

eventually, and there exists a nonempty set I E In such that u;(t) > 0 for t ~ t 0 , i E I and l:q;(t) > 0 fort ~ to. Then (2.6.33) has an eventually positive solution also.

iE/

Proof: Let x(t) be an eventually positive solution of (2.6.5). Then eventually n

n

x'(t) = - LP;(t)x(t- r;(t)):::;- L i=l

q;(t)x(t- u;(t)).

i=l

By Lemma 2.6.4, Eq. (2.6.33) has an eventually positive solution. We then consider (2.6.5) for the case that lim r;(t) = t-oo

D

r; and t-oo lim p;(t) = p;.

Chapter It

64 The "limiting equation" of (2.6.5) is n

z'(t) + :EiJ;z(t- r;) = 0.

(2.6.36)

i=l

lim T;(t) Assume that T;, Pi E C([to,oo),R+), t-ooo Theorem 2.6.8. lim p;(t) =Pi· Then (2.6.36) is oscillatory implies the same for (2.6.5).

= r;,

Proof: By Corollary 2.6.3 we have -A+ :Ep; exp (AT;)> 0 for all

A> 0.

(2.6.37)

Assume the conclusion of the theorem is not true. Let x(t) be a positive solution of (2.6.5). Then for any e > 0 there exists aT?: to such that n

x'(t) + :EC.r;- e)x(t- r; +e)~ 0,

t?: T.

i=l

In view of Corollary 2.6.5, it follows that n

x'(t) + :E(p;- e)x(t- r; +e)= 0 i=l

has a positive solution. Hence by Theorem 2.6.1 the characteristic equation n

F(JL)

= -JL + :E(.P;- e)e".,

i E In

t 0 E R+ and

(2.6.42)

lr;(t)

and n

L

M;ea;>.-

i=l

Then Eq. (2.6.39) has a positive solution.

>. :5 0.

(2.6.43)

Chapter 2

66

Proof: We will prove this theorem with condition (2.6.41). The theorem with condition (2.6.40) can be proved by a similar argument. Set v(t)=inf{s:s~t, r;(e)~t, for(~s,

iEln}.

Let t1 ~ v(to)· Let C([t 0 ,oo),R) be a space of all continuous functions defined on [to,oo) with the Frechet topology. Let n c C([t0 , oo ), R) be the subset satisfying: a)

x(r;(t)) a.\ . x(t) $ e ' , t ~ t1, ~ E In

b) x(t) = 1, t E [to,tt]

c) exp(-),. ] 111 q(s)ds) $ x(t) $ exp(l:::':, 1 ea;.\ J:, [p;(s)J- ds), t ~ t1 where [p;( s )]- = max (- p( s ), o). Clearly, n is a nonempty, closed and convex subset of C([t 0 , oo ), R). Define an operator T : n - C([t 0 , oo ), R) by

(Tx)(t)=

{ ( j exp-

t

;~ p;(s)x(T;(s))) ()

t,

X S

,t~t1

t E [t 0 , tt].

1,

n

It is easy to see that Tf! ~ and T is continuous. On every compact interval of [to, oo), (T x)(t), X E n is uniformly bounded and equicontinuous. By Ascoli theorem cl{Tf!} is compact in the topology of the Frechet space C([t 0 , oo ), R). By the Schauder-Tychonoff fixed point theorem, there exists an X En such that Tx == x, i.e.,

x(t) = exp

(i

n

t

;"£ p;(s)x(r;(s))ds)

-

( )

X S

11

,

which implies that x is a positive solution of (2.6.39) on [t 1 , oo ). n

Corollary 2.6.7.

and

Assume LP;(t)

~

0, t ~ T for a sufficiently large T ~ t 0 ,

i=l

sup{~ax [' (tPi(s)) ds} $ ~. t~T •Ein lt-T;(t) e j=l

67


Then Eq. (2.6.39) has a positive solution. In fact, choose n

q(t) = l:p;(t) i=l

and ). = e. Then all assumptions of Theorem 2.6.9 are satisfied. This corollary follows from Theorem 2.6.9. In particular, if n = 1, from Theorem 2.6.9 we obtain the following results. Corollary 2.6.8.

Assume that either

r(t) :5 t,

1

1

1

P1(s)ds

:5-,

r(l)

e

1

1

or

p(t) ;::: 0,

1

p(s )ds :5 -, e r(t)

Then (2.6.39) has a positive solution on [to, oo ), where p 1 (s) =max (p(s ), 0). Corollary 2.6.9.

Assume that:

(i) there exist holds:

E R, M;, i E

a;

In,>., t 0

p;(t) :5 M;,

tp;(t) :5 M;, t lntp;(t)::; M;,

E

Rt

such that one of the following

.

01 r·(t) > - t ''

t ;::: t 0 ,

or

r;(t) ;::: o:;t, t ;::: t 0 ,

or

r;(t) ;::: t

01 ' ,

t ;::: to;

n

(ii) L:M;e 01 '>.-). :50; i=l

(iii) one of (2.6.40) and (2.6.41) holds. Then Eq. (2.6.39) has a positive solution. This corollary follows from Theorem 2.6.9 by taking q(t) = 1, q(t) =

q(t)

= tfot respectively.

t and

Chapter£

68

Theorem 2.6.10. Assume that there exist nonnegative numbers M;, a;, i E In, .X, t 0 E R+ and a locally integrable function q : ~ --+ 14 such that

I{'

IP;(t)l5 M;q(t),

q(s)dsl5 a;,

t 2:: to,

i E In,

lr;(t)

and n

LM;exp(a;.X)- .X 50. i=l


The proof is similar to the proof of Theorem 2.6.9. We only point out that the functions in the set n satisfy

Corollary 2.6.10.

If for all sufliciently large t

l

1

t

lp(s)lds 5 -. e

r(t)


Corollary 2.6.11.

If there exist constants p;, r;, i E In such that

It- r;(t)l 5 r;

IP;(t)l 5 p;, and n

.X= Lp;e~r; i=l

has a positive root, then Eq. (2.6.39) has a positive solution.

69


2.6.6. Advanced equations

Finally, we would like to mention that most oscillation criteria for Eq. (2.6.5) can be extended to the equations with advanced type of the form n

x'(t) = Ep;(t)x(t + T;(t)).

(2.6.44)

i=l

For instance, we present the following results. Theorem 2.6.11. Assume that p;, T; E C([t0 , co), R+), with T;(t) uniformly bounded, i E In and such that

liminf{ inf t-oo

~>O

p;(t)exp(..\T;(t))} > 1. ~~ L...., A

(2.6.45)

i=l

Then every solution of (2.6.44) oscillates. Proof: Suppose the contrary, assume x(t) Set

x'(t)

..\o(t) = x(t),

> 0,

t ;::: T, is a solution of (2.6.44).

t;::: T.

Then

Define a set

!l = {..\ E C([T,oo),R): ..\(t);::: 0, t;::: T} and a mapping T defined on

n

(2.6.46)

Chapter 2

70

Clearly, Tf! and

c n,

A1 (t);::: A2(t);::: 0 implies (TA 1 )(t);::: (TA2)(t);::: 0 fort;::: T,

Ao(t) = (TAo)(t),

t;::: T.

Ao(t);::: 0,

In view of the assumption, there exists To > 0 such that IT;(t)l $ To, Form (2.6.45) there exists C > 1 such that 1

inf

>.e(O,oo)

n

:\ ~::>;(t)exp(AT;(t)) ;::: C. i=I

In particular, taking A= 1, we have fort;::: T n

n

C $ 'Lp;(t)exp(r;(t)) $

ero

'Lp;(t). i:l

i51

Hence for t ;::: T

C n "p;(t) ;::: ~

ero

•=I

=M

> 0,

n

Ao(t) = (TAo)(t);::: 'Lp;(t);::: M, i=l

and so n

Ao(t) = (TAo)(t);::: TM = LP;(t)exp(MT;(t));::: MC, i=l

Repeating this procedure we obtain Ao(t);::: C"M-+ oo,

as

k-+ oo,

which contradicts (2.6.46). The proof is complete. Corollary 2.6.12.

Assume that T;(t) $To, liminf t-+oo

i E In and

p;(t)T;(t) > ! . ~ e ~ i=l

t;::: T.

i E In.

71


Tben every solution of (2.6.44) oscillates.

Theorem 2.6.12. Assume tbat p;, >. • > 0, T ~ t 0 sucb tbat

r; E

C([to,oo),R+), and there exists

n

->.*+sup I>;(t)exp(>.*r;(t)) :::; 0. t;::T i=l

Tben (2.6.44) bas a positive solution on (T, oo ). Remark 2.6.3. Finding conditions for the existence of oscillatory solutions of (2.6.5) is an open problem.

2. 7. Equations with Forced Terms We consider the forced differential equations of the form

x'(t) + f(t,x(ri(t)), ... ,x(rn(t))) = q(t).

(2.7.1)

Assuming here q E C([t 0 ,oo),R) and let Q(t) E C 1 ((t 0 ,oo),R) such that Q'(t) = q(t) fort~ t 0 • The following result provides an oscillation criterion.

Assume tbat

Theorem 2.7.1.

i) j, q, r;, i E In, are continuous, ii) lim r;( t) = oo, i E In, t-+oo

iii) fort

~

T

~

t.

ud( t, ul! u2, ... , un) > 0 whenever u1 u; > 0, i E In and for flxed t, nondecreasing witb respect to u;, i E In, iv)

1 1

00

00

f(t,Q+(ri(t)), ... ,Q+(rn(t)))dt =

f is

00

f(t, -q-(ri(t)), ... , -Q-(rn(t)))dt = -oo

(2.7.2)

Chapter 2

72 wbere Q+(t) = HIQ(t)l + Q(t)), solution of (2. 7.1) oscillates.

Q_(t)

= HIQ(t)l- Q(t)).

Then every

Proof: Suppose the contrary, let x(t), be an eventually positive solution of (2.7.1). Then

(x(t)- Q(t))'

= - f( t, x( Tt (t)), ... , x( Tn( t))).

(2.7.3)

In view of condition ii), (x(t)- Q(t))' < 0 eventually. Condition iv) implies that Q(t) always changes sign for sufficiently large t. Therefore there exists a sequence tk --+ oo such that x(tk) > Q(tk), and then

x(t)- Q(t) > 0 for t

~

lim (x(t)- Q(t)) = t--+oo

l

T*,

(2.7.4)

which implies that

exists. From (2.7.3) we obtain that for sufficiently large T

!roo f(t,x(rt(t)), ... ,x(rn(t)))dt < oo. From (2.7.4), x(t) > Q+(t)

fort~

(2. 7.5)

T. In view of the monotonic property of

f and (2.7.5) we obtain (2.7.6) which contradicts (2.7.2). Similarly, we can derive a contradiction when Eq. (2.7.1) has an eventually negative solution. The proof is complete. Example 2.7.1. We consider the equation x'(t) + px(t- 1r) = sint + pcost

(2. 7. 7)

where pis a constant with 0 < p7re < 1, then the homogeneous equation x'(t) + px(t- 1r) = 0

(2.7.8)

73


has a positive solution by Corollary 2.1.2. For (2.7.7) Q(t) =-cost+ psint, it is easy to see that (2.7.9) By Theorem 2.7.1, every solution of (2.7.7) oscillates. In fact, y(t) =-cost is such a solution. Remark 2.7.1. From the proof of Theorem 2.7.1, it is easy to see that condition iv) can be replaced by iv),

liminf Q(t) = -oo t-+oo

(2.7.10)

lim sup Q(t) = +oo. t-+oo

We now consider the case that lim sup Q(t) = M < oo t-+oo

liminf Q(t) = m t-+oo

(2.7.11)

> -oo.

For the sake of simplicity we consider the equation of the form x'(t) + p(t)x(r(t))

Theorem 2.7.2.

= q(t).

(2.7.12)

Assume that

p E C([to,oo),~), r, q E C([t 0 ,oo),R), limr(t) = oo. t-+oo

Set Q'(t)

and

= q(t),

Q changes sign on (T,oo), where Tis any number. If

(2.7.13)

74

Chapter~

Then every solution of (2. 7.12) oscillates. Proof: Suppose the contrary, x(t) > 0, t C: t 0 , is a solution of (2.7.12), then there exists t1 C: to such that x(r(t)) > 0 fort C: t 1. As in the proof of Theorem 2.7.1 we know that x(t)- Q(t) > 0 and (x(t)- Q(t))' < 0 fort C: t1. which implies that

1,,

00

p(s)x(r(s))ds 0 there exists T C: t 1 such that l < x(t)- Q(t) < l

+ e,

and hence

-Q(t) < l+e,

t C: T.

Then -m

= -liminf Q(t) ~ l +e. t-+oo

(2.7.18)

On the other hand, there exists a sequence {tn} such that lim tn =co and n-+oo lim x(tn) = 0. From (2.7.17) we have

n-+oo

-Q(tn) > l- x(tn)

75


and hence -liminf Q(tn)

~f.

R-+00

and so -m ~f..

(2.7.19)

(2.7.18) and (2.7.19) imply that f.= -m. From (2.7.17), x(t)- Q(t)

> -m,

~

t

t1

and so x(t)

> [Q(t)- m]+, t

Choose t 2 > t 1 so large that r(t) x(r(t))

~

t1

fort~ t2.

~ t 1•

Then

> [Q(r(t))- m]+,

t ~ t2.

Substituting this into (2.7.14) we obtain

1

00

p(t)[Q(r(t))- m]+dt

< oo,

t2

which contradicts (2.7.13). Similarly, we can prove that (2.7.11) has no eventually negative solutions. The proof is complete. Example 2. 7 .2. Consider the equation x'(t) + p(t)x(t- '11") =-cost,

where p(t)=

{ ~t,

1.,.(--t), "

t E [0, ~] t E (,.2•'~~"1

and p(t + 2'11") = p(t) for all t ~ 0. In this case Q(t) = -sint and M = 1, m

= -1.

(2.7.20)

Chapter 2

76 Cleaxly, we have

ioo p(t)(1- Q(t- 1r)]+dt = oo, ioo p(t)[1 + Q(t- 1r)]+dt =

(2.7.21)

00.

By Theorem 2.7.2, every solution of (2.7.20) oscillates. We note that condition (2.7.2) does not hold for (2.7.20). In the next chapter we will allow a result for the existence of nonoscillatory solutions of (2.7.1).

2.8. Single Population Models with Delays 2.8.7. Delay logistic equation We consider the delay logistic equation with a constant caxrying capacity k

dx(t) = () ( )( _ x(t- r(t))) , k rtxt1 dt

(2.8.1)

where r, T axe positive continuous functions defined on [0, oo) and k is a positive constant. Denote r* = sup r( t). t>O

Motivated by the ~pplication of (2.8.1), we consider solutions of (2.8.1) with initial conditions of the type

x(s) =

~P(s)

is continuous on

where

IP ~ 0

[-r*, 0] and

~P(O)

> 0.

(2.8.2)

It is not difficult to show that solutions of (2.8.1) and (2.8.2) are defined for all t ~ 0 and remain positive fort ~ 0. A solution of (2.8.1) and (2.8.2) is said to be oscillatory about k if there exists a sequence {tn} such that lim tn = oo and k- x(tn) = 0, n = 1, 2, .... Set

n_.oo

y(t) =

x~t)

- 1,

t

~ -r•.

(2.8.3)

Then (2.8.1) becomes

y'(t) = -r(t)(1 + y(t))y(t- r(t)),

t > 0,

(2.8.4)

77

Oscillation.s of Fir.st Order Delay Equations

whose initial conditions are inherited from (2.8.2). By way of (2.8.3), the oscillation of x about k is equivalent to that of y (about zero) as in the usual sense. We note from the positivity of x(t) that 1 + y(t) > 0 fort ~ 0. We state some results on oscillation of Eq. (2.8.4) without proofs. Theorem 2.8.1. Assume that i) r, T E c(~.(O,oo)), and lim

t-oo

J0

(t- r(t))

= oo,

r(s)ds = oo. Then every solution of Eq. (2.8.4) is either oscillatory or converges to zero, eventually monotonically, as t-+ oo. ii)

00

Theorem 2.8.2.

In addition to the assumptions of Theorem 2.8.1 assume

r(s)ds >

liminflt t-r(t)

t-oo

~. e


Theorem 2.8.3.

Assume there exists a

i

to~

0 such that

1

t

r(s)ds

t-r(t)

~-,

e

t

~to.

Then Eq. (2.8.4) has a nonoscillatory solution.

=

r > 0, r(t) If r(t) Corollary 2.8.1. Eq. (2.8.4) is oscillatory if and only ifrre > 1.

=

> 0, then every solution of

T

Let us consider now the delay logistic equation of the form

dx(t) dt where n is a positive integer,

= T

r

(t) (t) [ 1 _ x(tx

nr)]

k(t)

> 0, r, k E C(R+,(O,oo)).

(2.8.5)

Chapter£

78

The following result is concerned with the oscillation of solution about k. Theorem 2.8.4. Assume that i) k is r-periodic and positive and a= max k(t), t~O

1

00

ii)

b =min k(t), t~O

r(t)[a- k(t)]dt

a > b,

= oo

(2.8.6)

(2.8.7)

and

1

00

r(t)[k(t)- b]dt

= oo.

(2.8.8)

Then every positive solution x of Eq. (2.8.5) is oscillatory about k(t), i.e., there exists a sequence {tn} such that lim tn = oo and x(tn) = f3(tn), n = 1, 2, .... n-oo

Proof: Assume the contrary, and let x(t) be a solution of (2.8.5) such that x(t) > k(t) fort ~ t 0 > 0. Then x'(t) < 0, t ~ t 0 , and hence lim x(t) = l ~ a t-oo exists. We claim l = a. Otherwise, from (2.8.5) we see that for t ~ t 1 ~ t 0 x'(t) :5

~ (1- ~)r{t).

Noting that {2.8.7) and {2.8.8) imply that x(t) :5 x(t 1 )

+~ 2

(1- ~) lt,ft

contradicting that x(t) > k(t) From (2.8.5)

a

f0

00

r(t)dt = oo. We have

r(s)ds-+ -oo,

as t-+ oo

~b.

a ] :5 - 1 r(t)(a- k(t)) a r{t) [1- k(t) x'(t) :5 2 2

for all large t, which leads to a contradiction. Similarly, (2.8.5) has no positive solution x(t) with x(t) < k(t) eventually. The proof is complete.

79


The following provides a sufficient condition for the existence of a solution of (2.8.5) with x(t) > k(t) eventually. Theorem 2.8.5. Assume 0 k( t) eventually. Proof: Choose T so large that

2b =

b* < k(t) 0 on [To, T) and hence y( t) Therefore y is a positive solution of Eq. (2.8.13) on [T, oo ).

We now consider the case:

>

0 for all t ;:: T0 •

ft': r(s).X(s)ds = oo. Using (2.8.18) we see that

Chapter 2

86 for sufficiently large t

f1

00

~

T

r( u )(1 + Caexp(- J; r(s )>.(s )ds )c::- 1 exp( -a J;(t) r(s )>.(s )ds) )du exp(- J;r(s)>.(s)ds)

:5 (1 + c .. )c::- 1 m < ~ < 1. Hence there exists a sufficiently large T1 such that

1 r( 00

u) ( 1 + c..exp ( -

lu

£

:5 exp ( -

r(s )>.( s )ds)

r( s )>.( s

)ds) ,

c::- exp (-a J:(u) r( s )>.( s )ds)) du 1

t

~ Tt.

By using the similar method to the previous one, we can show that (2.8.13) has a nonoscillatory solution. ii) Necessity. If (2.8.13) has an eventually positive solution then from Lemma 2.8.1 there exists a continuous positive function >.(t) such that

>.(t) = (1 + c .. exp( X

-1:

>.(s)r(s)ds))

c::- exp((1- a) ltr(t) r(s)>.(s)ds + lt

r(s).\(s)ds)

1

r(t)

1

~ c::- 1 exp((1- a) ir(t) r(s)>.(s)ds + lt

r(s)>.(s)ds).

r(t)

t1

(2.8.20)

.

Let m = cd-~ Then (2.8.20) implies (2.8.18). If (2.8.13) has an eventually negative solution, then

( i

>.(t) ~ (1-IC.. I)IC::-tlexp (1- a)

h

r(t)

>.(s)r(s)ds +

1t r(t)

)

>.(s)r(s)ds ,

where !C.. I < 1. Thus (2.8.18) is true also. The proof is complete. In the following we derive some explicit sufficient conditions for the existence of nonoscillatory solutions of (2.8.13).

87

Oscillations of Fird Order Delay Equations

If a > 1 and

Corollary 2.8.1.

(1- a)

l

r(t)

r(s)ds

+

1'

r(s)ds :5 A,

t

~ t0 ~

0,

(2.8.21)

r( t)

It

where A is some positive number. Then Eq. (2.8.13) has anonoscillatory solution. In fact, (2.8.21) follows from (2.8.18) by taking ~(t)

= ~ > 0.

Remark 2.8.1. H

loo

r(t)dt

< oo

or {

00

Jo0

r(t)dt = oo,

and

1'

r(s)ds :5 A,

r(t)

t

~ t0 ~ 0.

Then (2.8.21) is satisfied. Example 2.8.1. Consider

y'(t) =- _2_1 (1 + y(t))y 2 ( v't).

t+

(2.8.22)

Since

-

1

.,fi

to

ds

--+ 1 +S

j'

ds +S 1 ,fi

--=ln(1+t)-ln(1+t+?'i)+ln(v't;+1) :5ln (y't; + 1),

the condition (2.8.21) is satisfied. By Corollary 2.8.1, Eq. (2.8.22) has a nonoscillatory solution. In fact, y(t) = is such a solution.

t

2.8.9. Generalized delay logistic equation (II) Here we discuss the equation

x'(t) = r(t)x(t) 1 - x(t- r(t)) 1- cx(t- r(t))

(2.8.23)

88

Chapter 2

where c E (0, 1), r, T E C(R+, J4), r(t) :5 t, lim (t- r(t)) = oo. When c = 0, t--+oo (2.8.23) reduces to (2.8.1). From an ecological point of view, we will restrict our attention to the bounded positive solutions of (2.8.23). Denote

Eo= {t- r(t): t- r(t) :50, t

~

0} U {0}.

The initial conditions for (2.8.23) which we consider are c- 1 > x(8) = 4'(8) ~ 0,

8 E Eo,

cp(O) > 0

(2.8.24)

where cp(8) is continuous on E 0 • Assume that r(t) ~ T > 0 for all t ~ 0. It is known, by the method of steps, the initial problem (2.8.23) and (2.8.24) always (locally) has a unique positive solution. Make the change of variables 1- x(t) y(t) = 1- cx(t) ·

Then (2.8.23) and (2.8.24) reduce to

y'(t) =- r(t) (1- y(t))(1- cy(t))y(t- r(t)), 1-c

(2.8.25)

and

y(8) = [1- cp(8)]/[1- c4'(8)],

8 E E0 .

(2.8.26)

Obviously, -oo < y(8) :5 1, y(O) < 1. Thus, the study of the oscillatory behavior of solutions of (2.8.23) about 1 is equivalent to the study of the oscillatory behavior of solutions of (2.8.25) with respect to zero. It is easy to see that the solution of (2.8.23) and (2.8.24) is positive, which implies that the solution of (2.8.25) and (2.8.26) satisfies 1 - y(t) > 0 and 1 cy(t) > 0. We call a solution of a differential equation global, if it exists for all t ~ 0. In the following, we will consider global solutions of (2.8.23). For this we first give a sufficient condition to guarantee the existence of global solutions of (2.8.23). Theorem 2.8.8. Assume that t - r(t) is continuous and nondecreasing, limt--+oo(t- r(t) = oo, and r(t) ~ T > 0 fort~ 0. Assume further that


89

r(t) 2: r 0 > 0 is continuous,

0'

=sup t~O

{it

r(8) d8}

t-r(t)

is finite, e"' < c-I, and that 0 < cp(8) < c- 1 for 8 E [-T(O), 0) and cp(O) < c- 1 e-"'. Then the solutions x(t) of (2.8.23) are bounded, and limsupx(t) ~ e"'. t-+oo

Proof: Since r(t) 2: 0 ~ t ~ t, we have

T

> 0 fort 2: 0, there exists at> 0 such that t = r(t). For

[' 1 - x( 8 - r( 8)) ) x(t) = x(O)exp ( Jo r(8) 1 _ cx( 8 _ r( 8 )) d8 ·

Since 0 ::; cp(8) < c-I,

(2.8.27)

1- u(8- r(8))/(1- cu(8- r(8))) ::; 1, thus for

O~t~t

1 - x( 8 - r( 8 )) d8 < [' r(8)d8 < f' r(8) 1cx(8- r(8)) - } -

}0

0'.

(2.8.28)

0

Obviously, (2.8.27) and (2.8.28) imply that

x(t) ~ x(O)e"' < c- 1 , Hence we have shown 0 ~ x(t) t. Then for t ~ t1, we have

0 ~ t ~ t.

for

< c- 1 for -r(O)

~

t

~

t. Assume that t 1 -r(t1 ) =

Thus

x(t)

~ x(to)exp1 1 r(8)d8. to

From this we have

x(t- r(t)) 2: x(t)exp(

-it

t-r(t)

r(8)d8)

Chapter£

90 and hence

x(t- r(t))

~ x(t)e_.,..

This implies for t ~ t :5 t1 1- x(t)e_.,. ) x'(t) ~ r(t)x(t) ( 1 _ cx(t)e-... .

Since r(t)

~

ro

> 0, we see that all positive solutions of (2.8.29)

with initial values less than 6, for 6 > 0, e.,.< 6 < c- 1 e... , will be bounded by 6 and have e.,. as their limit. Therefore, forE~ t ~ tt, 0 ~ x(t) < c- 1 < c- 1 e.,.. By repeating this argument (assume that h > t 1 is such that t 2 - r(t 2 ) = t 17 we can prove that (2.8.29) holds for t 1 ~ t ~ t 2 • Define ti+ 1 - T(t;+t) = t;, then t;-+ +oo as i-+ +oo), we see x(t) is bounded and satisfies (2.8.29) for all t ~ t, thus the solution x(t) of (2.8.23) is bounded by x(O)e.,. < c- 1 and lim sup x(t)

~e.,..

t-oo

This completes the proof of the theorem. The corresponding result for Eq. (2.8.25) is the following.

Theorem 2.8.9. Assume that r(t) and T(t) are the same as described in Theorem 2.8.8, and the initial function y(8), 8 E [-T(O), 0] satisfies y(8) ~ 1, y(O) > (1 - c-le-.,.)/(1 -e-.,.). Then the solution y(t) of (2.8.25) is bounded, and liminfy(t) t-oo

~

(1- ea)/(1- ce.,.).

Remark 2.8.2. As we mentioned before, when c = 0 (2.8.23) reduces to the logistic equation. In general, from the ecological point of view c is small, so that equation (2.8.23) is close to the logistic equation, and hence e.,. < c- 1 holds naturally.

91


In the following we will consider only global solutions of (2.8.23).

Assume that r(t) and r(t) are positive and continuous, Theorem 2.8.10. lim (t- r(t)) = oo, Jo"', r(t)dt = oo. Then eve.ry global solution of (2.8.23) is t-+oo either oscillato.ry with respect to 1 or tends to 1 as t -+ oo. Proof: By Theorem 2.8.8, a global solution x(t) of Eq. (2.8.23) satisfies 0 < x(t) < c- 1 • Then x'(t) < 0 fort> T*, where T*- r(T*) > T, and hence lim x( t) = p. 2::: 1 exists.

(2.8.30)

t-+oo

If p. > 1, then we have 1-p.

x'(t) :5 r(t)p. -1 - - , - cp.

for

t 2::: T*,

which leads to limx(t) = -oo which contradicts (2.8.30). Therefore p. = 1. Similarly, we can prove if 0 < x(t) < 1, fort 2::: T, then lim x(t) = 1. t-+oo Next, we would like to discuss oscillation for Eq. (2.8.25). Theorem 2.8.11.

Let r,

T

E C(~,(O,oo)), with lim (t- r(t)) = oo, t-+oo

liminf {'

r(8)d8

t-+oo lt-r(t)

> 1- c . e

Then eve.ry solution of Eq. (2.8.25) oscillates. Proof: Let y(t) be a negative solution of (2.8.25), then

y'(t)

+ 1r~~

y(t- r(t)) 2::: 0.

(2.8.31)

By Theorem 2.1.1 this is impossible under the assumptions of the theorem. Now let y(t) be a positive solution of (2.8.25). Then y'(t) < 0 and y(t) -+ 0 as t -+ oo by Theorem 2.8.10. Dividing (2.8.25) by y(t) and integrating from t- r(t) tot we have ln y(t

CD(t)) = 1 ~

Y

[' C lt-r(t)

r(s)(1- y(s))(1- cy(s)) y(s

C~(s)) ds (2.8.32)

Y

S

Chapter!

92

Let w be defined by w(t) = y(t- r(t)) . y(t)

Clearly w(t)

~

1. From (2.8.32) we have

ln w(t) = w(e) 1- c

r

lc-..(t)

r(8)(1- y(s))(1- cy(8))d8

(2.8.33)

where e E (t- r(t), t). Similar to Lemma 2.1.3 we can prove that w is bounded. Taking the liminf on both sides of (2.8.33) we obtain ln R.

-JJ-

~

,

where R. = lim inf w( t). Since t-+oo the theorem. Theorem 2.8.12. that

. f -1li mm 1-+oo 1- c lnt t

1' ( r

t-.. (t)

8)d8

(2.8.34)

$ ! , (2.8.34) contradicts the assumptions of e 0

Suppose r(t) is bounded above, and there is a t 1 > 0, such

i

t

t- .. (t)

1-c r(9)d9 $ - e

for t

~ t1.

Then Eq. (2.8.25) has a bounded nonoscillatory solution on [t,oo). Proof: Assume r(t) $ M for all t ~ 0. Let BC denote the space of all bounded continuous functions defined on~. with the supernorm. Let n c BC denote the subset satisfying: is Lipschitzian and nonincreasing on

(i)

y(t)

(ii)

lip

y(t)IR+ $ Me(1- a)/(1- c)

(iii)

y(t)

= 1- a,

(iv)

e y(t- r(t)) ~ y(t)e ~ y(t- r(t)),

t E [0, t],

R+

a E (0, 1) t ~ t1

(v) (1- a)exp(- -1 e [' r(9)d9) $ y(t) $ 1- a, -c}c,

t

~ t1

O$cillation$ of Fir$t Order Delay

1-a IY'(t)I:5Me 1 _c'

(vi)

93

Equatio~

t~t1

tER+,

where lip y(t)IR+ denotes the Lipschitzian constant of y over~· Let

t E [0, t1]

1 - a,

Yo(t) = {

t

(1- a)exp(- l~c fc, r(8)d9), t ~ t1.

Then y 0 (t) E {}and Sis nonempty. It is not difficult to see that {}is convex and compact. Define a mapping T: n-+ BC as follows: 1

(Ty)(t)

={

tE[O,ti]

a

(1--

~)exp(-

l:c

J,t, r(8))

x (1- y(8)) (1- cy(9))

11 ( 11~;~ 11 >) d8, t ~ t 1 •

It is easy to see that Tfl c n, and T is completely continuous. Hence, by the Schauder-Tychonov fixed point theorem, we conclude that T has a fixed point y in S. Obviously, y* is a positive, bounded solution of Eq. (2.8.25) on [t, oo ). The proof is complete.

=

=

Corollary 2.8.2. Ifr(t) r, T(t) T, rand Tare positive constants. Then every solution of (2.8.25) is oscillatory if and only if rre > 1 - c.

2.9. Notes Lemma 2.1.1 is obtained by Koplatadze and Canturija [97]. Lemma 2.1.2 is due to Kwong [101]. Lemma 2.1.3 is taken from Yu, Wang, Zhang and Qian [184], also see Jian [92], Erbe and Zhang [44]. Lemma 2.1.4 is adopted from [92], also see Elbert and Stavroulakis [33]. Theorem 2.1.1 is obtained by Koplatadze and Canturiza [97], a special case is by Ladas and Ladde, see [110]. Theorem 2.1.2 is based on [92] and [101], and related work can be found also in [184]. Theorem 2.1.3 and Theorem 2.1.4 are taken from Zhou [219], Theorem 2.1.5 is new. The results in Section 2.2 are taken from Li [124]. Theorem 2.3.1 and 2.3.2 are new, Theorem 2.3.3 and 2.3.4 are taken from Gyori [72]. The materials of the Section 5.4 are based on [184]. Theorem 2.5.1

94

Chapter£

and 2.5.2 are taken from Zhang and Gopalsamy [203], also studied by Ladas and Qian [106]. Theorem 2.6.1 is obtained by Tramov and Ladas, etc., by a different method, see [110 or 76]. Theorems 2.6.2 and 2.6.3 are extracted from Huang [84]. The proposition is established by many authors such as Gyori [74], Kwong [101], Li [126], Philos [150], Chen and Huang [15] and Chuanxi, Ladas, Zhang and Zhao [25]. Theorems 2.6.4 and 2.6.5 are based on Chen and Huang [15]. Corollary 2.6.2 is obtained by Hunt and Yorke (85] using a different method. Lemma 2.6.4, Theorem 2.6.6, and Corollary 2.6.4 are taken from Philos [150]. Corollary 2.6.5 is obtained by several authors, Wei [170], Gyori and Ladas [76]. Theorem 2.6.7 is based on Lalli and Zhang [117]. Theorem 2.6.8 is an improved form of a result by Gyori and Ladas [76]. Theorems 2.6.9, 2.6.10 and Corollaries 2.6.7-2.6.10 are taken from Nadareishvili (139]. An oscillation criterion for Eq. (2.6.5) with oscillating coefficients can be found in Yu, Wang, Zhang and Qian [184]. The first study of the oscillation of differential equations with advanced type is due to Zhang and Ding [199], and Zhang (202]. The corresponding results for differential inequalities with advanced type is due to Onose, see (76]. Theorems 2.6.11 and 2.6.12 are found in Zhou [217]. Theorem 2.7.1 is from Zhang [191], also see Erbe and Zhang (44]. Theorem 2.7.2 is a new result by Yu and Zhang. Theorems 2.8.12.8.3 are taken from Zhang and Gopalsamy (208]. The global attractivity of the delay logistic equation is studied by Zhang and Gopalsamy [207]. Theorems 2.8.4 and 2.8.5 are obtained by J.S. Yu and B.G. Zhang. Theorems 2.8.6 and 2.8.7 are due to Li [125], also see Aiello [4]. Eq. (2.8.23) without delay as an ecological model of the single species is posed by Cui and Law [27]. Theorems 2.8.8 2.8.12 are taken from Kuang, Zhang and Zhao [70], in which some results for the existence of periodic solutions and the global attractivity of solutions are obtained. There are many important results for oscillation and nonoscillation of nonlinear delay differential equations in the literature. Some of then can be found in the monographs by Ladde, V. Lakshmikantham and Zhang [110], Gyori and Ladas [76]. Some further results for delay ecological equations can also be found in the monograph by Gopalsamy [51].

3 Oscillation of First Order Neutral Differential Equations

3.0. Introduction In general, the theory of neutral delay differential equations is more complicated than the theory of delay differential equations without neutral terms. For example, Snow (see also Gyori and Ladas, Chapter 6, [76]) has shown that even though the characteristic roots of a neutral differential equation may all have negative real parts, it is still possible for some solutions to be unbounded. In this chapter, we will present a systematic study for the oscillation theory of first order NDDEs, which contains some recent results and consequently is a useful source for researchers in this field. In Section 3.1 we consider the linear neutral delay differential equations with constant parameters, where the "characteristic equation" method plays an important role. Some necessary and sufficient conditions for the characteristic equations to have no real roots are presented. In Sections 3.2 and 3.3 we consider the neutral delay differential equations

95

Chapter 9

96 with variable coefficients of the form

(x(t) + p(t)x(r(t)))'

+ q(t)x(t- u(t))

= 0

(3.0.1)

where q(t) ;::: 0 and -1 :5 p(t) :5 0, p(t) := -1, p(t) :5 -1, or p(t) > 0, respectively. Since the behavior of solutions of Eq. (3.0.1) is closely dependent on the range of p(t), we deal with the case that -1 :5 p(t) :5 0 in Section 3.2, and the other cases in Section 3.3. Some sharp conditions for all solutions to be oscillatory and for the existence of positive solutions are presented. In Section 3.4, we present a comparison result for oscillation. In Section 3.5, we consider Eq. (3.0.1) in the unstable case. In Section 3.6, we discuss a class of sublinear neutral delay differential equations. In Sections 3. 7 and 3.8, we deal with the neutral delay differential equations with positive and negative coefficients of the form

(x(t)- c(t)x(t- r))'

+ p(t)x(t- r)- q(t)x(t- u) = 0.

Oscillation criteria, criteria for the existence of positive solutions and results of linearized oscillation are given. In Section 3.9, we are concerned with the neutral delay differential equations with a nonlinear neutral term. In particular, the equation

(x(t)

+ px

0

(t- r))' + q(t)xft(t- u) = 0

is considered, where a =f. 1, f3 =f. 1. The deviating arguments in the neutral differential equation (3.0.1) can be of various types. Usually, we investigate the following cases:

1) r(t) :5 t,

u(t);::: 0,

2) r(t) :5 t,

u(t) :50,

3) r(t) ;::: t,

u(t);::: 0,

4) r(t);::: t,

u(t) :50,

and the mixed type. In this chapter we mainly discuss case 1), which corresponds to the neutral delay differential equations. But the methods and techniques employed here can be used for equations of other types.

Oscillation of First Order Neutral Equations

97

3.1. Characteristic Equations We first state a basic result on the oscillation of the neutral delay differential equation with constant parameters

(x(t)+ where p;, r;, and q;,

~p;x(t-r;))' + ~q;x(t-u;),

u;

are given real numbers, and r;,

t

~to

u; ~ 0,

(3.1.1)

j E Im, i E Ik.

Theorem 3.1.1. Every solution of Eq. (3.1.1) is oscillatory if and only if its characteristic equation m

k

>. + >. :~:::>;exp( ->.r;) + j=l

L q; exp( ->.u;) = 0,

(3.1.2)

i=l

has no real roots. The above proposition can be extended to higher order NDEs. From Theorem 3.1.1 the characteristic equation (3.1.2) plays an important role in the investigation of the oscillation and asymptotic behavior of solutions of Eq. (3.1.1). But to determine if (3.1.2) has a real root is quite a problem itself. In the following we derive other necessary and sufficient conditions for the oscillation of Eq. (3.1.1) which can be easily applied to get some explicit sufficient conditions. For sake of convenience, we consider m

(x(t)- P1x(t- r1)- pzx(t- rz))'

+ L q;x(t- u;) = 0

(3.1.3)

i=l

where Pi (j = 1,2,), q;, u; (i = 1,2, ... ,m) E R+, T; > 0 (i = 1,2). From Theorem 3.1.1 Eq. (3.1.3) is oscillatory if and only if its characteristic equation

F(>.)

= >.(1- p1e-.Xr, -

m

pze-.Xr2 )

+ L q;e-.Xu; = 0 i=l

has no real roots, or, F(>.)

> 0 for all>.

E R.

(3.1.4)

Chapter 9

98

From (3.1.4) we see that if p 1 + p 2 = 1 then every solution of Eq. (3.1.3) is oscillatory. For the case that 0 1, we define the following three subsets Db, b = 1,2,3, of the space £1, and three functions hb(t) defined on Db, b = 1,2,3, · 1y: (u' · t h e expansiOns · b elow c;-(k+ ) = (-k-1)(-k-2) (-le-i) respective vve h ave m i! ··· 1 i,k=1,2, .... ) D1 = {S = (S;;~c): (ijk) E 12, S;;~cC~(k+l)[(k + i

+ 1)-r1- i-r2- u;] 2:: 0,

E2S;;1c = 1}, D 2 = {S = (S;;~c): (ijk) E 12, S;;~cC~(k+ 1 )[(k E2S;;~c

= 1},

D 3 = {S = (S;~c): (j, k)

e 13 ,

h (S) 1

= E2

(k

S;;~c

+ i + 1)7"1 -

i7"2 -

S;~c[(k

+ i + 1)-r2- i-r1- u;] 2:: 0,

+ 1}-rt- u;] 2:: 0,

EaS;1c = 1},

O"j

; x ln ( eC -(lc+l)P 1-(k+i+l) p;2 q; [(k + z. + 1) T1

-

. - u; ]/Sij/c ) , z-r2 (3.1.7)

h (S) = E 2 2 (k + i X

S;;1c

+ 1)7"2- i7"1- O"j

; ; -(k+i+l) q; [(k + z. + 1) 1"2 ln ( e C -(k+1)P1P2

.

- ZT1 -

u; ]/Sijk ) , (3.1.8)

(3.1.9) where12={(ijk): j=1, ... ,m; k, i=1,2, ... }, 1a={(jk):j=1, ... ,m, k = 0, 1, ... }, E2 = L: , L:a = L: , where S;;1c = 0 or S;k = 0 imply that (ijk)E/2

(jk)Ela

the corresponding terms vanish. Different from the series in (3.1.5), we cannot guarantee that all the series in (3.1.7) - (3.1.9) are convergent in Db, instead, as shown in the Remark 3.1.1, we see there exists at least one series which is convergent on Db. Theorem 3.1.3.

Assume p 1

+ P2 > 1.

Then

i) for some b = 1,2, or 3, hb(t) has a maximum at a point S(b) = (S~Ji), b = 1, 2, or s< 3 > = (Sm) on Db, which is completely determined by one of

Chapter 3

100 the conditions that 1

(k

+ i + 1h X 1n

ir2- ui

; -(k+i+I) ; [(k (c -(k+l)Pl p 2 qj + t. + 1)T1

.

- tT2 - U j

]/S(l)) ijk , (3.1.10)

1

(k

+ i + 1)r2 X 1n

ir1 -

Uj

; ; 2-) < oo for some b = 1, 2, or 3. Theorems 3.1.2 and 3.1.3 can be applied to get a series of sufficient conditions for oscillation of Eq. (3.1.3). To use Theorem 3.1.2 we need to find a t• E D such that f(t'") > 0, whereas, to use Theorem 3.1.3, we need firstly to choose a suitable function hb which is convergent on Db, and then find aS* E Db such that hb(S'") > 0. The following corollaries are derived from Theorems 3.1.1 and 3.1.3, where m )1/m q= ( q; ,

II

i=l

Corollary 3.1.3. Assume Pt + P2 < 1. Then each one of the following is sufficient for Eq. (3.1.3) to be oscillatory:

(3.1.13)

Oscillation of First Order Neutral Equations 00

ii)

mq

k

101

.

2:: 2:: Cip~p~-'[iTI + (k- i)T2 + u];::::

~,

k=O i=O

Corollary 3.1.4. Assume P1 i) Let T1 f. T2, and E3 ( 2pl ) - ( k+l) qi [(k

+ P2 > 1.

+ 1) T1 -

-ln u i ] e (.,- •• 1

llP2 )((k+l)r1 -a,·]

= 1.

(3.1.14)

Then Eq. (3.1.3) is oscillatory if and only if (3.1.15)

Otherwise, each one of the following is sufficient for Eq. (3.1.3) to be oscillatory: " L.J'

ii)

ci-(k+l)Pl-_ -, 1

(3.1.16)

e

(ijk)E/2

"' L.J c;-(k+l}Pl- 0, (k + i + 1)Tl -

iT2- O"j

> 0,

(ij)EJ2

U

l> 1 _ e

(3.1.17)

where

(k + i + 1)T2 - iT1 -

O"j

J2 = {(ik): G~(k+l) > 0, (k+ Z.+1) T2

-

· tT1 -

U

> 0,

and

)k+ 2i+l (k + i + 1)T2- iTt- u·1 ( Pl . . P2 ( k + t + 1 )TI - t T2 - u i

> 1}

(k + i + 1)Tl- iT2- U > 0, >O

an

d

(Pl)k+ 2i+l(k+i+1)T2-iTI-0" (k . . P2 + z + 1)Tl - tT2 - u

iii) (3.1.16) and (3.1.17) bold ifp1 and p2 , respectively.

T1

and

T2

> 1}

exchange their positions

102

Chapter 8

It is easy to see that 12 I 0 (empty set) and has infinitely many terms provided PI > P2, or PI = P2 and TI > r2. To prove Theorem 3.1.2 we need the following lemmas.

Lemma 3.1.1.

Assume 0 0 for F defined by (3.1.4) and all A E R. In particular, m

F(Ao)

= >.o(1-ple-.>.or1 -P2e-.>.or2) + Lq;e-.>.o"'i

>0

j=l

where Ao satisfies 0 < Ple-.>.0 '"1 + P2e-.>.o.-, < 1 by Lemma 3.1.1. Hence m

GI(Ao)

= Ao + Lq;e-.>.o"'i (1- Ple-.>.o.-1- P2e-.\or2)-1 > 0. j=l

From Lemma 3.1.2 we have f(t 0 ) = G 1(>. 0) > 0. On the other hand, if f(t 0 ) > 0, by Lemma 3.1.2, G 1 (>. 0 ) > 0. Hence G1(A) > 0 for all A E (>.'",oo), where A* is given by Lemma 3.1.1. If Eq. (3.1.3) is not oscillatory, then there exists a A1 E R such that m

F(At)

= >.1(1- Ple-.>.

1 - p2e-.>.1'"2)

1 '"

+ Lqie-.>.1"'i

= 0.

(3.1.23)

j=l

Obviously A1 < 0 and thus 0 .1n (3.1.23) gives us that

+ P2e-.>.1'"2
.1"'i (1- P1e-.>.1n - P2e-.>.1'"')-1 =

G 1(>. 1)

j=l

which contradicts G1(A)

> 0 on (A*,oo).

0

To prove Theorem 3.1.3 we need the following lemmas. Lemma 3.1.3.

Assume Pl

+ P2 > 1. Define m

G2(1J)

= -I'+ Lq;e-P"'i (p1e-P'"1 + P2e-P'"2 -

1) -1,

j=l

and let IJ* > 0 be such that p 1 e_,..,.1 !Jo E ( -oo, IJ*) such that

+ p2 e_,..,. = 1. 2

Then there exists a

(3.1.24)

105


and G2{Po) is the minimum value of G2(p) in {-oo, p*). 0

Proof: Similar to Lemma 3.1.1. Assume p 1 i) la.e-l'o(r,-rt) < 1 and

Lemma 3.1.4.

p 0 is defined by Lemma 3.1.3. Let

'

P1

ii)

+ P2 > 1,

sg! = C~(k+l)P~(J:+i+ 1 )p~qj((k + i + 1)r1 -

ir2 - O"jjel'o[(k+i+l)r1-i1'2-CTjJ,

(ijk) E 12.

Then i) S< 1> =

(sgl) E D1,

ii) (3.1.10) has the same value for all (ijk) E 12, iii) h1 (S< 1>)

sm =/:

0,

= G2(Po),

iv) h 1(S< 1>) is the maximal value of h1(S) on D1. Proof: By (3.1.24) and condition i), there exists a neighborhood U of po such that 0 < (p1e-""1 + P2e-""2)- 1 < 1, and ~e-"("2 -rt) < 1 for p E U. m

G2(p)

= -p + Lqie-""'; (p1e-""1 + P2e-""2)-1 [1- (p1e-""1 + p 2 e-""2)- 1]-1 j=1 m

= -p

+L j=l

co

L qie-1'"'; (p1e-""1 +P2e_,.,.,) -(1:+1) k=O

(3.1.25) P-(J:+l)p;q·e"[(J:+1)r1-ir,-cr;] ~ Ci -- _,_,... + "-'2 • 2 1 -(k+1) 1

{3.1.26)

The rest of the proof is similar to that of Lemma 3.1.2 and hence is omitted. 0 Lemma 3.1.5. Let the conditions of Lemma 3.1.4 hold ifp1 and P2, r 1 and are replaced by s~J!. Then the 7'2 exchange their positions, respectively, and

sZ!

Chapter 9

106

conclusion of Lemma 3.1.4 holds if s(l>, D~, h~, and (3.1.10) are replaced by D 2 , h2 , and (3.1.11), respectively.

s< 2>,

Lemma 3.1.6. Assume p 1 i) l!.a.e-l'o(r.-rt) = 1 and Pt

+ p 2 > 1,

p. 0 is defined by Lemma 3.1.3. Let

'

ii) S~!) = (2pi)-(k+l)qj[(k+1)TJ-O'j]e~' 0 [(k+l)rt-D'i], (jk) E 13.

(3.1.27)

Then

i)

s< 3> = (SW) E D3,

ii) (3.1.12) has the same value for all (jk) E 13, S~!)

f 0,

iii) h3(S< 3>) = G2(P.o),

iv) h3 (S< 3 >) is the maximum value of h 3 (S) on

D3.

Proof: From (3.1.25) and condition i) (3.1.28) The rest of the proof is omitted.

Remark 3.1.1. There exists at least one of hb which is well-defined on Db (b = 1, 2, 3). In fact, there is at least one expansion of (3.1.26), its dual form, and (3.1.28), of G 2 (p.), which converges at p. = p.0 • Therefore, at least one of hb(s< 6>), (b = 1, 2, 3), has a finite value and satisfies hb (S< 6>) = G 2 (p. 0 ). Proof of Theorem 3.1.3: The first part is shown by Lemmas 3.1.4- 3.1.6. Assume Eq. (3.1.3) is oscillatory. Then F(p.) > 0 for all p. E R. In partieular, m

F(p.o) = P.o(1- Pte-l'ort- P2e-l'or•)

+ Lq;e-I'OD'i > 0 i=l

where p. 0 satisfies (3.1.24). Hence m

G2(P.o)

= -p.o + Lqie-l'oD'i (Pt e-l'ort +P2e-l'or2 j=l

1) -1

> 0.

107


By Remark 3.1.1 there exists b = 1, 2, or 3, such that hb(S(b)) = G2(!-Lo). Therefore, 0 < hb(S(b)) < oo for some b = 1, 2, or 3. On the other hand, if 0 < hb(S(b)) < oo for some b = 1, 2, or 3, by Remark 3.1.1, G2(fLo) > 0. Hence G2(fL) > 0 for all fL E ( -oo,fL*). Assume Eq. (3.1.3) is not oscillatory. Then there exists a ILl E R such that m

F(fLl) = /Ll (1- Ple-,.,r,- P2e-"'r2 )

+ Lqie-l'ta;

=

0.

j=l

Obviously, /Ll

> 0, and thus Pl e-"' r, + P2e-~'' r 2 > 1, i.e.,

/Ll E ( -oo, fL*), and

m

G2(/Ll) = -ILl

+ Lqie-"'a; (ple-"' r, + p2e-"' r

2 -

1) -l = 0,

j=l

contradicting that G 2(fL)

> 0 on ( -oo, fL*).

0

Proof of Corollary 3.1.3: i) Choose

such that l:1tijk = 1. By (3.1.13) we have 0

< c ~ 1. Hence

i k-i qic 1n -1 > ~ c;kP1P2 f(t *) = .c..1e _ o. c

Noting that t• does not make (3.1.6) have the same value for (ijk) E I, we see t* =f. t 0 • So f(t 0 ) > 0. By Theorem 3.1.2, Eq. (3.1.3) is oscillatory. The proofs for ii) and iii) are similar. In fact, for ii) we choose

0 Proof of Corollary 3.1.4:

i) The conditions of Lemma 3.1.6 are satisfied, where fLo = _l_lnl?.!.. P2 rt -r2

Chapter 9

108

Hence from (3.1.9) and (3.1.27)

Thus h 3 (S< 3 >)

> 0 if and only if (3.1.15) holds.

ii) Without loss of generality we may assume the left hand side of (3.1.16) is finite, for otherwise we can replace I2 by its suitable subset in (3.1.16) such that the above assumption holds. Choose S'!'.k IJ

i [(k +'· + 1)Tt = { eC i-(lc+t)Pt-(Hi+I) P2q; 0,

such that 'E 2 Si;~c From (3.1.7)

= 1, i.e., s• = (Si;~c) E D2.

h I (s *)

u; ]c, (.l:J"k) E I*2 (ijk) E !2\!2

By (3.1.16) we have 0 < c :5 1.

i In ~ 1 > e ci-(k+t)Pt- p2q;c - o.

" L.J'

=

·

-tr2-

(ijlc)E/2

From (3.1.8) and the definition of !2

ln

[( PI) k+2i+l (k(k + i. + 1)r2 1)

+I +

P2

ir1 .

u;] > 0.

Tt - lT2 - Uj

By Remark 3.1.1 we get that there exists a b = 1 or 2 such that h6(S< 6>) < oo. Noting that S* does not make (3.1.10) have the same value for (ijk) E 12 , hence hb(S< 6>) > hb(S*) ~ 0. By Theorem 3.1.3, Eq. (3.1.3) is oscillatory. iii) The proof here is similar and so is omitted. 0 If we let p 1

= p,

P2

= 0,

r1

= r, then Eq.

d dx [x(t)- px(t- r)]

(3.1.3) becomes

m

+ ~q;x(t- u;) = J=l

0,

(3.1.29)

109

Oscillation of First Order Neutral Equations and the above results can be reformulated as follows. Define two sets

D 1 ={t=(t;k):t;k~O,j=1, ... ,m;

k=0,1, ...

;L Lt;k=1, j=l m

oo

}

k=O

D 2 = { S = (S;k): S;k((k + 1)r- u;};:::: 0, j = 1, ... ,m; k = 0, 1, ... ;

L j=l m

oo

}

L:S;k=1 . k=O

On D 1 we define a function

on D 2 we define a function m

oo

h(s)=L L

S jk

J=l =0 (k + 1)rl - u 3· .

ln(ep-(k+llq;((k+1)r-u;)/S;k)·

k

Theorem 3.1.4. Assume 0 < p < 1. Then i) f(t) has a maximum at a point t 0 = (t~k) on D 1 which is determined by the condition that

has the same value for all j = 1, ... , m, k = 0, 1, ... , t~k ii) Eq. (3.1.29) is oscillatory if and only if f(t 0 ) > 0.

:f:. 0.

Theorem 3.1.5. Assume p > 1. Then i) h(S) has a maximum value at a point SO = (SJk) on D2, which is determined by the condition that

Chapter 3

110

has the same value for all j = 1, ... , m, k = 0, 1, ... , SJk f- 0, (k+1)T-l7jf-0. ii) Eq. (3.1.29) is oscillatory if and only if h(S0 ) > 0.

Corollary 3.1.5. Assume 0 < p for Eq. (3.1.29) to be oscillatory: m

oo

I: I: pkqi(kr + l7j) 2::

i)

< 1. Then each of the following is sufficient

~'

j=I k=O 00

I: pk(kr + u) 2::

ii) mq

~'

k=O

m

oo

I: I: pkqj,

iii) let a=

•

a;k = k~.:~., j = 1, ...

j=l k=O

m

oo

E I:

~ ln(ea(kr

,m, k = 0, 1, ... , and

'

+ u;)] 2:: 0.

j=l k=O

Corollary 3.1.6. Assume p > 1. Then each one of the following is sufficient for Eq. (3.1.29) to be oscillatory:

i)

there is a k; for each j = 1, ... , m, such that (k;

ii)

there is a k0 such that ( k0 00

mq

E

k=ko

+ 1)r -

+ 1)r- l7j

> 0, and

l7 > 0, and

p-(k+I)[(k + 1)r -u] 2:: ~·

3.2. Equations with Variable Coefficients (I) 3.2.1.

We consider the linear neutral differential equations of the form

(x(t)- p(t)x(t- r))'

+ q(t)x(t -u(t)) = 0,

t 2:: to.

(3.2.1)

Clearly, the oscillatory behavior of solutions of (3.2.1) depends on the range of p. In this section, we discuss the case that 0 ~ p ~ 1.

111


The following result reduces the oscillation problem of NDE (3.2.1) to a corresponding problem of delay differential inequality of type (2.1.1).

Assume that

Theorem 3.2.1.

i)

T

E (O,oo),

p, q, u E C([to,oo),R+) such that 0

1

00

~

q(t)dt

= oo

and

~

lim (t- u(t))

t-oo

p(t)

~

1,

= oo;

ii) there exists a positive integer N such that the differential inequality N-1

y'(t)+q(t)[1+p(t-u(t))+ ..

·+IT p(t-u(t)-ir)]y(t-u(t)) ~ 0 (3.2.2) i=O

has no eventually positive solutions. Then every solution of (3.2.1) is oscillatory. Proof: Assume the contrary, and let x(t) be an eventually positive solution of (3.2.1). Set

z(t) = x(t)- p(t)x(t- r).

(3.2.3)

Then z'(t) ~ 0. If z(t) < 0 eventually, then x(t) ~ p(t)x(t- r) ~ x(t- r) eventually, which implies that x(t) is bounded. Hence z(t) is bounded, and lim z(t) = e exists. From (3.2.1), we obtain that t-oo

1

00

q(t)x(t- u(t))dt < oo.

to

This together with condition i) derives that liminf x(t) = 0. Hence there exists t-oo a sequence {tn} such that lim tn = oo and lim x(tn) = 0. Since n-+oo

n-+oo

+ r)- z(tn)) = n~~ (x(tn + r)- (p(tn + r) + 1)x(tn) + p(tn)x(tn- r))

0 = lim (z(tn n-oo

we see that

Chapter 9

112

which implies that lim p(tn)x(tn- r) = 0 and hence n--+oo

R.

= n--+oo lim (x(tn)- p(tn)x(tn- r))

= 0.

This contradicts the assumption. Therefore z( t) > 0 eventually. Obviously, x(t) > z(t) for all large t. Considering that z(t) is decreasing we have

x(t) = z(t) + p(t)x(t- r)

2:: [ 1 + p(t) + .. · +

!!

N-1

]

p(t- ir) z(t)

(3.2.4)

for all large t. Substituting (3.2.4) into (3.2.1) we obtain

z'(t) + q(t) [ 1 + p(t- u(t))

+ · ·· +

!!

N-1

]

p(t- a(t)- ir) z(t- a(t)) 50

D

which contradicts assumption ii).

Remark 3.2.1. Some sufficient conditions for Eq. (3.2.2) to have no eventually positive solutions have been presented in Theorems 2.1.1 and 2.1.2. In the following we shall give a nonoscillation result for Eq. (3.2.1).

Theorem 3.2.2. and T E (O,oo), u(t) T ;::: t 0 such that

Assume that p E

C([t 0 ,oo),~),

q E C([t 0 ,oo),(O,oo)),

= u E [O,oo). Further assume that there exist A* E (O,oo) and

sup [p(t- u) ( q(t) ) e.\"r t~T q t- T

+ ,1

q(t)e.\""] 5 1.

A*

Then Eq. (3.2.1) has a positive solution on [T, oo ).

(3.2.5)


113

Proof: First we claim that the integral equation

w(t)

tc:} )

= p(t- u) q t

r

w(t- r)exp(

+ q(t)exp 1~, w(s)ds,

t 2::: T

r' w(s)ds) lt-T + m,

(3.2.6)

possesses a positive continuous solution on [T, oo ), where m =max{ r, u }. To this end, set w1 (t) = 0, t 2::: T, and fork= 1,2, ...

p(t- u) 9 (~~)T)wk(t- r)exp(J,'_T wk(s)ds) Wk+t(t)

={

+q(t)exp(J,'_, wk(s)ds), Wk+t(T+m),

t 2::: T

+m

(3.2.7)

T:::;t 0 = w 1 (t) fort 2::: T and w2(t):::; ..\* fort 2::: T. Assume for some positive integer k, Wk- 1 (t) :::; Wk(t) for t 2::: T. Then by (3.2.7) and (3.2.5), wk+ 1 (t) 2::: Wk(t) fort 2::: T, and

Wk+l (t) :::; p(t - u) q(:~)r) ..\ * e>.•T + q( t)e>.•" :::; ..\*sup [p(t- u) t;::T

q(t) e>.•T + _!_q(t)e>.•,] :::; ..\*. ..\*

q(t- r)

By induction we see that for t 2::: T

Then it follows that lim wk( t) = w( t) exists and is positive on [T, oo ). By taking k-oo limits on both sides of (3.2. 7) and by using the Lebesgue's monotone convergence theorem we see that w(t) is a solution of (3.2.6) on [T,oo). We also see that the sequences {wk(t)}f:1 converge uniformly on [T, T+m]. Then it follows by (3.2.6) that w(t) is a continuous function on [T,oo). Set z(t) = exp(- J;w(s)ds). Then w(t) =- £ill Z(t} and z'(t) < 0 fort 2::: T. Thus (3.2.6) reduces to

z'(t) = p(t- u) (q(t) ) z'(t- r)- q(t)z(t- u). q t- r

(3.2.8)

Chapter 9

114

Now define x(t) = - z'(t + u)/q(t + u). It is easy to see that x(t) is a positive solution of Eq. (3.2.1) on [T, oo ). 0 Theorem 3.2.3.

Assume that

i) p, q E C((to,oo),R), (O,oo);

0 :5 p(t) :5 p

0 such that

+ jq(t)!e""" :5p,,

p(t)p.e~'r

t;:::: to.

(3.2.9)

Then Eq. (3.2.1) has a positive solution on [t 0 ,oo). Proof: Set p(t)

= p(t 0 ),

t :5 t 0 , and

t + m- to ( ) - - - - q to , to - m :5 t :5 to q(t) = { m 0, t :5 t 0 - m where m = max{r,u}, thenp,q are well defined on Rand i) and ii) hold on R. Define a set of functions on R as: X={>.:>. E C(R,R), I.X(t)l :5p, fort;:::: t 0 where the norm of >. is II.Xll =

-

m, .>.(t)

=0

fort :5 t 0

-

m}

sup I.X(t)le- 2 ~-'t. Then X is a Banach space. t>to-m

Define a mapping on X as follow;;

('T.X)(t) = q(t) [ exp Clearly, ('T >. )( t)

!~.,. >.(s)ds + ~

= 0 for t :5 t 0 -

m.

U

p(t- u- jr)exp

j~u-ir >.(s)ds].

(T >. )( t) is continuous and

I(T>.)(t)! :5 jq(t)l [exp !~.,. l.X(s)lds

+~

U

p(t- u- jr)exp

:5lq(t)!(e" 0 such that q1 , ••• , qt-1 are defined, and

00

sq(s) /.

00

•

qt- 1 (u)duds

= oo.

(3.2.16)

Then every solution of (3.2.13) is oscillatory. Proof: Assume the contrary, and let x(t) be an eventually positive solution of (3.2.13). Set

z(t) = x(t)- x(t- r).

(3.2.17)

Then it is easy to see that z(t) > 0 eventually. Hence there exists M > 0 such that x(t) ~ M,

t

~

T ~to.

(3.2.18)

Substituting (3.2.18) into (3.2.13) we have

z'(t) + Mq(t) $ 0.

(3.2.19)

If (3.2.15) holds, (3.2.19) leads to

lim z(t) = -oo,

t-oo

(3.2.20)

Chapter 9

118

a contradiction. If

roo q(t)dt < oo,

(3.2.21)

fto from (3.2.19) we obtain

z(t) 2:: M

1

00

q(s)ds,

t;::: T.

(3.2.22)

Hence

x(t) 2:: x(t- r) ;::: M (

;::: M [ t

+M

1

00

q(s)ds

roo q( s )ds + roo q( s )ds + ... + roo

},

ft-r

~

T] 1 q(s)ds, 00

ft-{['~T]-l}r

q( s )ds)

t 2:: T,

where [·] denotes the greatest integer function. Hence there exist M1 T1 2:: T such that

> 0 and

(3.2.23) Substituting {3.2.23) into (3.2.13) we have

If lim t-+oo

roo sq( s) /.• Jr

00

q( u )duds = oo,

(3.2.24)

we obtain (3.2.20). Otherwise

Repeating above procedures f-1 times we get a contradiciton with (3.2.16).

0


119

As an example we consider the equation

(x(t)- x(t- r))'

+t

0

x(t- 0') = 0.

(3.2.25)

From Theorem 3.2.4 we have the following result. Corollary 3.2.2.

If a > -2, then every solution of (3.2.25) is oscillatory.

In fact, there exists an integer n > 2 such that 1

a> n - -2+-. Let i

= n,

(3.2.26)

then (3.2.16) holds. Corollary 3.3.2 follows from Theorem 3.2.4.

Remark 3.2.2. We will see from Theorem 3.2.6 that if a < -2, then (3.2.25) has a bounded positive solution. This shows that condition (3.2.26) is sharp. Example 3.2.1. Consider the equation ( xt -xt-1 ) ,

()

(

1

( ) =0 . ) + 2(t- 1)Vt ( Jt + Vt=l) xt-1

(3.2.27)

x(t) = .Jt is a solution of (3.2.27), which implies a > -2 is the best possibility for oscillation of (3.2.25). Example 3.2.2. Consider the equation

(x(t)- x(t- 1))' + ___.;_ x(t- 1) = 0, tln t

t

~ 2.

(3.2.28)

It is easy to see that (3.2.16) with i = 1 is satisfied. Therefore every solution of (3.2.28) oscillates by Theorem 3.2.4. We note that (3.2.15) does not hold here. In the following we present a necessary and sufficient condition for the existence of a bounded positive solution of Eq. (3.2.13). Theorem 3.2.5.

Eq. (3.2.13) possesses a bounded positive solution if and

Chapter 3

120

only if

L: },roo . q(t)dt < 00

(3.2.29)

00.

to+•r

i=O

Proof: St£fficiency. By (3.2.29) there exists T ~to such that fort~ T

f: 1~ q(t)dt s

1.

q(s)ds,

t~T

(3.2.30)

t+tr

i=O

Define a function

f1

00

H(t) =

{

(t- T

+ r)H(T)/r,

T- r S t < T

t < T-

0,

(3.2.31)

T.

It is easy to see that HE C(R,R+)· Let 00

y(t)

= 2: H(t- ir),

t ~ T.

(3.2.32)

i=O

Then y E C([T, oo), R+)· Condition (3.2.30) implies that 0 < y(t) S 1 From (3.2.31) we have

y(t)=y(t-r)+H(t),

fort~

T.

t~T+r.

Define a set X as follows: X= {x: x E C([T,oo),R),

0 S x(t) S y(t),

t

~

T}.

The set X is considered to be endowed with the usual pointwise ordering

0 for t ;::: t 0 • Hence x(t) > x(t- r), t 2:: to, and there exist M > 0 and t 1 2:: t 0 such that x(t) 2:: M, t 2:: t 1 • From (3.2.13), z'(t) :5 -Mq(t), t 2:: t 1 + u. Integrating it from t to oo for t 2:: t1 + 0' we have

x(t) 2:: x(t- r) + M

1

00

q(s)ds.

Then

x(t1 +

n joo

0'

+ nr) 2:: x(t1 + u) + M ~

t,+u+ir

q(t)dt.

122

Chapter 3

Since x(t) is bounded, from the preceding inequality we obtain that

~ 00

i=l

100

. q(s)ds

< oo.

0

t,+.,.+•r

The following result will lead to an equivalent version to condition (3.2.29). To this end we consider the second order delay equation x"(t) + q(t)x(t- r) = 0.

(3.2.33)

Theorem 3.2.6. Assume that q E C([t 0 , oo ), R+ ), three propositions are equivalent

r

> 0. Then the following

(i) Every bounded solution of Eq. (3.2.33) is oscillatory;

(ii)

f 1';tq(t)dt = oo;

Proof: (i) ==> (ii). It is sufficient to show that Eq. (3.2.33) has a bounded positive solution if

1

00

tq(t)dt

< 00.

(3.2.34)

to

In fact, from (3.2.34), there exists T > t 0 such that J;' tq(t)dt ~ t· We introduce the Banach space X of all bounded continuous functions x : [to, oo) ~ R with the sup norm Uxll = sup{lx(t)i: t E [to,oo)}. We consider the subset n of X defined by

!l={xEX:

l~x(t)::=;2,

t2::to}.

(3.2.35)

It is obvious that n is a bounded, closed and convex subset of X. We define the operator 7 on n by 1 + J;(s- T)q(s)x(s- r)ds

(Tx)(t)

={

+ (t- T) J00 q(s)x(s- r)ds, 1

(Tx)(T),

t 2:: T to:::; t:::; T.

(3.2.36)

123

Oscillation of First Order Neutral Equations In view of (3.2.35) we have 1

~ (Tx)(t) ~ 1 + 2(

£

00

(s- T)q(s)ds)

~ 1 + t ~ 2,

i.e., Tn ~ n. Let X}, X2 be elements of n. Then

I(Txi)(t)- (Tx2)(t)l

~

£

(s- T)q(s)lx1(s- T)- x2(s- T)lds

+ (t- T)

1

~ 1lx1- x21l

00

q(s)lx1(x- T)- x2(s- T)ids

(£

~tllx1-x21l,

(s- T)q(s)ds

+ (t- T)

1

00

q(s)ds)

t~T.

Hence

That is, T is a contraction operator on Tx=x,or

n. Then there exists an X

E

n such that

1 + J;(s- T)q(s)x(s- T)ds

x(t) =

{

+(t- T) J100 q(s)x(s- T)ds,

t

to~

x(T1),

(3.2.37)

~T,

t

~

T.

which implies that x(t) is a bounded positive solution of (3.2.33). (ii) ==> (i). If not, let x(t) be a bounded positive solution of Eq. (3.2.33). Then x"(t) ~ 0, x'(t) > 0 and x(t) > 0 eventually. Thus lim x(t) = o: > 0 t-+oo exists, because x(t) is bounded. Integrating (3.2.33) twice we have

x(t)

~

£

(s- T)q(s)x(s- T)ds

+ (t- T)

1

00

q(s)x(s- T)ds

(3.2.38)

where Tis a sufficiently large number. Then condition (ii), implies that lim x(t) = t-+oo oo which contradicts the boundedness of x.

Chapter 9

124

(i) => (iii). It is sufficient to show that Eq. (3.2.33) has a bounded positive solution if

f: 1

00

q(t)dt

•

(i). If not, let x(t) be a bounded positive solution of Eq. (3.2.33). Then x"(t) $ 0, x'(t) > 0, x(t) > 0 eventually. Integrating (3.2.33) we have x'(t)

2::

1

00

q(u)x(u- r)du,

t;::: T,

(3.2.41)


125

where T is a sufficiently large number. Integrating (3.2.41) from t - r to t we have

x(t)

~ x(t- r) +

lt 1

00

t-T

~ x(t- r) + r

1

00

q(u)x(u- r)du,. t

~ x(t- r) + ar

~ T + r. (3.2.42)

Hence there exists a> 0 such that x(t)

x(t)

q(u)x(u- r)duds

•

~a fort~

1

00

T. From (3.2.42) we have t

q(u)du,

~ T + r.

By induction

x(T + (n + 1)r) ~ x(T) + ar

n+l

roo

i=l

T+ar

L J'l

.

q(u)du.

Since x(t) is bounded, it follows that

which contradicts (iii).

0

Combining Theorems 3.2.5 and 3.2.6 we obtain the following result. Theorem 3.2.7. only if

Eq. (3.2.13) possesses a bounded positive solution if and

roo tq(t)dt
u;

+ r- u))ds > 0;

iii) there exists a positive and continuous function H(t) such that

liminf t-oo

f t

t+r-a

H(s)ds > 0,

liminf{q(t)/p(t + r - u)H(t)} > 0; t.-oo

Oscillation of First Order Neutral Equations iv) there exists T

+r

~ t0

.m f

{

t?;T,.X>O

+

127

such that

jt+r H( s)ds) ( jl+r-a )}> )H( )exp .X H(s)ds

q(t)H(t + r) (' exp" p(t + T - u)q(t + r)H(t)

(t) .X ( q_ pt+r u

t

1

1.

t

(3.2.47)

Then every solution of (3.2.1) is oscillatory. Proof: Assume the contrary, and let x(t) be an eventually positive solution of Eq. (3.2.1). Set z(t) = x(t)- p(t)x(t- r). From condition (i) we have z(t) < 0 and z'(t) < 0 fort~ t 1 ~ t 0 • Eq. (3.2.1) becomes

z'(t)- p(t- u) tt) ) z'(t- r) + q(t)z(t- u) = 0, q t- T

t

~ t0,

or equivalently

z'(t)

=

p(t

q(t)

+ r- u)q(t + r)

z'(t

+ r)

q(t)

+ p(t +r-u )z(t+r-u), Set .X(t)H(t)

= z'(t)fz(t),

for

t

~ t 1.

t~t 0 -r.

(3.2.48)

Then

(3.2.49)

and .X(t) > 0 fort ~ t1. Substituting (3.2.49) into (3.2.48) we have

(Jl+r .X(s )H(s )ds) q(t) (jt+r-a .X(s)H(s)ds), (3.2.50) + p(t + u)exp q( t) ( jt+r-a ) 1 >p(t+r-u)exp .X(s)H(s)ds,

.X(t)H(t) = p(t +

q(t) u)q(t + r) .X(t + r)H(t + r)exp

7 _

7 _

1

1

1

t~t -r.

128

Chapter 9

By Lemma 3.3.2 (see Section 3.3) we have

liminf t-oo

l t+T-cr >..(8)H(s)d8 < oo. t

It is not difficult to see that liminf >..(t) t-oo

= >.. 0 E (O,oo).

From (3.2.47) there exists a a E (0, 1) such that . f

0 t;?!:~.).>O

{

q(t)H(t + T) (>.. p(t+T-u)q(t+T)H (t)exp q(t)

+ \1 ( "' p

t +T

_

( )H( ) exp >..

t

U

lt+T H( 8)d8) t

lt+T-cr H(8)d8 )}> 1.

(3.2.51)

t

There exists a t 2 ~ max{t 1 ,T} such that >..(t) > a>.. 0 , into (3.2.50) we have

q(t)H(t + T) ( >..(t)H(t) > a>..o ( )( pt+T-uqt+T ) exp a>..o q(t)

(

+ p(t + T _ u) exp a>..o

t

~

t 2 • Substituting it

lt+T H(8)d8 ) t

lt+T-cr H(8)d8) , t

t ~ t2• (3.2.52)

Thus, for

8 ~ t2

>..(8) > inf { a>..op( t;?!:t2

q(t)H(t + T) ( t + T - U )q (t + T )H() t exp a>..o

lt+T H(8)d8 ) t

)} + p(t + T _ u)H(t) exp ( a>..o lt+T-cr t H(8)d8 . q(t)

Taking inferior limits in

8

we have

q(t)H(t + T) ( >..o ~ t;?!:t2 inf { a>..o ( )( )H() exp a>.. 0 p t +T - U q t +T t q(t)

+ p(t + T _ u)H(t)

(

exp a>..o

lt+T H(8)ds ) t

lt+T-cr H(8)d8 )}. t

(3.2.53)


129

Letting .X 1 = a.Xo in (3.2.53) we have

. f { a 1n t~t2

q(t)H(t + r) exp ('"1 p(t+r-u)q(t+r)H(t) + \

1 p

"1

(

q(t) ( )H( ) exp .X1 t + T - 0' t

lt+r H( s )ds) t

lt+r- 0. (3.2.54) contradicts (3.2.51).

(3.2.54)

0

From Theorem 3.2.9 we can obtain different sufficient conditions for oscillation of Eq. (3.2.1) by different choices of H(t). For instance, if we choose H(t) = q(t)jp(t + T - u) or H(t) = 1, then (3.2.47) becomes {

inf

t~T,.X>O

1

p(t + 2T- u)

1 + \ exp(g.X

exp ( .X

lt+r- 0, (3.2.55) and (3.2.56) lead to the following corol-

laries. Corollary 3.2.6.

In addition to condition (i) of Theorem 3.2.9 further assume

that lim inf { q( t) / p( t + t-+oo

T -

0')} > 0

(3.2.57)

and liminf { t-+oo

1

p( t + 2T -

+ e 0')

lt+r- 0 and T

+T

q(t~

-

(7

( qt

+T

~to

) e>."r

+ T - u)}

> 0;

such that

+ ,1 A•

) e>."(r-rr)} < 1. (3.2.60)

( q(t) p t +T -

-

(7

Then Eq. (3.2.1) bas a positive solution on [T + r- u, oo). Proof: At first, we show that the integral equation

.X(t) =

q(t) ) .X(t + r)exp ) ( ( pt+r-uqt+r q(t)

)exp ( + pt+r-u

Jt+r .X(s)ds t

Jt+r-u .X( s )ds

(3.2.61)

t

has a positive solution. To this end we define a sequence {Ak(t)} as follows: .X 1 (t)

=

0,

t

~

T,

and

Ak+l(t)

( q(t) ).Xk(t)exp ) ( = p(t +r-uqt+r

Jt+r Ak(s)ds) t

) .Xk(s)ds , ( q(t) )exp (Jt+T- 0, ( q t +a

t 2:: T- a.

(3.2.65)


(x(t)- p(t)x(t- r)) 1 + q(t)x(t- a)= 0. That is, we have found a positive solution x(t) of Eq. (3.2.1).

D

3.2.4.

=

=

a > 0, r > 0, q E p f=: 1, a(t) Assume that p(t) Theorem 3.2.11. 00 C( [to, oo ), R+) and j 0 q( t )dt < oo. Then Eq. (3.2.1) bas a positive solution.

Chapter 9

132

Proof: Let BC be the Banach space of all bounded continuous functions on [t 0 ,oo) with the super nonn. i) Consider the case that 0 < p < 1. Let t" be so large that t* - T ~ t 0 , t* - u ~ t 0 and q( s )ds $ ~ . Set f2 = {x E BC: 1 $ x(t) $ 2, t ~ t 0 }. Then n is a bounded, closed and convex subset of BC. Define a mapping T: n-+ BC as follows:

J,':'

(Tx)(t) = { 1- p + px(t- r) (Tx)(t"), Clearly, T is continuous and Tn

+ J1

00

q(s)x(s- u)ds,

t ~

t0 $

c n.

For

Xt, Xz

E

t"

t $ t*.

n and t ~ t*

I(Tx1)(t)- (Txz)(t)i $ plx1(t- r)- x2(t- r)l

+

1

00

q(s)lx1(s- u)- xz(s- u)lds

1-p 1+p Sllx1- x21!(p+ - 2- ) = - 2-llxi- xzll. Hence

1 + p/2 < 1 implies that T is a contraction. Then there is a Tx = x. It is easy to see that x(t) is a solution of (3.2.1) on [t",oo).

X

E

n that

ii) Consider the case that p > 1. Lett* be so large that t*- a~ to and ft':'+r q(s)ds $~·Define a subset n of BC and a mapping Ton n as follows:

p-1 fl={xEBC: - 2 -$x(t)$p,

t~to}

and

(Tx)(t) = {

p -1

+!. P

(Tx)(t"),

x(t + r)- l. f 100 +r q(s)x(s- a)ds, t ~ t* P

to$ t $ t*.

133

Oscillation of First Order Neutral Equations It is easy to show that

Tn c

n and

for xl, X2 E n, i.e., Tis a contraction on n. Then there is a X E n such that Tx = x, or x(t) is a positive solution of (3.2.1) on t ~ t•. The rest can be discussed similarly. We only give an outline for those. iii) The case that -1 < p:::; 0. Choose t• so that t• - r ~ t 0 , t• - u ~ t 0 and ft~ q( s )ds :::; ~. Define

n = {x E BC:

2( 1 + p) t > t } 0 for all t

~ t0 -

x(t) = p(t)x(t- r) +

l

t + (1- -) > 0.

to

m. Set x(t)

oo

= y(t)e-ort. Then

n

q(s)

t

II xor (s- u,)ds, 1

t

~to.

i=l

Consequently, n

(x(t)- p(t)x(t- r))' + q(t)

IJ xor (t- u;) = 0, 1

t ~ t0

i=l

I.e. x(t) is a positive solution of Eq. (3.3.1) and x(t) approaches zero exponentially. D

143


Remark 3.3.1. When p(t) (3.3.23) becomes

= p,

pe"'r

q(t)

+ fexp a

= q are both positive constants, condition

(at a;a;) : ;

(3.3.24)

1

i=l

for some a > 0. From Theorems 3.3.1 and 3.3.2 condition (3.3.24) provides a sufficient and necessary condition for the existence of a nonoscillatory solution of Eq. (3.3.1). Example 3.3.2. Consider the equation (3.3.25) where q(t) = Zettl•(t-l)11 7•(t-z)t/a , t ~ 3. It is not difficult to see that all assumptions of Theorem 3.3.21 are satisfied. Therefore, Eq. (3.3.25) has a nonoscillatory solution which tends to zero as t - t oo. In fact, x(t) = te- 1 is such a solution. Corollary 3.3.3.

Assume that there exist positive constants p and q such that (3.3.26)

and that the linear equation

(x(t)- px(t-

r))' + qx(t-

taw;)=

0

(3.3.27)

has a nonoscillatory solution. Then Eq. (3.3.1) also has a nonoscillatory solution. In fact, from Theorem 3.1.1, (3.3.27) having a nonoscillatory solution implies that the characteristic equation of (3.3.27)

). - >.pe-).r

+ q exp ( - >.

t a;a;)

= 0

(3.3.28)

•=1

has a negative real root >.. Let a = ->. > 0, then (3.3.24) is satisfied. The conclusion of the corollary follows from Theorem 3.3.2.

Chapter 9

144

The following result is for Eq. (3.3.1) with oscillating coefficients p(t) and

q(t). Theorem 3.3.3. Assume that p E C 1 ([t 0 ,oo),R), q E C([t 0 ,oo),R) and there exists a positive number p. such that (3.3.29)

Then Eq. (3.3.1) has a positive solution fort;::: t 0 • Proof: Set

t;::: to

p(t) PI(t) = { t-t~±r p(to),

0,

t0

p'(t),

P2(t) = {

t-t~±r

to-T$ t $to -

m-

T

$ t :5 t 0

-

r,

t ;::: t 0

p'(to), to - T :5 t $to

0,

to -

q(t),

t 2: to

t-t~±r q(to),

to-T :5 t $to

0,

t 0 - m- T $ t $to-T.

m -

T $ t $ to - r,

and

Pa(t)

Then p;, i

={

=1,2,3, are continuous on [to-m- r,oo). From (3.3.29) we have P.IPI(t)le"'r

+ IP2(t)le"'r + IPa(t)l exp(p.

t a;u;)

$ p.,

•=1

t 2: to-m- T.

(3.3.30)

We introduce the Banach space X of all bounded continuous functions x: [to-m- r,oo)-+ R with the norm llxll = sup jx(t)je-11t where 71 > 0 t;:::to-m-T


145

satisfies the following inequality IPl(t)le"r(e-"r + ~) + e"r IP2(t)l + .!IP3(t)lexp(p. ~

~

~

t

for

ta;u;) ~1

:5

~to-r.

~ (3.3.31)

We consider the subset 0 of X as follows

n ={.X EX:

I.X(t)l :5 p.,

t

~to-m-

r}.

Clearly, f1 is a bounded, closed and convex subset of X. Define an operator Ton

n as follows:

.X(t- r)p1(t) exp(J,'-r .X(s)ds) + P2(t) exp(ft'-r .X(s)ds)

+ P3(t) exp(:E a; ft-tr· .X(s)ds), i=l • t

n

(T.X)(t) =

to - m - r :5 t :5 t 0

0,

t ~to-r

r.

-

In view of (3.2.41), we have 1(7-X)(t)l :5 P.IPI(t)le"r

+ IP2(t)le"r

+ IP3(t)lexp(p.

t a;u;)

:5 p. for

which shows that T maps 0 into itself. Next, we show that T is a contraction on and t ~to-r

t

n. In fact,

~to-r for any AI' A2 E

n

I(T.XI)(t)- (7.X2)(t)1 :5IPI(t)II.XI(t- r)

exp(l~r .X1(s)ds)- .X2(t- r) exp(l~r .X2(s)ds) I

+ IP2(t)ll exp(l~r .X1(s)ds) -

exp(l~r .X2(s)ds) I

+ IP3(t)llexp( ~a; l~tr; .X1(s)ds) -

exp(

t

a;

l~tr; .X2(s)ds) I

Chapter 3

146

::::; IPt(t){e"rl>.t(t- r)- >.2(t- r)l

+ J.Le"r l~r 1>-t(s)- >.2(s)lds)

+ e"riP2(t)lj~r 1>-t(s)- >.2(s)lds + IP3(t)l exp(J-1. ~ a;u;) ~a; 1~"' ::::; { e"riPt(t)le'IT [e-'IT

+ ~(1- e-'lr)] + ~e"riP2(t)l(l- e-'IT)e'IT

+ ~IP3(t)l exp(J-1. ~a;u;) ::::; e"t{ IPt(t)le"r ( e-"r

1>-t(s)- >.2(s)lds

~a;(l- e-""')e"1}11>.1- >.2ll

+ ~) + e~r IP2(t)l

+ ~IP3(t)1 exp(J-1. ta;u;)}ll>-t- >.211 TJ

•=1

Hence

IIT>.t- T>.2ll =

sup

I(T>.t)(t)- (T>.2)(t)le-" 1

t. E Q such that T >. = >.. That is,

>.(t- r)pl(t)exp( fLr >.(s)ds) n

+ P2(s)exp(fLr >.(s)ds)

t

+ P3(t)exp(L: a; ft-cr· >.(s)ds), t::?: to-r, i=l

>.(t) =

I

0,

to - m - r ::::; t ::::; t 0

-

r.

147

Oscillation of First Order Neutral Equations According to the definition of p;, we have

>.(t)=>.(t-r)p(t)exp(i~r >.(s)ds) +p(t)exp(i~r >.(s)ds) + q(t)exp (~a; l~u,. >.(s

)d.s),

t

~to.

Set x(t) = exp(- f 110 >.(s)ds). Then we can find that x(t) is a positive D solution of Eq. (3.3.1) on [to, oo ). Example 3.3.3. Consider 2sint )' ( x(t)---x(t-1) t-1

2cost + 1-1-t x(t-1)=0.

(3.3.32)

It is easy to see that (3.3.29) with p. = 1 is satisfied for t ~ T, where T is a sufficiently large number. Then Eq. (3.3.32) has a positive solution on [T, oo). In fact, x(t) = t is such a solution of Eq. (3.3.32).

3.3.6. We now consider the neutral delay differential equations with several delays of the form n

(x(t)- px(t- r))' + Lq;(t)x(t- a;)= 0,

t ~ t0

(3.3.33)

i=l

where p E [0, 1], q; (i = 1,2, ... ,n) E C([t 0 ,oo), R+) and r;,a; (i = 1, 2, ... ,n) E (0, oo ). Denote m =max{ r, a1, ... , an}· Lemma 3.3.3.

Assume that liminf J.11_ cr,. q1 (s)ds > 0 and t-+oo qi(t-r)$qj(t),

t~t 0 +m,

j=1,2, ... ,n.

(3.3.34)

Let x(t) be an eventually positive solution of (3.3.33) and set z(t)

= x(t)- px(t- r).

(3.3.35)

Chapter 3

148

Then eventually z(t) > 0,

z'(t) < 0, and n

z'(t)- pz'(t- r)

+ Lq;(t)z(t- u;)

~ 0.

(3.3.36)

i=l

Proof: As in the proof of Theorem 3.2.1, it is easy to see that z(t) > 0, z'(t)

0, and = r, 0"}, 'O"n

e

li~~f { pe~'r

0.

0

n + f.p.1 ~ e~'"'

it+l t

q;( s )ds

}>

1.

(3.3.37)

Then every solution of Eq. (3.3.33) is oscillatory. Proof: Assume (3.3.33) has an eventually positive solution x(t). By Lemma 3.3.3 there exists aT;::: to such that z(t) > 0, z'(t) < 0, and (3.3.36) holds fort ;::: T. Let w(t) =t;::: T. Then w(t) > 0, t;::: T, and (3.3.36) becomes

Z:N?,

w(t);::: pw(t-

r)exp(i~r w(s)ds) + tq;(t)exp(l~.,, w(s)ds)

(3.3.38)

fort;::: r + m. We now define a sequence of functions {w k( t)} for k = 1, 2, . . . and t ;::: T, and a sequence of numbers {p.k} fork= 1, 2, ... , as follows:

Oscillation of First Order Neutral Equations and for k = 1, 2, ... , t

~ T

149

+ km

= pwk(t- T) exp(l~T WA:(s)ds)

WH 1 (t)

+ tq;(t) exp(l~.r; Wk(s)ds) and p. 1

= 0, and fork=

Jlk+l = inf

t~T

(3.3.39)

1,2, ...

{ PJlkellkT min t=r,trt,···•trn

n lt+t } + £1 :~:::el'ktrk q;(s)ds .

t

i=l

(3.3.40)

We claim that

i) 0 = Jll < Jl2 < ... , ii) WA:(t) ~ w(t) fort~ T + (k -1)m and k = 1,2, ... , iii) J,t+t Wk(s )ds ~ Jlk fort~ T + (k + 1)m, l = 1, 2, ... , and l = T,aJ, ... ,an. In fact, i) and ii) follow from (3.3.39) and (3.3.40) by induction. Clearly iii) is true for k = 1. Assume iii) is true for some k. Then (3.3.39) and (3.3.40) imply that fort~ T + km, l = T,a1, ... ,an

t

f1

(1s-T Wk(8)d8 ds n lt+t (18 ) + 18 q;(s) exp s-tr; wk(8)d8 ds

lt+t t Wk+J(s)ds = f1 lt+t t Wk(S- T) exp 1

8

)

t

Hence iii) holds. Let p.* = lim Jln· From (3.3.37) and (3.3.40) there exists an a > 1 such n-+oo that JlA:+I ~ ap.k, k = 1, 2, ... , which implies that p.* = oo. In view of ii) and iii) lim J,t+trt w(s)ds = oo. From the definition of w, we t-+oo have

z

(

z(t)

t + al

) = exp

lt+trt t

w(s)ds.

Chapter 9

150

Thus

r

z(t)

t2-~z(t

+ u1)

(3.3.41)

= oo.

From (3.3.33) n

z'(t) =- Eq;(t)x(t- u;) $ -q1(t)x(t- u 1 ) i=l

By Lemma 2.1.3, z(~(t)t) is bounded above which contradicts (3.3.41).

Assume there exists a p* > 0 and a T

Theorem 3.3.5. l=T,u, ... ,un

~

0

to such that for

(3.3.42)

Then (3.3.33) has a positive solution on [T + m, oo ). Proof: First we claim that the integral equation

l

t

n

lt

v( t) = pv( t - T)exp t-r v( s )ds + ~ q;( s )exp ,_,7 , v( s )ds has a positive solution on [T + m, oo ). To this end set v 1 ( t) k = 1,2, ...

= 0,

(3.3.43)

t

~

T, and for

pvk(t- T)exp(JLr Vk(s)ds) Vk+t(t) =

{

+ E:=I q;(t)exp(f/-.,., vk(s)ds), Pk+t(t)

t~T+m

(3.3.44)

T$t n

1.

(3.3.48)

Corollary 3.3.5. If there exist i E In = {1, ... , n} and .e > 0 such that liminft-oo q;(s)ds > 0 and T = m 0 l, u; = m;l for some integers m;, i = 0, 1, ... , n. Furthermore

J,'_.,.,

q;(t)

= g;(t) + h;(t),

i

i

= 1, 2, ... , n,

where g;( t) are l-periodic functions with ftt+l g;( s )ds = gj, h;{ t) are non decreasing with lim h;(t) = hi, i = 1, 2, ... , n. Then (3.3.33) is oscillatory if and t-oo

153

Oscillation of First Order Neutral Equations only if for all p.

>0

For the case that q;(t) becomes

=0

and

h;(t)

= hi,

= 1, 2, ... , n,

(3.3.49)

(3.3.50)

this implies that Eq. (3.3.33) with constants parameters is oscillatory if and only if its characteristic equation has no real roots. If h;(t) = 0, i = 1, 2, ... , n. Eq. (3.3.33) becomes n

(x(t)- px(t- r))'

+ Lg;(t)x(t- a;)= 0

(3.3.51)

i=l

where g;(t) are £-periodic functions. Then (3.3.51) is oscillatory if and only if its "characteristic equation" n

A(1- pe->.r)

+ Lgie->. = 0

(3.3.52)

17 ;

i=l

t J/+l

has no real roots, where gi = g;(s )ds, i = 1, 2, ... , n. By employing the above technique in Eq. (2.7.1) the condition (2.7.3) in Theorem 2.7.1 can be improved to assuming that for all A> 0 and j = 1, 2, ... , n n 1 liminf { - - " ' " ' t.-oo

Corollary 3.3.6.

ATj(t)

£;:

ft+r;(t) t

p;(s)e>.r;(s)ds

If r; ( t) are non decreasing for i n

ft+r;(t)

liminf L t-oo

Then Eq. (2. 7.1) is oscillatory.

i=l

t

}

> 1.

(3.3.53)

= 1, 2, ... , n, and

1 p;(s)ds > - . e

(3.3.54)

Chapter 9

154

From Theorem 3.3.4 we derive some explicit conditions for oscillation of Eq. (3.3.33). Theorem 3.3.6. Assume that the assumptions of Lemma 3.3.3 bold. Furthermore, for f.= r, O'J, ••• ,un (3.3.55)


> 0, and then Eq. (3.3.33)

Proof: We shall show that (3.3.37) is true for all p. is oscillatory by Theorem 3.3.4. In fact

q;(s)ds )e "''(1- pe"r)- 1 f -1 L (1jt+l n

1

/1- i=I

t

1 =-

LL n

00

(

f1

jt+l q;(s)ds)

/1- i=l k=O

pkep(kr+u;)

t

Thus (3.3.55) implies that 1 liminft--.oo p.

-f. q;(s)ds )e~-' 17'(1- pe"r)- 1 > 1. L (1jt+l n

i=l

Hence (3.3.37) holds for all p.

t

> 0 and f.

= r, u 1 , ... , u n. The proof is complete. 0

For the case p Theorem 3.3. 7. i) p ~ 1, ii)

T

>

O'j,

~

1, we can prove the similar results using above techniques.

Assume that

q;(t) :5 q;(t- r), fori= 1, 2, ... , nand all large T; t+ r-maxc:r; n i

= 1,2, ... ,n, liminfJ 1 t-<X>

2

L: q;(s)ds > 0;

i=l

155


= r,T- u;, and i = 1,2, ... ,n

l

iii) for all fl.> 0,

it+l

1 1 n liminf { -epr + -l::>P(r-a;) p

t--+oo

lpfl i=l

q;(s)ds

}>

1.

(3.3.56)

t

Then every solution of (3.3.33) is oscillatory. Theorem 3.3.8. i) p

~

1,

Assume that n

liminf ).*

> 0 and T

1 •r sup { -e>. t>T

-

q;(t) > 0,

T

> u;,

= 1, 2, ... , n;

i

i=l

t-oo

ii) there exist

l:::

P

~to,

for f.=

T, T -

n • 1 l:e>. + --. (r-a;)

fp).

u;,

it+l

i = 1, 2, ... , n

q;(s)ds

}:s;

1.

(3.3.57)

t

.

•=1

Then Eq. (3.3.33) bas a positive solution on [T, oo ). Corollary 3.3.4. Assume that there exist l > 0 and positive integers m 0 ,m 1 , ••• ,mn such that T =mol, u; = m;l and mo > m;, i = 1,2, ... ,n, and q;(t) are £-periodic functions. Then every solution of (3.3.33) is oscillatory if and only if its "characteristic equation"

>.(1- pe->.r) +

n

L qie->.a; = 0

(3.3.58)

i=l

bas no real roots, where

i

Remark 3.3.2.

= 1,2, ... ,n.

The following condition is sufficient for (3.3.56) to hold: for

f=T,T-O"j, i=1,2, ... ,n 00

11t+l

lie~fLL i n

i=l k=O

(

t

q;(s)ds

) p

1

k+ 1

1

((k+1)r-u;)>;·

(3.3.59)

156

Chapter 9

3.4. Comparison Results We consider neutral delay differential equations of the form n

(x(t)- px(t- r))'

+L

q;(t)x(t- u;(t))

= 0,

t;::: to.

(3.4.1)

i=l

In the following we present a necessary and sufficient condition for oscillation of all solutions of Eq. (3.4.1) by a comparison method.

Theorem 3.4.1.

Assume that

(i) p, -r are constants, 0 ~ p < 1, -r > 0; (ii) u;EC(R+,R+), lim(t-u;(t))=oo, 1-+00

i=1, ... ,n;

(iii) q; E C(~, R+), i = 1, ... , n. Then every solution of Eq. (3.4.1) is oscillatory if and only if the differential inequality n

(x(t)- px(t- r))'

+L

q;(t)x(t- u;(t)) ~ 0

(3.4.2)

i=l

has no eventually positive solutions. Let us now compare Eq. (3.4.1) with the equation n

(x(t)- px(t- -r))'

+ Lq;(t)x(t- u;(t))

= 0,

t;::: t 0 •

(3.4.3)

i=l

Theorem 3.4.2. Assume that all the assumptions of Theorem 3.4.1 hold. Further assume that 0 ~ p:::; p < 1, Mt);::: q;(t);::: 0, i = 1, ... , n. Then the oscillation of Eq. (3.4.1) implies the oscillation of Eq. (3.4.3). Since we will establish more general results in Chapter 5, we omit the proofs here. We now consider a more general neutral differential equation n

(x(t)-px(t--r))' +Lq;(t)x(t-u;(t))+F(t,x(gt(t)), ... ,x(gm(t))) = 0. (3.4.4) i=l

157


Theorem 3.4.3. In addition to all assumptions of Theorem 3.4.1 we assume g; E C(~,R), FE C(R+ x Rm,R) and F(t,Yl>····Ym)Yt ;:: 0 whenever y 1 y; > 0, j = 1, 2, ... , m. Then the oscillation of Eq. (3.4.1) implies the oscillation of Eq. (3.4.4). Proof: Hnot, assume Eq. (3.4.1) is oscillatory, and Eq. (3.4.4) has an eventually positive solution x(t). Then x(t) satisfies the differential inequality n

(x(t)- p:c(t- r))'

+L

q;(t)x(t- u;(t)) $ 0.

(3.4.5)

i=l

By Theorem 3.4.1, Eq. (3.4.1) has a nonoscillatory solution, which contradicts 0 the assumption. For example, we consider the mixed neutral differential equation m

n

(x(t)- px(t- r))' + L q;x(t- u;) + L r;x(t + 6;) = 0,

(3.4.6)

j=l

i=l

where 0 < p < 1, r, q;, u; are positive constants, r;,C; are nonnegative constants, j = 1, 2, ... , m. According to Theorem 3.4.3 we have the following conclusion. If every solution of n

(x(t)- px(t- r))' + L q;x(t- u;) = 0

(3.4.7)

i=l

is oscillatory, then every solution of Eq. (3.4.6) is also oscillatory. Using Theorem 3.4.1 we obtain the following "Linearized oscillation" result for the nonlinear equation n

(x(t)- px(t- r))' + Lqd;(x(t -u;)) = 0. i=l

Theorem 3.4.4. Assume (i) p E (0, 1), r, q; E (O,oo), u; E (O,oo), i = 1, ... ,n; (ii) /;EC(R,R), x/;(x)>Ofor:cf:O, i=1, ... ,n;

(3.4.8)

Chapter 9

158

(iii) there exists an e > 0 such that f;~u) 2: 1 for lui $ e, i = 1, ... , n. Then every solution of (3.4.8) is oscillatory if every solution of the "linearized equation" (3.4. 7) is oscillatory. Proof: Let x(t) be a positive solution of (3.4.8). Then we have lim x(t) = 0, t-oo

and n

(x(t)- px(t- r))'

+L

q;x(t- cr;) $ 0.

i=l

By Theorem 3.4.1, (3.4.7) has a nonoscillatory solution. This contradicts our assumption. Remark 3.4.2.

Theorem 3.4.4 is true for (3.4.8) with variable coefficients n

q;(t) E C(R+,R+), i

= 1, ... ,n, satisfying j 10 I: q;(t)dt = oo. 00

i=l

We then compare the oscillation of two neutral delay differential equations

(x(t)- p(t)x(t- r))'

+ q(t)x(t- cr) = 0

(3.4.9)

(x(t)- p(t)x(t- r))'

+ q(t)x(t- cr) = 0.

(3.4.10)

and

Theorem 3.4.5. 0; p(t) ~ 1, q(t)

> 0,

Assume that p, p, q, q E C([t 0 ,oo),R+), f 1";q(t)dt = oo; and

r > cr 2:

Then (3.4.9) is oscillatory implies that (3.4.10) is oscillatory. Proof: If not, let x(t) be an eventually positive solution of (3.4.10). Set z(t) x(t)- p(t)x(t- r). Then z(t) < 0, z'(t) < 0 eventually, and

'( ) Z

q(t)

1

q(t)

t = p(t +r-crq ) (t +r ) Z ( t + T) + pt+r-cr ( )z( t + T

-

CT ).

=

159

Oscillation of First Order Neutral Equations Set .X(t)

= :(w .Then .X(t) > 0 fort;:: T .X(t) =

p( t + T

+ ;::

(

-(t) q ( /(t + r) exp U )q t + T

-

(t) q ) (

pt+r-sqt+r (t)

it+T .X(s)ds t

1t+T-I1 .X( s )ds it+T .X(s )ds ) .X( t + r) exp

q( t) exp p(t+r-u)

+ pt+r-u ( q ) Define .X 0 (t) = .X(t),

and

t

t

exp

it+T-11 .X(s)ds.

(3.4.11)

t

t;:: T, and for n = 0, 1, 2, ... and t;:: T

An+l(t) = p(

q(t)

) ( /n(t t+r-uqt+r

q(t)

+ pt+r-u ( ) exp

+ r) exp

it+T An(s)ds t

(3.4.12)

it+T-11 An(s)ds. t

Clearly, 0 $ An(t) $ An-l(t) $ · • · $ Ao(t) Hence lim An(t) n-oo

= .X(t),

= .X*(t) exists and .X*(t) ;:: 0, t ;:: T.

t;:: T.

By the Lebesgue's domi-

nated convergence theorem to (3.4.12) we obtain that

.X*(t) =

it+T .X*(s)ds

q(t)

(t ) ( ).X*(t + r) exp p +r-uqt+r . q(t) + p(t+r-u)

exp

t

it+T-11 .X*(s)ds. t

(3.4.13)

Set

y(t)

= -exp

£

.X*(s)ds < 0.

(3.4.14)

Then y'(t) = .X*(t)y(t), and (3.4.13) becomes

'( ) _ q(t) 1 q(t) y t - p(t+r-u)q(t+r)y(t+r)+ p(t+r-u)y(t+r-u).

(3.4.15)

ChapterS

160 Let x(t) =- ~g::?

> 0. Thus (3.4.15) reduces to

(x(t)- p(t)x(t- r))' This means that Eq. assumption.

+ q(t)x(t -u) = 0.

(3.4.15) has a positive solution x(t), contradicting the 0

3.5. Unstable Type Equations We consider neutral differential equations of the forms

(x(t)- px(t- r))'- q(t)x(g(t)) = 0,

t ~to

(3.5.1)

and

(x(t)- px(t- r))'- q(t)x(g(t)) = F(t), where p ~ 0, r lim g(t) = oo.

> 0, q E C([to,oo),%),

t ~ t0 •

(3.5.2)

g, FE C([to,oo),R), g(t)::; t and

t-+oo

We first show some properties of nonoscillatory solutions of Eq. (3.5.1). Lemma 3.5.1. Let x(t) be a positive solution of (3.5.1). Then x(t) is of one of the following types of asymptotic behavior: (a) lim x(t) = 0, t-+oo

(b) lim x(t) = t-+oo

ei

(c) lim x(t) =

0,

t-+oo

00.

To prove this lemma we shall use the following facts: Lemma 3.5.2.

Let x, z E C([t,oo),R) satisfy

z(t) = x(t)- px(t- r),

t

~

t0

+ max{O, r },

where p, r E R. Assume that X is bounded on [to,oo) and lim z(t) = t-+oo Then the following statements hold: (i) Ifp = 1, then£= 0; (ii) Ifp ;f ±1, then lim x(t) exists. t-+oc

e exists.

161

Oscillation of First Order Neutral Equations The next lemma is for the first order linear difference equation x(t + 1) = p(t)x(t) + y(t),

(3.5.3)

x(to) = xo

where to and t are integer-valued, p andy are given functions with p(t) :/= 0 for all t. The corresponding homogeneous equation to (3.5.3) is

z(t + 1) = p(t)z(t),

Lemma 3.5.3. Let p(t) solution of Eq. (3.5.3J is

:f. 0 and y(t)

(3.5.3)'

z(to) = zo.

be given fort= t 0 , t 0 + 1, .... Then the

t-1

z(t) = z(to)

IT p(s),

t =to+ 1, to+ 2, ...

•=to to-1

where

IT

p(s)

= 1; and the solution of Eq. (3.5.3) is

•=to

(3.5.4)

to-1

where

E

p(s)

= 1.

•=to

Proof of Lemma 3.5.1: Set z(t) = x(t)- px(t- T), then z'(t) ~ 0. First we assume that z(t) > 0 eventually. Then lim z(t) = l t-+oo

~

co. If

l =co, then x(t) ~ z(t)-+ co, as t-+ co, i.e., (c) holds. If l < co and x(t) is bounded, Lemma 3.5.2 implies that lim x(t) exists t-+oo

when p :f. 1, i.e., either (a) or (b) holds. When p = 1, x(t) ~ x(t- T) + ~ ~ · · · ~ x(t- nT) + n · ~ -+co as n-+ co. This is impossible since x(t) is bounded. H l 0 fort 2:: t 0 • Set x(t) = y(t)v(t). Then

r' q(s)x(g(s))ds + 1 , t;::: T. x(t)- px(t- r) = lr 2

(3.5.11)

Therefore, x(t) is a positive solution of (3.5.1) fort;::: T. This proves (i), (ii) and the first part of (iv). In the case that p > 1 we have

x(t)

~

px(t- r) 2:: · · · ~ pnx(t- nr),

or

x(t) ~ x(to)exp (JL(t- to)),

for

t ~to,

¥

where Jl = > 0, which shows that (iii) is true. In order to prove the second part of (iv), we let x(t) be a bounded positive solution of Eq. (3.5.1) and define z(t) = x(t)- px(t- r). Then lim z(t) = t-eo

e

165

Oscillation of First Order Neutral Equations exists. If£> 0, then x(g(t)) ~£fort~ T1 ~ t 0 • Hence

z(t)- z(T1 ) = [' q(s)x(g(s))ds

lr,

~£

[' q(s)ds.

lr,

J;,

q( s )ds ---+ oo as t ---+ oo, we have a contradiction. In view of (3.5.11) Since we assume that p ::j; 0. Now for p E (0, 1) it is impossible that £ < 0, thus 0 lim z(t) = 0 and hence lim x(t) = 0. This completes the proof. t-oo

t-.oo

Theorem 3.5.2. Assume that p E [0, 1), positive integer k such that

u'(t)

+ q(t) L"

1

i+l

u(g(t)

ft'; q(t)dt =

oo, and there exists a

+ (i + 1)r) ~ 0

(3.5.12)

i=O p

has no eventually negative solutions. Then every bounded solution of (3.5.1) is oscillatory. Proof: From the proof of Theorem 3.5.1, we see that if x(t) is a bounded positive solution of (3.5.1), then z(t) < 0 eventually. Also by induction we see that 1 " x(t-r)>-2: i+lz(t+ir). i=O p

Substituting it into (3.5.1) we obtain

z'(t) ~ -q(t)

L"

1

i+l

z[g(t)

+ (i + 1)r]

i=O p

which contradicts the fact that (3.5.2) has no eventually negative solutions. Example 3.5.1. Consider

(x(t)- ~x(t- 1))' = q(t)x(t- 10)

(3.5.13)

where q E C([to, oo ), R+ ). If the delay differential equation

z: 8

y'(t) + q(t)

i=O

2i+ 1 y(t- (9- i)) =

o

Chapter 9

166

is oscillatory, then every bounded solution of (3.5.13) is oscillatory. In fact, we may choose k = 8. Then all assumptions of Theorem 3.5.2 hold. Theorem 3.5.2. Assume that p < 0, p =f:. -1, and every bounded solution of (3.5.1) is oscillatory.

f

1";

q(t)dt = oo. Then

Proof: Let x(t) > 0 be a bounded solution. Set z(t) = x(t)- px(t- r), then z'(t) ~ 0. Thus lim z(t) =f. exists and f.> 0. From (3.5.1) we have t->oo

1

00

q(t)x(g(t))dt

< oo.

to

In view of the assumption we have that liminf x(g(t)) sequence {tn} such that lim tn n-oo

z(g(tn))- z(g(tn)-

= oo and

t-oo

= 0. Then there exists a

lim x(g(tn)- 2T)

n~oo

= 0. Noting that

r)

= x(g(tn))- px(g(tn)- r)- (x(g(tn)- r)- px(g(tn)- 2-r)),

and letting n --. oo we have liminf (x(g(tn))- (p + 1)x(g(tn)- r)) n-oo

If p

= 0.

< -1, then liminf x(g(tn)) n~oo

=0

and

liminf x(g(tn)- r) = 0, n~oo

and so liminf z(tn) = lim z(t) = 0, contradicting that f.> 0. t-+oo

n-+oo

If -1 0 again.

Example 3.5.2. Consider (x(t)

+ ~x(t- 27r))' =

~x(t- ~11").

(3.5.14)


167

It is easy to see that the conditions in Theorem 3.5.2 are satisfied. Therefore,

every bounded solution of (3.5.14) is oscillatory. In fact x(t) = sint is such a solution. For the case that p = -1, we need the following lemma. Lemma 3.5.1. Let p < 0. Assume that x E C([to, oo), R+), x(t)- px(t- r) is increasing, and x(t) - px(t - r) ~ f. > 0, for t ~ T ~ to. Then for every t 1 E [T, oo) the set

E = {t I t1 $ t $ t1 satisfies mes (E) ~ r, where ,8 =min measure.

+ 2r, {f,

-px(t- r)

~

,8}

-~}, and mes (E) denotes Lebesgue

In Chapter 4, we will prove a more general result, so we omit the proof here.

JE

Theorem 3.5.4. Let p = -1. Assume q(t)dt = oo for every closed subset E of [to, oo) whose intersection with every interval of the form [t, t + 2r], to $ t < oo, has a measure not Jess than T. Then every bounded solution of (3.5.1) is oscillatory. Proof: Let x(t) be a bounded positive solution of (3.5.1). As shown before,

1

00

q(t)x(g(t))dt < oo.

to

Obviously, the conditions of Lemma 3.5.1 are satisfied. The intersection of the set E ={tIT$ t < oo, x(t- r) ~ f} with the interval [s,s + 2r], s E [T,oo), has a measure not less than r. Therefore,

["" q(t)x(g(t))dt

jT

~

r q(t)x(g(t))dt ~ ~2 jEr q(t)dt =

jE

This contradiction proves the theorem.

00.

D

Now we briefly discuss an oscillatory property of the forced equation (3.5.2).

Chapter 9

168 Theorem 3.5.5. and

Assume that there exists a function

liminf f(t) = -oo t-oo

f such that F(t)

and lim sup f(t) t--+oo

= oo.

= f'(t),

(3.5.15)

Then every bounded solution of (3.5.2) is oscillatory. Proof: Let x(t) be an eventually positive and bounded solution of (3.5.2) and set z(t) = x(t)- px(t- r). Then

(z(t)- f(t))'

= q(t)x(g(t))

~ 0.

If z(t)- f(t) ~ 0 eventually, then z(t) ~ f(t) eventually, which contradicts (3.5.15) and the boundedness of z. Hence z(t)- f(t) > 0 eventually, which is impossible since z(t) is bounded. 0 Example 3.5.2. The equation

(x(t) + x(t -11")) 1 - tx(t- 211") = -tsin t

(3.5.16)

satisfies the assumptions of Theorem 3.5.5. Hence every bounded solution of (3.5.16) is oscillatory. In fact, x(t) = sint is such a solution.

3.6. Sublinear Equations Consider the nonlinear neutral differential equations of the form

(x( t) + p(t)x( r(t))) 1 + f(t, x(gi (t)), ... , x(gN( t)))

= 0.

(3.6.1)

Definition 3.6.1. f(t, Yb ... , YN) is said to be strongly sublinear if there exist constants (3 E (0, 1) and m > 0 such that izi-Pif(t, z, ... , z)i is nonincreasing in

lzl for 0 < lzl

~

m.

The following notations will be employed:

= 19~N max g;(t) R(g.,r) = {t E [t 0 ,oo): g*(t)

to S g*(t) < r(t)}.

169


Theorem 3.6.1.

Assume that

(i) p E C([t0 ,oo),(O,oo)),

1 < p.

~

p(t)

~ 11

where p. and

11

are constants;

(ii) r E C([t 0 ,oo),R) is strictly increasing, and satisfies that r(t) limt_. 00 r(t) = oo; (iii) g; E C([to,oo),R), limt- 00 g;(t) = oo, (iv)

(v)

< t, and

i = 1,2, ... ,N,

f

E C([t 0 ,oo) x RN,R) is nondecreasing in each y; and strongly sublinear, andy;J(tt,yt, ... ,yN)?.O, tOforyty;>O, 1~i~N;

fn(g•,r) lf(t, a, ... , a)idt = oo for every constant

a=/= 0.

(3.6.2)

Then every solution of Eq. (3.6.1) is oscillatory. Proof: Assume the contrary and let x( t) be a nonoscillatory solution of Eq. (3.6.1 ). Without loss of generality we may assume that x(t) is eventually positive. Set

z(t) = x(t) + p(t)x(r(t)). Then z(t) > 0 and z'(t)

~

(3.6.3)

0 for all large t. From (3.6.3) and using (i), we have

(3.6.4) for all large t, where r- 1 is the inverse function of rand r- 2 (t) = r- 1 (r- 1 (t)). Setting y(t) = ~z(t) and noting that f is sublinear, from (3.6.1) and (3.6.4) we have

(3.6.5) provided T 2:: to is sufficiently large. From (3.6.2) and (3.6.5), it is easy to show that ~~~z(t) = 0. Thus there exists T1 ;::: T such that y(r- 1 (g*(t))) ~ m for

Chapter 9

170

t

~

T1 • In view of (iv), we have

f (t, y( ,.-I (gi (t))), ... , y( ,.-I (gN( t)))) ~ f(t,y(T-I(g*(t))), ... ,y(T-l(g*(t))))

~ m-.8(y(T- 1 (g*(t)))).B f(t,m, ... ,m),

t ~ T1.

{3.6.6)

From (3.6.5) and (3.6.6) we have

(3.6.7) Dividing {3.6.7) by (y(t)).B and integrating it over R(g*,,.) n [T1 ,oo), we have

1

R(g• ,r)n{T1 ,oo)

f(s,m, ... ,m)ds~

pvm.B(y(TI)) 1 -P ( _ 1)( 1 -.8) 0 for t ~ T, 1 ~ i ~ N. Then z(t) defined by (3.6.3) is bounded and nonincreasing. Integrating (3.6.1) we have

£""

f(t, c, ... ,c)dt

~ z(T)- z(oo) < z(T) < oo

which contradicts (3.6.2). Sufficiency. Let c > 0 and T > to be such that pc ~ b and To=min {T(T), inf {g 1 (t)}, ... ,inf {gN(t)}} ~to l~T

l~T


171

and

Loo f(t,p.c, ... ,p.c)dt ~ (p. -1)c.

(3.6.9)

Define a set XC C[To,oo) as follows:

e C[T0 ,oo):

c ~ x(t) ~ J-LC fort 2: T, x(t) = x(T) for To~ t ~ T}. (3.6.10) For every x E X we define

X= {x

t 2: T (3.6.11)

H0 (t)

= 1,

i-1

H;(t) = flp(ri(t)),

i = 1,2, ....

(3.6.12)

j=O

Then 0 < x(t)

~

t 2: T. Define a mapping Ton X as follows:

p.c,

c+ ]', f(s,x(gi(s)), ... ,x(gN(s)))ds, t;::: T 00

(Tx)(t) = {

Tx(t),

To~

(3.6.13)

t

~

T.

In view of (3.6.9), T X C X. It is not difficult to prove that T is continuous and T(X) is relatively compact in the topology of the Frechet space C[T0 ,oo). Therefore, by the Schauder Tychonoff fixed point theorem, there exists an x E X such that x = T x. That is

(3.6.14) From (3.6.11) we have

x(t) + p(t)x(r(t)) = x(t), and thus

x(t)+p(t)x (r(t))=c+

1

00

f(s,x(gi(s)) , ... ,x(gN(s)))ds,

t?_T. (3.6.15)

Chapter 3

172

Differentiating (3.6.15) we have that the function x(t) defined by (3.6.11) satisfies Eq. (3.6.1) fort;::: T and (3.6.16) Therefore, x(t) is a bounded nonoscillatory solution of Eq. bounded away from zero.

(3.6.1) which is 0

Combining Theorems 3.6.1 and 3.6.2, we obtain the following result. Corollary 3.6.1. Let g•(t) < -r(t) and (i)-(iv) in Theorem 3.6.1 bold. Tben, (3.6.2) is a necessary and sufficient condition for oscillation of all solutions of Eq. (3.6.1).

Example 3.6.1. Consider the neutral equation

(x(t) where Jl

+ JlX(t- 211'))' + (1 + Jl) cosi txl (t- ~ 11') =

0

(3.6.17)

> 1 is a constant.

It is easy to see that all the assumptions of Theorem 3.6.1 are satisfied. Therefore every solution of (3.6.17) oscillates. In fact, x(t) = sint is such a solution. The following example shows that condition (3.6.2) is not sufficient for oscillation of linear equations.

Example 3.6.2. Consider the linear equation

(x(t)

2 + 2x(t- 1')) + 3ft(t- _4f2)+2 4 x(t) = I

0.

(3.6.18)

It is easy to see that all assumptions of Theorem 3.6.1 are satisfied. But (3.6.18) has a positive solution x(t)

=t .

We now consider the nonlinear neutral differential equation described by n

(x(t)- px(t- -r))' + Lqi(t)/;(x(gi(t))) = 0 i=l

(3.6.19)

173


where p;:::: 0, r > 0, q; E C([to,oo),~), g; E C([to,oo),R), oo, !; E C(R,R), xf;(x) > 0 for x =J 0, i = 1,2, ... ,n. f is said to be locally sublinear near x = 0 if there exists a > 0 such that f(x) is increasing in (O,a) and (-a,O) and

r

dx Jo f(x) < oo and

0 /_ -at

dx f(x) < oo.

(3.6.20)

Theorem 3.6.3. Assume that p E [0, 1), there exists i 0 E In= {1, ... , n} such that j; 0 (x) is locally sublinear near x = 0, and %(t)dt = oo. Then (3.6.19) •o is oscillatory, where

JA·

A; 0 = {t E [to,oo): to :5 g; 0 (t) :5 t}.

(3.6.21)

Proof: Let x(t) be an eventually positive solution of (3.6.19), say, x(t) > 0, x(g;(t)) > 0 for t 2:: t1 2:: to, i = 1, ... , n. Set z(t) = x(t) - px(t - r), then z'(t) :50. If z(t) > 0, t 2:: t2 2:: t1, we have limt-+oo z(t) = l1 2:: 0. If l 1 > 0, since l1 :5 z(t) :5 z(t2) fort~ t 2 and

N(t)-1

x(t) =

L

pi-lz(t- (i -1)r) + pN(t)- 1x(t- (N(t) -1)r)

i=l where N(t) is a positive integer such that T- r < t- (N(t) -1)r:::; T, we have

and

From (3.6.19) we see

-z'(t) 2:: %(t)!; 0 (x(g; 0 (t))) 2:: q; 0 (i)L.

Chapter 9

174

Integrating it from t 2 to t we have

z(t2)- z(t)

~L

lt

%(s)ds.

t,

Letting t -+ oo we obtain

JA;o

q; 0 (t)dt = oo. which contradicts that If z(t)-+ 0 as t-+ oo, then x(t)-+ 0 as t-+ oo. Then there exists t3 ~ t2 such that 0 ~ x (gio ( t)) ~ a and z( t) ~ a for t ~ ta. Since f is nondecreasing for t E A; 0 n [t3, oo)

From (3.6.19)

Then we have rz(ts)

Jz(t)

it

du fio(u) =

t3

~

1

-z'(s) J;.(z(s)) ds

A; 0 n(ts,t]

-z'(s) ds J;.(z(s))

~

1

q;.(t)dt.

A; 0 n(t 3 ,t]

Letting t -+ oo we have

JA·

q; 0 (t)dt = oo again. which contradicts •o lim z(t) = The last possibility is that z(t) < 0, z'(t) ~ 0, t ~ t 2 • Then t-oo exists a there so unbounded, is x(t) then oo, = £ -£2, £2 > 0 or £2 = oo. If 2 sequence {tn} satisfying lim tn = oo, x(tn) = max x(t), and lim x(tn) = n-+oo

t2 1~p < 0 0 contradicting the positivity of x.

175


Remark 3.6.1. The local sublinear condition for J; is important here. In fact, Theorem 3.6.3 is not true for linear equations. For example, consider the equation

(x(t)- ~x(t -1)) where q(t) = !:2(~~1~ solution x(t) =

t.

,

+

- 4t + 4 4t 2 (t _ 1 ) x(t -1) = 0

t2

(3.6.22)

. Clearly, J:"; q(t)dt = oo. But (3.6.22) has a nonoscillatory

Using Tychonoff-Schauder fixed point theorem we can prove the following result for the existence of nonoscillatory solutions. We omit the proof here. Theorem 3.6.4.

n

< 1 and L: ftoo q;(t)dt
O

E lt i=l

q;(t)dt

= oo.

to

3. 7. Equations with Mixed Coefficients In this section we consider neutral delay differential equations with positive and negative coefficients of the form

(x(t)- c(t)x(t- r))'

+ p(t)x(t- r)- q(t)x(t- a)= 0.

The following is an oscillation criterion for ( 3. 7.1). Theorem 3.7.1.

Assume that

(i) r E (O,oo), and r, a E (O,oo), and c,p,q E C((t 0 ,oo),R+); (ii) r ~ a; (iii) p(t) ~ q(t- T +a) and p(t) ¢. q(t- r +a) fort~ t 0 + r- a;

(3.7.1)

Chapter 9

176 (iv) 1- c(t)- J11_r+tt q(s)ds ~ 0 for all sufficiently large t; (v) every solution of the delay differential equation

q(s- T)ds) y(t- T) = 0 y'(t) + (p(t)- q(t- T + u)) (1 + c(t- T) + 1' t-r+a (3.7.2) is oscillatory. Then every solution of (3. 7.1) is oscillatory.

Proof: Assume the contrary, and let x(t) be an eventually positive solution of Eq. (3.7.1). Denote 1

q(s)x(s- u)ds. y(t) = x(t)- c(t)x(t- r) - 1 t-r+a Then it is easy to see that y'(t) :5 0 and (3.7.3) and (3.7.1),

y'(t)

(3.7.3)

y(t) > 0, eventually. In view of

= -(p(t)- q(t- T + u))x(t- T).

(3.7.4)

From (3.7.3) we have

x(t)

~ y(t) + c(t)y(t- r) + i~r+a q(s)y(s- u)ds ~

(1+c(t)+ ('

lt-r+tt

q(s)ds)y(t).

(3.7.5)

Substituting (3.7.5) into (3.7.4) and noting condition (iii), we have 1 q(s-T)ds) y(t-T) :50. (3.7.6) y'(t)+(p(t)-q(t-T+ u)) (1+c(t-T)+1 t-r+a

Hence, (3.7.2) has a positive solution, which contradicts (v).

Assume that > u;:::: 0, c,p,q E C([t 0 ,oo),R+)i

Theorem 3. 7.2.

i) r

> 0,

T

0

177

Oscillation of First Order Neutral Equations ii) there exists t1

~ t0

such that fort

0::; c(t)

p(t)

~ t1

+ 1~~" q(s + u)ds::; 1,

= p(t)- q(t- r + u) ~ 0;

J,11_ r p( s )ds > 0; there exists T;?: t 1 + max{r, T}

(3.7.7)

iii) lim inf t-oo

iv)

inf

t~T,A>O

such that

1 {~exp(.X1t-r p(s)ds)+c(t-r)exp(.xjt p(s)ds) t-r A

(3. 7.8)

Then every solution of (3. 7.1) is oscillatory. To prove this theorem we need the following lemmas.

Lemma 3.7.1. Assume that conditions i)-iii) in Theorem 3.7.2 hold. Let x(t) be an eventually positive solution of (3. 7.1). Set u(t) = x(t)- c(t)x(t- r)

-lt- 0: u'(t) + ..\p(t)u(t):::; 0

holds eventually}.

Then A =f:. 0, and has an unifonn upper bound which is independent of x(t).

Proof: From Lemma 3.7.1 and its proof, there exists t 2 2:: t 1 such that u'(t):::; 0, u(t) > 0, fort ;::: t2, and

u'(t)

= -p(t)x(t- r):::; -p(t)u(t- r):::; -p(t)u(t),

t;::: t 2

+ r,

(3.7.11)

179


i.e., 1 EA. Now we show that A has an upper bound which does not depend on x(t). Set

liminfj t-oo

1

t-r

p(s)ds

= 3k.

From condition iii) in Theorem 3. 7.2 we see k > 0. Hence there exists t3 such that

~

t2

+T

j~rp(s)ds>2k, t~t3. By Lemma 2.1.3 this together with (3.7.11) implies that u~~)) ::; p, t ~ t 3 . From x(t) = 0. (3.7.1) and condition (iii) in Theorem 3.7.2, it is easy to see that liminf t-+oo lim sn = oo, and Hence there exists a sequence {sn} such that sn ~ t 3 + 2r, n-+oo x(sn- r) = min{x(s- r): t3 ::; s::; sn}, n = 1, 2, .... Integrating (3.7.10) from Sn - T to sn we have

u(sn)- u(sn- r) = -1:-r p(s)x(s- r)ds

1

4n

::; -x(sn- r) an-r p(s)ds::; -2kx(sn- r),

n = 1,2, ... ,

and hence u(sn- r) > 2kx(sn- r). Substituting it into (3.7.10) we obtain

u'(sn) = -p(sn)x(sn- r) 1 1 >- 2k p(sn)u(sn- r) >- 2k3 p(sn)u(sn)

which means that ~xw.

2h- EA, i.e., 2h- is an upper bound of A which does not depend

o

Proof of Theorem 3.7.2: Let x(t) be an eventually positive solution. By Lemmas 3.7.1 and 3.7.2 u'(t) ::; 0, u(t) > 0, t ~ t 2 • We claim that there exists an a > 1 such that ..\ 0 E A implies that a..\ 0 E A.

ChapterS

180

In fact, from condition (iv), there exists a> 1 such that inf inf { ~exp (.xjt p( s )ds) + c( t - r )exp (.xjt p( s )ds)

A>O t~T

A

+

1-T

l~~u q( s -

1-T

r +

q

1

)exp ( >.1 p( tt )du) }

~ a > 1.

(3.7.12)

By the definition of A we see that eventually

u'(t) + >.0 p(t)u(t) :::; 0. Set z(t) = u(t)exp(>.o

It: p(s)ds).

(3.7.13)

Then

z'(t) = [u'(t) + >.op(t)u(t)] exp (>. 0 it p(s)ds) :::; 0. I,

From (3.7.10) and (3.7.13) we have x(t- r) (3.7.12)

~

>. 0 u(t), and then from (3.7.9) and

u'(t) = -p(t)x(t- r)

= -p(t) [u(t- r) + c(t- r)x(t- T - r) + 1t-r-u q(s + q)x(s)ds] t-2r

:5 -p(t) [u(t- r) + >.oc(t- r)u(t- r) + >. 0 1t-r-u q(s + q)u(s + r)ds] l-2r

= -p(t) [u(t- r) + >. 0 c(t- r)u(t- r) + >. 0

l~~u q(s + q- r)u(s)ds]

= -p(t) [ z(t- r)exp(- >. 0 1:-r p(s)ds)

+ >.oc(t- r)z(t- r)exp(- >. 0 1:-r p(s)ds) + >.o

1~~u q( s + q -

:5 -p(t)[exp(>.o

T

)z( s )exp ( - >. 0 1: p( u )du) ds]

1~rp(s)ds) +>. c(t-r)exp(>.o 1~/(s)ds) 0

181

03cillation of Fir3t Order Neutral Eqv.ation3

i~~(f q(s + u- r)exp( >.

+ Ao

~-

(>-o

inf exp

t~T

+ >.o ~

i'

t-r

0

1'

p( s )ds) + >.o c(t -

i~~(f q(s + u- r)exp( >.o

T

p(u)du )ds] u(t)

)exp

1'

(>-o it-r t p( s )ds)

p(u)du )ds] p(t)u(t)

-a>.op(t)u(t).

This implies that a>. 0 E A. Repeating this we obtain that am >.o E A for any positive integer m, which contradicts the boundedness of A. D Theorem 3.7.3.

i)

r

> 0,

T

Assume that

>u

~

0,

c,p,q E C([to, oo),R+);

ii) p(t) ~ 0, fort~ t 1 ~ t 0 and ftc;'p(t)dt = oo, where pis defined in Theorem 3.7.1; iii) For some T

~ t1

+max {r, T }, there exists,\* > 0 such that

sup { ;. exp (,\ * t2:T

+

i'

t-r

p( u )du) + c( t -

i~~(f q(s- r + u)exp( >.*

T )exp (,\•it

1'

p(u)du) ds}

Then Eq. (3.7.1) has a positive solution x satisfying lim x(t) t-+oo

x(t)

~ exp(- >.•i:+r p(u)du)

p( s )ds)

t-r

for all large

~ 1.

(3.7.14)

= 0 and

t.

(3.7.15)

To prove this theorem we first show the following lemma. Lemma 3. 7 .3.

Let conditions i) and ii) in Theorem 3. 7.2 hold. FUrther assume

Chapter 9

182

that the integral inequality

+ l~~w q(s + u)z(s)ds

c(t)z(t- r)

1:

+

[p(s

+ r)- q(s + u)]z(s)ds :5 z(t), (3.7.16)

lim z( t) = 0 where has a positive solution z : [t 1 - m, oo) -+ (0, oo) such that t-oo m = max { r, r }. Then there exists a positive function x : [t 1 - m, oo) -+ (0, oo) satisfying the integral equation

c(t)x(t-r)+

1~~" q(s+a)x(s)ds

+ 1:[p(s + r)- q(s + u)]x(s)ds

= x(t), (3.7.17)

and 0 < x(t) :5 z(t),

t ~ t 1•

Proof: Choose T ~ t 1 such that z(t) > z(T) for t 1 functions as follows:

n ={wE C([t 1 -

m,oo),R+): 0 :5 w(t) :5 z(t),

and define a mapping T on

{

~ t 1 - m}

t

+ Jt"~; q(s + a)w(s)ds t1

-

0 :5 Xn(t) :5 Xn-l(t) :5 · · · :5 x1(t) :5 z(t),

= x(t) exists fort ~ t 1 -

m.

(3.7.19)

m :5 t < T.

By (3.7.16) Tn c n. Define a sequence {xn} on T Xn-1! n = 1, 2, .... It is easy to see that

n-+oo

t ~T

+ ft~r (p(s + r)- q(s + u))w(s)ds, (Tw)(T) + z(t)- z(T),

Then lim xn(t)

(3.7.18)

n by

c(t)w(t- r) (Tw)(t) =

m :5 t < T. Define a set of

-

t

= z,

n

by Xo

~

t1- m.


183

It follows from the Lebesgue's dominated convergence theorem that x(t) satisfies (3.7.17). Clearly, x(t) is continuous fort~ t1- m. Since x(t) > 0 on [t 1 - m, T), it follows that x(t) > 0 for all t ~ t 1 - m.

Set

Proof of Theorem 3.7.3:

= exp(- A• £+r p(s)ds ).

z(t)

t

~ T- r.

(3.7.20)

It is obvious that z(t) is continuous fort ~ T- r and lim z(t) = 0. We shall t.-oo show that z(t) satisfies the integral inequality (3.7.16) fort~ T. From (3.7.14), we have

1 ( .\•exp .\•

lt(+r p(u)du ) rt+r-u

+ },

(

it+r

(

f.t+r

+c(t)exp A• t+r-/(s)ds

q(s- r + a)exp A• •

)

)

p(u)du ds $ 1,

t

~

T- T. (3.7.21)

We observe that roo p( S +

lt-r

T

)exp (A •

f.t+ r p( U )du) ds •+r

(3.7.22) and

j t+r-a q(st

=

T

(

+ a)exp A•

lt-rt-u

f.t+r 8

(

)

p(u)du ds

q(s + a)exp A•

jt+r p(u)du ) ds. s+r

(3. 7.23)

ChapterS

184 Substituting (3.7.22) and (3.7.23) into (3.7.21) we have c(t)exp ( A* 1

+

1""

) t+r p(u)du t+r-r

P(s

t-r

+ ~t-a q(s + u)exp ( A* 1t+r p(u)du ) t-r

ds

~+r

+ r)exp(A* {t+r p(u)du)ds ~ 1, la+r

t;::: T- r.

(3.7.24)

From (3.7.24) and (3.7.20) we obtain c(t)z(t- r)

+~~~a q(s + u)z(s)ds +

1:

p(s

+ r)z(s)ds

~ z(t),

t;::: T- r,

i.e., (3.7.16) has a continuous positive solution z(t) defined by (3.7.20). By Lemma 3.7.3, (3.7.17) has a positive solution x(t) on [T, oo) satisfying x(t) ~ z(t). Thus xis a positive solution of (3.7.1) satisfying lim x(t) = 0. 0 t-oo

Remark 3.7.1. Condition (3.7.14) is sharp in the following sense: if c,p,q are constants with c > 0 and p > q, condition (3.7.14) is not only a sufficient condition but also a necessary condition for the existence of a positive solution which tends to zero as t-+ oo. This is based on the fact that for the autonomous case, Eq. (3.7.1) has a positive solution which tends to zero if and only if its characteristic equation (3.7.25) has a negative real root. In the same sense, condition (3.7.8) is sharp.

Corollary 3. 7 .1.

Assume that conditions i) and ii) in Theorem 3. 7.2 bold,

and

li~inf{e t

00

1'

t-r

p(s)ds

+ c(t- r) + ~t-a q(s- r + u)ds} > 1.

(3.7.26)

1-r

Tben every solution of (3.7.1) is oscillatory.

Example 3. 7 .1. Consider (x(t)-

t x(t- 2))' + (1 + ~ + sint)x(t -1r)- ~ x(t- f)= 0

(3.7.27)

185

Oscillation of First Order Neutral Equations and

(x(t)-

e x(t -1)- ~ x(t- t) = 0. k x(t -1)) , + 2(13+e)

(3.7.28)

It is easy to see that for (3.7.27) and (3.7.28) all assumptions of Corollary 3.7.1 are satisfied. Therefore, these equations are oscillatory.

3.8. Linearized Oscillation The following linearized oscillation result reveals that, under some assumptions, certain nonlinear differential equations have the same oscillatory behavior as an associated linear equation with constant coefficients. Consider the neutral differential equation with positive and negative coefficients

with the following assumptions G,H~,H2 E C(R,R);

(1) P,Q1,Q2 E C([to,oo),R+), (2)

T

E (0, oo ),

(3) lim sup P(t) t~oo

u~, u2 E [0, oo ),

= P2 E (0, 1);

(5) 0 ~ G~u) ~ lfor u (6) ~~~=~

;:

1 and 0

.o"'•J

!f >0.

0

Lemma 3.8.2. Assume that FE C([T,oo),( O,oo)), HE C(R+,~), r E (O,oo), u 11 u 2 E [O,oo), c 11 c 2 E [O,oo); H(u) is nondecreasing in a neighborhood of the origin and u 1 ~ 0'2. Let m = max{ r, ut} and suppose that the integral inequality

F(t)z(t-r)+c1 1~~~· H(z(s))d s+c21:

1

H(z(s))ds :5 z(t),

t

~T

(3.8.6)

has a continuous positive solution z: [T- m, oo)-+ (0, oo) such that lim z(t) 1-ooo

0.

=

Then there exists a positive solution x : [T- m, oo) -+ (0, oo) of the corresponding integral equation

F(t)x(t-r )+ct

1~~~· H(x(s))d s+c21:

1

H(x(s))ds = x(t),

t

~ T.

(3.8.7)

Proof: Choose a Tt ~ T and a 6 > 0 such that z(t) > z(T1 ) forT- m :5 t < Tt, 0 < z(t) < 6 fort~ T1 - m and H(u) is nondecreasing in [0, 6]. Define a

Chapter 9

188

set of functions

n ={wE C([T- m,oo),R+): and define a mapping T on

0:5 w(t) :5 z(t),

t ~ T- m}

n as follows:

F(t)w(t- r) + c1 Jt~:,_2 H(w(s))ds (Tw)(t)

={

+ c2 ft~rr, H(w(s))ds (Tw)(Tl)

(3.8.8)

+ z(t)- z(T1),

T-m:5too •

and let {tn} and {t~} be two sequences in the interval [t1,oo) such that lim tn = oo,

n-+oo

lim t~ = oo,

n-+oo

lim x(tn) = 0

t-+oo

lim x(t~) = p,.

and

t-+oo

Since z(tn) :::; x(tn), then£:::; 0. On the other hand, by choosing sufficiently small > 0 we have

E: 1

z(t~);::: x(t~)- p(t~)x(t~- r)- M(q2 + 77) 1~~~~ 2 x(s)ds 2:: Letting n

--+

x(t~)- p(t~)(p,

+ c')- M(q2 + 77)(u1- u2)(p, + c').

oo we obtain

£ 2:: p,- P2(!-L + c')- M(q2 + 77)(u1- u2)(p, + c'). Since

E: 1

can be arbitrarily small, we have

£ 2:': 1-L- P2/-L- M(q2

+ 77)(u1- u2)!-L = [1- P2- M(q2 + 77)(u1- u2)]p,

which, together with£:::; 0 and (3.8.12), implies£= p, = 0. Hence lim z(t) = 0 and lim x(t) = 0. Then t-+oo

t-+c:x:>

(3.8.16) Now by using (3.8.14) and (3.8.16) we obtain

x(t) 2:: p(t)G(x(t- r))

+ (q2 + 77) 1~~~ 2 H2(x(s))ds

+ (qt- q2- 277) =p(t)

1:.

H2(x(s))ds

r-cr•

G(x(t- r)) x(t-r) x(t-r)+(q2+77) lt-cr,

+ (qt- q2- 277)

1

00

t-u1

H 2(x(s)) x(s) x(s)ds

H2(x(s)) x(s) x(s)ds.

191

Oscillation of Fir.st Order Neutral Equations Choose 0

< e < -21 min {P2,

}. Then by assumptions (3), (5), (6) __!L_+ 92 'I

2

x(t) 2: (P2- e)x(t- r) + (q2 + 77)(1- e) ~~~~ x(s)ds

+ (q1 -

q2- 277)(1- e)

1:

1

x(s)ds,

t;::: t2

where t 2 is sufficiently large. By Lemma 3.8.2, the equation

2 v(t) = (P2- e)v(t- r) + (q2 + 77)(1- e) ~~~~ v(s)ds

+ (q1- q2- 277)(1- e)

1:

1

v(s)ds

has a continuous positive solution v : [T - m, oo) ---. (0, oo ), where T 2: t2 is sufficiently large. Clearly, v(t) is a positive solution of the neutral equation

where e1 = 77 + eq1 - 71e and e2 = 77 - q2e - 71e are sufficiently small positive numbers. Hence, by Lemma 3.8.1., Eq. (3.8.2) has a positive solution. This 0 contradiction proves the theorem.

3.9. Equations with a Nonlinear Neutral Term Consider the nonlinear neutral differential equations of the form

(x(t)- g(x(t- r)))' + h(t, x(t- u)) = 0,

t;::: t 0

(3.9.1)

where r > 0, u 2:0, g E C(R, R), hE C([to, oo) x R, R). Theorem 3.9.1. Assume that (i) g is nondecreasing, xg(x);::: 0 and there exists a E (0, 1] such that either

lim sup lzi--ooo

IYI (xl )I =MER+, X"'

Chapter :J

192

whenever o: E (0, 1), or lim sup g(x) = c E [0, 1), lxl-+oo X

(ii) his nondecreasing in x,

h(t,x)x

~

roo jh(t, c)jdt =

ito

if o:

= 1;

0 for x =/: 0, and

00

for any c =/: 0;

(iii) There exists a positive integer n such that the differential inequality

y 1(t)

+ h(t,y(t- u) + g(y(t- r)

+ g(y(t- u- 2r) + ··· + g(y(t- 0'- nr)) ... )))

::; 0 (3.9.2)

bas no eventually positive solutions and

y'(t) + h(t, y(t- u)

+ g(y(t- u- r)

+ g(y(t- u- 2r) + ··· + g(y(t- u- nr)) ... )))

~ 0

(3.9.3) has no eventually negative solutions. Then every solution of Eq. (3.9.1) is oscillatory.

Proof: Assume the contrary, and let x(t) be an eventually positive solution of Eq. (3.9.1), say x(t) > 0, t;:::: T;:::: t 0 • Set

z(t)

= x(t)- g(x(t- r)).

(3.9.4)

Then z'(t) ::; 0, t ~ T + m, m = max {r,u}. If z(t) < 0 eventually, then there exists a t 1 ~ T + m such that z(t) ::; -f.< 0 fort 2: t 1 • Thus, from (3.9.4), g(x(t- r)) ~ £ > 0 fort;:::: t 1 • Integrating Eq. (3.9.1) and noting condition (ii) we have lim z(t) = -oo. Consequently, lim g(x(t- r)) = oo, which implies that t-+oo

t-..oo

lim x(t) = oo. Thus there exists a sequence {en} such that lim

t-+oo

n-+co

en =

oo and

193

Oscillation of First Order Neutral Equations x(~n) =

max {x(t)}-+ oo,

t,9::;;en

n-+ oo. Then

z(~n) = x(~n)- g(x(~n- T));::: x(~n)- g(x(~n))

= x(~n)[1- g(x(~n))/x(~n)]-+ oo,

as

n-+ oo,

which contradicts the fact that z(t) is eventually negative. Therefore z(t) > 0 for t ;::: T + m. Hence

+ g(x(t- T)) + g(x(t- 2T))) ;::: z(t) + g(z(t- T) + g(z(t- 2T) + · · · + g(z(t- nT)) ... ))

x(t) = z(t)

= z(t) + g(z(t- T)

(3.9.5) fort;::: T

z'(t)

+ nT.


+ h(t, z(t- u) + g(z(t- u- T)

+ g(z(t- u- 2T) + · · · + g(z(t- u- nT)) ... ))) :50 (3.9.6) for t ;::: u + T + nT, which implies that (3.9.2) has an eventually positive solution. This is a contradiction. A parallel argument is true if we assume that Eq. (3.9.1) has an eventually negative solution. 0 Theorem 3.9.2. Assume that (i) g is nondecreasing, xg(x);::: 0 and there exists ad> 0 such that g(d) < d, (ii) his nondecreasing in x, h(t,x)x;::: 0 and

r= lh(t, c)idt
0 such that

/3 + g(d)

0.

Proof: We denote by BC the space of all bounded continuous functions defined on [to, oo). Define a distance in BC by l!x- Yl! = suplx(t)- y(t)l, for x,y E t>to

BC. Let U:::: {x E BC: c $ x(t) $ ldl, t ~to} .;here 0 < c 0. t-+oo

3.11.

Notes

Theorem 3.1.1 can be seen from Arino and Gyori [5], also from Gyori and Ladas [76]. The rest of Section 3.1 is from Erbe and Kong [40]. Theorem 3.2.1 is based on Jaros [87]. Theorem 3.2.2 is taken from Chuanxi, Ladas, Zhang and Zhao [25]. Theorem 3.2.3 is adopted from Zhang and Yu [211]. Theorem 3.2.4 is based on the results of Zhang and Gopalsamy (204]. Theorems 3.2.5 and 3.2.6 are taken from Zhang and Yu [214]. Theorem 3.2.9 is from Zhang, Yu and Liu. Theorem 3.2.10 is adopted from Gopalsamy, Lalli and Zhang [55], also see Yu [177]. Theorem 3.2.11 is taken from Yu, Wang and Qian [181], also see Wang [166], Kitamura and Kusano [94]. Theorem 3.2.12 is based on Ladas and Sticas [76]. Lemma 3.3.1 is from [25]. Lemma 3.3.2 is obtained by Gyori [75]. Theorem 3.3.1 to Theorem 3.3.3 are taken from Zhang and Yu [212]. Theorem 3.3.4 to 3.3.6 are adopted from Erbe and Kong [38]. Condition (3.3.54) is new. Theorem 3.3.7 and Theorem 3.3.8 are taken from Zhang and Gopalsamy [204]. The content of Section 3.4 is based on Lalli and Zhang [117] and Gopalsamy


201

and Zhang [56]. Lemma 3.5.2 is from Gyori and Ladas [76]. Theorem 3.5.1 is adopted from Lalli and Zhang [116]. Theorem 3.5.4 is new. Theorem 3.5.5 is based on Erbe and Zhang [44]. Theorem 3.6.1 and 3.6.2 are from Jaros and Kusano [89]. Theorem 3.6.3 is adopted from Wang [165]. Theorem 3.7.1 is from Yu and Wang [179]. Theorem 3.7.2 is a modification ofYu [176]. Theorem 3.7.3 is by Gopalsamy and Zhang [203]. Theorem 3.8.1 is taken from Chuanxi and Ladas [22]. The content of Section 3.9 is taken from Zhang and Yu [211]. Theorem 3.10.1 is adopted from Lu Wudu [132].

4 Oscillation and Nonoscillation of Second Order Differential Equations with Deviating Arguments

4.0.

Introduction

In Section 4.1 we introduce the linearized oscillation results for a certain class of second order nonlinear delay differential equations. In Section 4.2 we present results on the existence of oscillatory solutions of the delay differential equation. In Section 4.3 we introduce Sturm comparison theorems which are very useful for the oscillation theory as well as the boundary value problems. In Section 4.4 we establish some oscillation criteria for second order neutral differential equations. By way of these criteria the oscillation problem for neutral differential equations can be reduced to the same problem for the corresponding delay equations and sometimes to the corresponding ordinary differential equation. In Section 4.5 we introduce the classification of nonoscillatory solutions for second order nonlinear neutral differential equations, various existence results of nonoscillatory solutions of different types are given. In Section 4.6, we present several bounded oscillation 202

Second Order

Equation~

with Deviating

Argument~

203

criteria for the second order neutral differential equation of unstable type. Section 4. 7 deals with the forced oscillation. In Section 4.8 equations with nonlinear neutral terms are investigated. In the last section, advanced type equations are investigated, also a relation between the oscillation of equation of the advanced type and the oscillation of the corresponding ordinary differential equation is established.

4.1. Linearized Oscillation We consider the second order nonlinear delay differential equations

x"(t) + A(t)x'(t) + B(t)f(x(t- r)) = 0,

t~0

(4.1.1)

x"(t)- A(t)x'(t) + B(t)f(x(t- r)) = 0,

t~0

(4.1.2)

A,B E C(R+,(O,oo)), f E C(R,R) B(t) =be (0, oo). lim A(t) =a E (0, oo), tlim ...... oo t-+oo

(4.1.3)

and

where

r > 0,

The sunflowing equation

. y ( t - r) = 0, y "( t ) + -a y '( t ) + -b sm r

r

t~O

( 4.1.4)

is a special case of (4.1.1). Under some assumptions the following equations are called the linearized limiting equations of (4.1.1) and (4.1.2), respectively:

x"(t) + ax'(t) + bx(t- r) = 0,

t~0

(4.1.5)

= 0,

t~0

(4.1.6)

and

x"(t)- ax'(t) + bx(t- r) respectively.

We shall establish the relations between the oscillation of (4.1.1), (4.1.2) and that of their linearized limiting equations, ( 4.1.5), ( 4.1.6), respectively.

204

Chapter 4

Theorem 4.1.1. Assume that (i) uf(u) > 0 for u -:f. 0, lui~ H, where HE (0, oo) and lim u~o (ii) The characteristic equation of Eq. (4.1.5)

J(>.)

= >.. 2 +a>.+ be-).r = 0

f(u) u

= 1, (4.1.7)

has no negative roots. Then every solution of Eq. (4.1.1) whose graph lies eventually in the strip R+ x [- H, H] is oscillatory. Theorem 4.1.2. Assume that (i) uf(u) > 0 as u "I 0, lim fsu) = 1,

JuJ--oo

(ii) The characteristic equation of ( 4.1.6)

(4.1.8)

has no positive roots. Then every solution of Eq. (4.1.2) is oscillatory. The methods of the proofs of Theorems 4.1.1 and 4.1.2 are similar. So we only show the proof of Theorem 4.1.1 in the following. We first prove some Lemmas which will be used in the proofs of Theorems 4.1.1 and 4.1.2. Lemma 4.1.1. Assume that a, b, r E (0, oo) and every solution of Eq. (4.1.5) is oscillatory. Then there exists an e E (0, b) sucl1 that every solution of the equation

z"(t) +(a+ e)z'(t) + (b- e)z(t- r) = 0

(4.1.9)

is oscillatory also.

Proof: From Theorem 6.0.1 we will see that ( 4.1.5) is oscillatory is equivalent to the statement (4.1.7) has no real roots. It is easy to see that>. ;:::: 0 does not satisfy (4.1. 7). Therefore, is oscillatory is equivalent to ( 4.1. 7) having no negative roots. Since f(O) = b > 0, f(-oo) = oo, and (4.1.5) is oscillatory, we have m =min{!(>.):>. ~ 0}

> 0,

205

Second Order Equationa with Deviating Argumenta

and hence J(>.) = ).. 2

+a>.+ be->.r 2:: m for >. ~ 0.

From the fact that

we know that there exists a

>.o < 0 such that for >. ~ >.o

Set

Then for

>.

~

>. 0

For>. E (>.o,O] >. 2 +(a+ e)>.+ (b- e)e->.r

= J(>.)- e(e->.r- >.)

Therefore, >. 2 +(a+ e)>.+ (b- e)e->.r = 0 has no negative roots. Consequently every solution of (4.1.9) is oscillatory. Lemma 4.1.2. Assume that A, BE C(~,(O,oo)), f E C(R,R), uf(u) > 0 as u -:f:. 0 and lui~ H where HE (O,oo), and f is nondecreasing in [-H,H]. If y(t)2::

1 1: 00

B(u)f(y(u-r))exp(

-1

8

A(v)dv)duds,

t2::T (4.1.10)

206

Chapter

bas a positive solution y(t) : [T- r, oo)

z(t) =

lX> lr"

--+

4

(0, H], then

B(u)f(z(u -r))exp(

-1·

(4.1.11)

A(v)dv)duds

bas a positive solution z(t) on [T- r,oo) and 0 < z(t) ~ y(t).

(4.1.12)

Proof: Let C([T - r, oo ), R+) be the space of all continuous and nonnegative functions on [T - r, oo ). Define a set X by

X={wEC([T-r,oo),R+):

O~w(t)~1,

t~T-r}.

(4.1.13)

Define an operator T on X by ylt)

(Tw)(t) =

{

fo00 J;B(u)f(y(u- r)w(u- r)) X

exp(-

~ (Tw)(T)

J: A(v)dv)duds,

t

+ (1- ~),

It is easy to see that TX C X, (Twt)(t) ~ (Tw 2 )(t).

~

(4.1.14)

T

T- r

~

t

~

T.

Tis continuous, and w 1 (t) ~ w 2 (t) implies that

Define a sequence of functions {wk(t)}, k

=1, wk(t) = (Twk-d(t),

=0, 1, 2, ...

as follows:

w 0 (t)

k = 1,2, ....

(4.1.15)

By induction 0 ~ Wk(t) ~ Wk-t(t) ~ · · · ~ wo(t) = 1, Then the limit w(t)

= k-+oo lim wk(t),

t ~ T- r.

t ~ T- r exists and 0 ~ w(t) ~ 1,

t ~

207

Second Order Equations with Deviating Arguments T- r. By the Lebesgue's monotone convergence theorem

yfu f:X' J;B(u)f(y(u- r)w(u- r)) w(t) =

{

t ~T

exp(- J: A(v)dv)duds, ~w(T)+(1-~),

(4.1.16)

T-r5:t5:T.

w(t) is continuous on [T- r, oo) and w(t) > 0 fort E [T- r, T). Then w(t) > 0 for t ~ T - r. Set (4.1.17)

z(t) = y(t)w(t). Then z(t) is a positive solution of (4.1.11) and satisfies (4.1.12).

0

Proof of Theorem 4.1.1: Assume the contrary. Let x(t) be an eventually positive solution of (4.1.1) with 0 < x(t) 5: H, t ~ T. Set

u(t) = x'(t)exp(1t A(s)ds). Then

u'(t) = -B(t)f(x(t- r))exp(1t A(s)ds) < 0,

t

~ T1 = T + r.

(4.1.18)

There are two possibilities for u :

(i) u(t) > 0,

t

~

Tb (ii) there exists T2 ~ T1 such that u(t)

< 0 fort~ T2.

For (i), x'(t) > 0 fort~ T1 . Hence

x"(t) < -B(t)f(x(t- r)) 5: -CB(t),

t ~ T1 + r

where 0 < C =min {f(x): x(TI) 5: x 5: H}, which implies that lim x(t) = -oo, t-oo a contradiction. For the case (ii), x'(t) < 0 fort~ T2. Then lim x(t) = i exists. We show t-oo that i = 0. In fact, if f > 0

B(t)f(x(t-

r))-+ bf(i) > 0,

as

t-+ oo,

Chapter 4

208 and hence there exists a Ta ;::: T2 such that

B(t)f(x(t- r)) >

bf~£)

and

A(t):::; a+ 1,

t;::: Ta.

for

From ( 4.1.1) we obtain

x"(t) +(a+ 1)x'(t) + ~ f(£):::; 0,

t;::: Ta

which implies that

x'(t) +(a+ 1)x(t)--> -oo,

as

t--> oo,

and hence x'(t) --> -oo, as t --> oo. This is impossible. So lim x(t) = 0. It is t .... oo

easy to see that lim x'(t) = 0 also. t .... oo

From condition (ii) and Lemma 4.1.1 there exists an c E (0, b) such that

>. 2 +(a+ c)>.+ (b- c)e-.\r > 0 for

>. < 0.

(4.1.19)

From (4.1.3) and condition (i) there exists a T4 such that

A(t) < a+t:,

and

B(t)

f(x(t- r)) ( ) > b-t:, xt-r

for

t;::: T4.

Substituting them into (4.1.1) we obtain

x"(t) +(a+ t:)x'(t) + (b- t:)x(t- r):::; 0,

t;::: T4 ,

and hence

which leads to the integral inequality

By Lemma 4.1.2, the integral equation

z( t)

= (b-

c)/,""£"

z( u- r )e(a+~)(u-s) duds,

t ;::: T4

Second Order Equationil with Deviating Argumentil

has a positive solution and 0 < z(t)

~

209

x(t). It is easy to see that

z"(t) +(a+ c)z'(t) + (b- c)z(t- r) = 0 which contradicts (4.1.9). Similarly we can prove that (4.1.1) has no eventually D negative solution with 0 > x(t);:::: -H. The following results are about the existence of nonoscillatory solutions, where condition (4.1.3) is no longer required. Theorem 4.1.3. Assume that (i) there exist a> 0, b > 0 such that A(t);:::: a, B(t) ~ b, t;:::: 0; (ii) there exists an H > 0 such that uf(u) > 0, as u E (O,H], f(u) ~ u for u E [0, H], and f is nondecreasing on [0, H]; (iii) The characteristic equation (4.1. 7) has a real root. Then Eq. (4.1.1) has an eventually positive solution x(t) lying in the strip R+ x (0, H] eventually. Theorem 4.1.4. Assume that (i) uf(u) > 0 as u :f: 0; (ii) there exists an M > 0 such that f(u) ~ u for u;:::: M (iii) there exist a > 0, b > 0 such that A(t) ;:::: a, B(t) ~ b, non decreasing on [0, oo ); (iv) (4.1.8) has a real root. Then (4.1.2) has an eventually positive solution.

t ;:::: 0; and f

is

Proof of Theorem 4.1.3: Let Ao be a real root of (4.1. 7). Clearly, Ao < 0. Then y(t) = e>-ot is a solution of (4.1.5). Choose T so large that y(t) ~ H for t;:::: T- r. From (4.1.5)

Integrating it twice we obtain

Chapter 4

210

Then

y(t);:::

1

00

B(u)f(y(u-r))exp(

;;

-1"

A(v)dv)duds,

t;:::T.

-1"

A(v)dv)duds,

t;:::T

By Lemma 4.1.2, the equation

z(t)=

1

00 ;;

B(u)f(z(u-r))exp(

has a positive solution z(t) and 0 satisfies

z"(t)

< z(t) :5 y(t) :5 H, t;::: T-

+ A(t)z'(t) + B(t)f(z(t- r))

= 0,

r. Clearly z(t)

t;::: T,

i.e., z(t) is a positive solution o£(4.1.1) on [T,oo) and 0 < z(t)

:5 H, t;::: T.

0

The proof of Theorem 4.1.4 is similar to the above, we omit it here.

4.2. Existence of Oscillatory Solutions In the oscillation theory of the differential equations with deviating arguments the problem of the existence of oscillatory solutions is a difficult problem.

4.2.1.

We consider the second order delay differential equations of the form

x"(t)

+ a(t)x(t) + h(t,x(gi(t)), ... ,x(gn(t))) = 0

(4.2.1)

compared with the equation

y"(t) + a(t)y(t)

=0

where a E C 1 (R+, R), g; E C(R+, R), t - r 1,2, ... ,n, r > 0, and hE C(R+ X Rn,R).

(4.2.2)

< g;(t)

~

t,

t ;::: 0,

t

=

Assume that (i) there exist two constants A and B sucll tllat 0 < A :5 a(t) :5 B, t E R+, a' (t) is of a constant sign; (ii) lh(t, u1, u2, ... , un)l :5 b(t)F(Iull, !u2!, ... , lunl)

Theorem 4.2.1.

211

Second Order Equations with Deviating Arguments where bE C(R+,R+), FE C(R+,R+), and F (iii) ifO < u; :5 v;, i = 1,2, ... ,n, then

and

1

oo

o

b(t)dt < oo,

1

> 0 when u; > 0,

ds = F(s,s, ... ,s)

00

0

1,2, ... ,n;

00.

Then Eq. (4.2.1) has an oscillatory solution.

Proof: First we show some details of the behavior of the solutions of Eq. ( 4.2.2). It is well known that condition (i) implies that every solution of (4.2.2) is oscillatory. Let Y1(t),y2(t) be solutions of (4.2.2) which satisfy the initial conditions Yl(O) = 1, yHO) = 0, Y2(0) = 0, y~(O) = 1. Then

I

Yl(t) Y2(t) y~(t) yHf)

1-=

1,

t

~

0.

(4.2.3)

Suppose y(t) is a solution of (4.2.2) and define

w1(t,y,y')

=y

12

+a(t)y2

2 1 ,, ') ( =a(t)y +y. w2t,y,y

Then along any solution of Eq. ( 4.2.2)

dw~t(t)

= a'(t)y2(t),

(4.2.4)

dw2(t) = _ a'(t) [ '( )] 2 a2(t) y t · dt If a'(t)

< 0' < 0' then .!!!!!1. dt -

-

.!!!!!2. dt

0 and so > ' -

+ a(t)y~(t) :5 [y~(oW + a(O)y~(O) = a(O) [y~(t)] 2 + a(t)y~(t) :5 [y~(OW + a(O)y~(O) = 1,

[y~(tW

(4.2.5)

Chapter 4

212 and

a~t) [y~(tW + y~(t) ~ a(10) [y~(oW + y~(o) = 1, (4.2.6)

a~t) [y~(tW + y~(t) ~ a!o) [y~(0)]2 + y~(O) = If a'(t) > 0 ' then we have .!!!!!1. > 0' dt -

[YWW + a(t)y~(t) ~

.!!!!!.2. < 0 ' and so dt -

[y~(oW

+ a(O)y~(O) =

+ a(t)y~(t) ~ [y~(oW + a(O)y~(o) =

[y~(tW

a!o).

a(O) (4.2.7) 1;

and

att) [y~ (t)j2 + y~(t) att) [y~(tW

~ a(~) [y~ (0)]2 + y~(O) =

+ y~(t) ~

a(10)

1,

[y~(oW + y~(O) = a!o) .

From (4.2.5) and ( 4.2.8) we have y2(t)< 1

-

a(O)

(k + 1).

lto

Since (1>( u) is increasing we have

k + 2Mi

1' to

b(s)H(s)ds $ k + 1.

(4.2.18)

From (4.2.16) and (4.2.18) we obtain lx(t)l $ k + 1 for t 0 $ t < T. On the other hand, for to-r$ t $to, lx(t)l = IPYl(to) + Y2(to)l :::; M1p + M1 = k $ k + 1.

215

Second Order Equations with Deviating Arguments

Then

lx(t)i ~ k + 1,

t0

ix(g;(t))i ~ k + 1,

-

to~

r ~

t < T,

and

t < T.

(4.2.19)

In view of (4.2.10), (4.2.13), (4.2.14) and (4.2.19) we have

lx'(t)i ~PlY~ (t)i

+

+ IY;(t)i

Jt iYI(s)y;(t)- Y2(s)y'(t)iih(s, x(gi(s)), ... ,x(gn(s)))ids to

~PM{+ M{ + 2MIM{

J'

b(s)F(ix(gi(s))i, ... , ix(gn(s))i)ds

to

~PM{+ M{ + 2M M{F(k + 1, ... , k + 1) 1

~PM{

= (P

1'

b(s)ds

to

+M'+M{

+ 2)M{,

t E [t 0 , T).

(4.2.20)

Now we claim that for the maximal existence interval of x(t), [to, T), we have T = oo. Otherwise, T < oo. Then rewrite ( 4.2.1) in the form

x'(t)

= z(t)

z'(t) = -a(t)x( t) - h (t, x(gi (t)), ... , x(gn(t))). Define Q = C([-r,O],R 2 ) and

f(t,r):=f(t,cp,'I/J)= (

llrll =

sup -r$8$0

ir(B)i, for

r = (cp,¢)

(4.2.21)

En.

Denote

¢(0) ) , -a(t)cp(O)- h(t,cp(gi(t)- t)), ... ,cp(gn(t)- t)

and u(t) = (

x(t)) z(t)

,

Ut

= u( t + 8).

Then ( 4.2.21) becomes u'(t)

= f(t,u 1 ).

(4.2.22)

Chapter

216

4

By (4.2.19) and (4.2.20) we have

llutll

$ k + 1 + (P + 2)M~,

(4.2.23)

t E [to,T)

and

I -a( t) 0 for u; > 0, i = 1,2, ... ,n, and

Proof: Let F( u1, ... , un)

lh(t, UI, ••. , Un)l =

I~ b;(t)u;- f(t)l

~ b(t) ( ~ lu;l + 1) =

lb(t)l F(lu1l, ... , luml).

Therefore, all assumptions of Theorem 4.2.1 are satisfied and hence Eq. (4.2.24) has an oscillatory solution. Example 4.2.1. Consider

X

t X (t "( t ) + ax (t ) + -2sin -t +1

where a > 0, T > 0. Since has an oscillatory solution.

f0

00

T

)= 0

(4.2.25)

I ;J~i ldt < oo, by Corollary 3.2.1. Eq. (4.2.25)

Chapter

218

4.2.2.

4

We consider the initial value problem of the equation (4.2.26)

y"(t) = p(t)y(g(t)) where p,g E with

C([t 0 ,oo),~),

y(s)

g(t) :5 t,

= 1. t-oo

(4.2.29)

}g(t)

Then the initial value problem for Eq. (4.2.26) with y(t) = cp(t) fort E Et 0 has a unique oscillatory solution. In fact, (4.2.28) implies that a = {3. (4.2.29) guarantees that the corresponding solution is oscillatory. Finding the conditions so that a < {3 is an open problem.

4.3. Sturm Comparison Theorems We consider the second order linear delay equations of the form (p(t)x'(t))'

+

t

q;(t)x(r;(t))

+

lt

k(s, t)x(s)ds = 0,

t E [a, b)

(4.3.1)

r(a)

i=l

and (p(t)y'(t))'

+ tQ;(t)y(r;(t)) +

1;a) K(s,t)y(s)ds

= 0,

t E [a, b)

(4.3.2)

where a < b :5 oo, p E C 1 ([a, b), (0, oo)), q;, Q;, K(s, t), k(s, t) are continuous functions over [a, b) and {(s,t): s :5 t, a :5 t < b}, respectively. Also, r;(t) :5 t, where r; is continuous, i = 1, ... , n, and r(t) = min{r;(s), s;;::: t, i = 1, ... ,n}.

(4.3.3)

For a given initial function cp E C[r(a), a], there exists a unique solution x(t) to Eq. (4.3.1) in [a, b) with x(t)

= cp(t)

t E [r(a),a],

and

x'(a+)

= cp'(a-).

(4.3.4)

Chapter 4

220

Let .,P(t) E C[r(a),a] be an initial function for Eq. (4.3.2), and y(t) be the corresponding solution to Eq. ( 4.3.2) with the initial condition given by .,P( t). For Eqs. (4.3.1) and (4.3.2) assume the following comparison conditions hold. (At) Q;(t) ~ lq;(t)l, ~

(A2) K(s,t)

= l, ... ,n,

i

s,t E [a, b),

lk(s,t)l, ~

(A3) .,P(t)f.,p(a)

t E [r(a),a].

l 0), which satisfies the modified delay equation

(p(t)y~(t))' + tQ;(t)ye(r;(t)) + i=l

1'

K(s,t)ye(s)ds +e = 0

(4.3.12)

r(a)

and the initial condition (4.3.13) From {4.3.12), we get the modified Riccati inequality

R~(t) > R~(t) + p(t)

t

i=l

lq;(t)l Ye(r;(t))

Ye(t)

+

1'

r(a)

lk(s,t)IYe(s) ds. Ye(t)

(4.3.14)

The general result then follows from the specialized result by letting e-+ 0. We shall show that under the hypothesis of strict inequalities, the conclusion (4.3.6) actually holds with a strict inequality sign. Suppose the contrary, let q >a be the first point at which R(q) = r(q). Then for all t E [a,q), R(t) > r(t). It follows that

x'(t) y'(t) y(t) < x(t)'

forall

tE[a,q).

(4.3.15)

Integrating it over the interval [s,t] C [a,q], we obtain

y(s)>lx(s)l x(t) y(t)

(4.3.16)

for all s < t E [a, o"]. Combining this with the third comparison condition, we see that (4.3.16) actually holds for all s < t E [r(a),q]. Hence the right hand side of (4.3.10) is not smaller than that of (4.3.8) for all t E [a, (7]. The various sign requirements in the comparison conditions are imposed to ensure that the product of the coefficient and the fraction in each of the terms of ( 4.3.8) and (4.3.9) do inherit the desirable comparison property. We have now concluded that R'(t) > r'(t) for all t E (a,q). In particular, R'(q) > r'(u). This contradicts the assumption that R(u) = r(u) and R(t) > r(t) fort < u and completes the 0 proof of the theorem.

223

Second Order Equations with Deviating Arguments Associate ( 4.3.2) with the delay equation n

(p(t)z'(t))'

+L

['

Q;(t)z(r;(t))

+ }.

K(s, t)z(s)ds = 0

(4.3.17)

T(a)

i=l

where

'f;(t)

~

r;(t)

~

t,

i

= 1, ... ,n.

(4.3.18)

We assume that the initial condition ( 4.3.19)

holds. Furthermore, assume that

Q;(t),

~

K(s, t)

(4.3.20)

0

and

.,P'(t) 0 eventually. Suppose that lim x'(t) = p. > 0 and hence x(t) "'p.t, t-+ oo. From ( 4.3.25) t-+oo

x"(t) "'p.t8(t). The convergence of x'(t) implies the integrability of x"(t). Hence

1

00

t8(t)dt
0,

y"(t) 50 on [T, oo).

227

Second Order Equations with Deviating Arguments Then for each k E (0, 1) there is a Tk 2: T such that

y(g(t));::: k

g~t)

t 2: Tk.

y(t),

( 4.4.2)

Proof: It suffices to consider only those t for which g(t) t > g(t) 2: T,

< t. Then we have for

y(t)- y(g(t))::; y'(y(t))(y- g(t)), by the mean value theorem and the monotone properties of y'. Hence

~ < 1 + y'(g(t)) y(g(t))

y(g(t)) -

(t- g(t))

'

t > g(t);::: T.

Also,

y(g(t)) 2: y(T) so that for any 0

+ y'(g(t)) (g(t)- T)

< k < 1 there is a Tk ;::: T

with

t _> Tk.

y(g(t)) > k (t) y'(g(t)) - g ' Hence, ~

< t + (k -1)g(t)

contradicts the fact that lim z(t) = -oo. Therefore, z'(t) > 0 fort ;::: T. There 1--+oo

are two possibilities for z(t) :

(a) z(t)>Ofort;:::T1 ;:::T, (b) z(t) < 0 fort;::: T1. For case (a), by Lemma 4.4.1, for each k E (0, 1) there is a Tk ;::: T such that

z(t- u(t));::: Since 0

k(t-u(t)) t z(t),

t;::: Tk.

< z(t) < x(t), from Eq. (4.4.1) we have z"(t) + p(t)f( k(t -tu(t)) z(t)) :50.

Using Theorem 7.4 in [86] we see that ( 4.4.3) has an eventually positive solution. This contradicts the assumption. For the case (b), as mentioned before, we lead to that lim x(t) = 0. 0 t--+oo

Theorem 4.4.1. that

1 t

limsup t--+oo

In addition to the conditions of Lemma 4.4.2, assume further

if lim J(yl = 1 (u-(t-u(t)+r))q(u)du >

{

t-cr(t)+r

p,

O,

y--+0

(4.4.4)

if lim !JJtl y--.0


y

y

= oo.

Second Order Equation3 with Deviating Arguments

229

Proof: As in the proof of Lemma 4.4.2, it is sufficient to show that z(t) < 0 for t :;:: Tis impossible under the assumptions. Suppose that x(t) > 0, z"(t) ::; 0, z'(t) > 0, and z(t) < 0 eventually. Then

z(t- u(t)

+-r) > -px(t- u(t)).

Substituting this into Eq. (4.4.1) we have

z"(t)- q(t) f(z(t- u(t) + r)) ::; 0. p

Integrating (4.4.5) from s to t for t

11'

z'(t)- z'(s)--

p •

(4.4.5)

> s we have q(u)f(z(u- u(u) + r))du::; 0.

(4.4.6)

Integrating (4.4.6) ins from t- u(t) + r tot, we have

z'(t)(u(t)- r)- ['

dz(s)

lt-tT(t)+r

11'

(u- (t- u(t) + r)]q(u)f(z(u- u(u) + r))du::; 0.

--

t-tT(t)+r

P

Hence for t sufficiently large

z(t- u(t) + r)- z(t) 11t --

P t-17(t)+r

[u- (t- u(t) + r)]q(u)f(z(u- u(u) + r))du::; 0.

Dividing the above inequality by z(t- u( t) + r) and noting the negativity of this term, we have

1_

z(t) z(t-u(t)+r)

(4.4.7)

- pz (t - 0'1(t) + T ) We note that z(t)

lt

0, z(t) > 0,

(b) z'(t) < 0, z(t) < 0,

(c) z'(t)

> 0, z(t) < 0

eventually. (a) Assume z'(t) > 0, z(t) > 0, t ~ t 0 ~ 0. Then (4.4.10) holds and for any t E Rr0 = {t; t + u(t) ~To}, there exists a positive integer n such that T0 $ t- u(t)- nr iz(t- u(t)- ir), i=O

from Eq. (4.4.1) we have

+ pnx(t- u(t)- nr)

233


In view of (4.4.10) we have

z"(t) + q(t)f (

tk ~p'(t- u(t)- ir)z(t) n

.

)

50,

i.e., ) kr n n+l k z"(t) + q(t)f [ ( -(t- u(t)) p _ - - L pi z(t) 50. t i=l p 1 t ]

(4.4.14)

Since

We have k

p"+l - 1

-(t- u(t)) p -1 t

kr

n

.

--Lip' t i=l

= (p-k1)2t [(t- u(t))(p"+2- pn+l- p + 1)- r(npn+2- (n + 1)pn+l + p)] k

= (p _ 1)2 t [(t- u(t)- nr)p"+2- (t- u(t)- (n + 1)r)p"+ 1

- (t-u(t)+r)p+ (t-u(t))] k

~ (p _ 1)2t[Topn+2- Topn+l- (t- u(t)

+ r)p + (t- u(t))] (4.4.15)

for some A E (0, 1) if T0 and t are sufficiently large. Substituting (4.4.15) into (4.4.14) we have

z"(t) + q(t)f ( tA p!=..!..ill z(t) ) 50. T

Noting that z(t), z(To) are upper and lower solutions of (4.4.12) and by a known result in [86] we see there is a solution y(t) of (4.4.12) satisfying z(T0 ) 5 y(t) 5 z(t), contradicting the fact that Eq. (4.4.12) is oscillatory for all 0 0 eventually. For otherwise, there exists a fl. > 0 such that z'(t) ~ -fJ,, t ~ T2 • On the other hand, x(t- r) ~ -~ z(t), thus

z"(t)

+ q(t)f(- ~z(t- a(t) + r))

$0.

Integrating it from T2 to t we get

z'(t) + Noting that z(t- a(t)

£.

q(u)f(-

+ r)

~z(u- a(u) + r))du $0.

$ -f.(t- a(t)

£. q(u)f(~(u!(~) £. z'(t) +

+ r)

a(u)

q(u)f(u- a(u)

we have

+ r))du

+ r)du

$ 0,

$ -z'(t) $ fl,

(4.4.17)

235

Second Order Equations with Deviating Arguments which contradicts ( 4.4.16). Therefore, z' ( t) ( 4.4.17) there is a T,. such that

On Ec

< - J.l for all

J.l

> 0 eventually. From

n [T,., oo),

Thus

J(!!:_)p;tfr [' (t- J.L)q(u)f(u- u(u) + r)du::::; 1, P

lr~

and Ptfr

~~ (t- u)q(u)f(u- u(u) + r)du::::; !(\)'

which contradicts ( 4.4.13) since J.l (c) Assume z'(t) > 0, z(t) obvious.

( 4.4.18)

> 0 is arbitrary and f( oo) = oo. < 0, t ~ t 0 ~ 0. Then x(t) < px(t- r) is 0

Corollary 4.4.1. In addition to the assumptions of Lemma 4.4.3, further assume that 0 and z(t) < 0, t-+oo

t ~ t 0 ~ 0, hence z'(t) ~ 0, z(t) ~ 0 as t ~ oo. If liminf x(t) > 0, then t-+oo x(t) ~a> 0, t ~ t 1 ~to. Integrating Eq. (4.4.1) twice we get z(t)+

1

00

(u-t)q(u)f(a)du

contradicting ( 4.4.19). Thus

> 0 and liminf x(t) = 0.

lim sup x(t) t-+oo

Then we can choose tz

> t1

t-+oo

such that x(tz- u) > x(t1- u). We claim that

~to

liminf x(t 2 - u + nr) > 0.

(4.4.20)

n-+oo

In fact n

x(ti- u + nr) = L:z(tj- u + ir) + x(tj- u),

j = 1,2.

i=l

Since z(tz- u

+ ir) ~

z(t 1 - u

+ ir) fori=

liminf x(t 1 n-+oo

-

u

1, 2, ... , n, and

+ nr) ~ 0.

We have liminf x(tz- u n-+oo

Now, choose to :::,; t1

+ nr) ~ x(tz- u)- x(t1

- u) > 0.

< tz < t3 such that for any T E [tz, t 3],

x(tt- u) < x(tz- u):::,; x(T- u). From the above discussion, we see that (4.4.20) holds, i.e., there exists a p. such that x( t 2 - u + nr) ~ p. for all n. It is easy to see that for T E [t 2 , t 3 ], n

x(T-u+nr)

= Lz(T-u+ir)+x(T-u) i=l n

~ L:z(t 2 -u+ir)+x(t 2 -u) i=l

=x(tz-u+m)~p..

>0

237

Second Order Equations with Deviating Arguments From {4.4.1) we have

-z'(s) + z(t 0 )

+

J.t q(u)f(x(u- a))du ~ 0,

f\u- t 0 )q(u)f(x(u- a))du

lto

to~ s ~ t,

~ 0,

t

~to.

{4.4.21)

Hence

z(to)

+ f(J.L)

t

ita~ir (u- to)q(u)du ~ 0, t•+•r

i=O

and then

contradicting {4.4.19).

D

Corollary 4.4.2. In addition to the assumptions of Lemma 4.4.4, assume further that a is a positive constant,

1

00

(4.4.22)

(u- t)q(u)du = oo

and oo

rT+ir+or

L f(pi) } T+•r i=O

.

1

hold for any T E R+ and 0 < a (4.4.1) tend to zero as t--+ oo.

~ T.

(u- T)q(u)du

= oo

( 4.4.23)

Then all nonoscillatory solutions of Eq.

Proof: If not, similar to the proof of Corollary 4.4.1 we see that there exists an eventually positive solution x(t) satisfying lim sup x(t) t-+oo

> 0 and liminf x(t) t-+oo

= 0.

From the proof of Lemma 4.4.4 this can only occur when z"(t) ~ 0, z'(t) > 0, and z(t) < 0, t ~ t 0 . Choose t2 > t 1 ~to such that x(t2- a) > x(t 1 -a).

238

Chapter

4

Since n

x(t2- 17 + nr)

=LPn-iz(t2- 17 + ir) + p"x(t2- 17), i=l n

x(t1- 17

+ nr) =

LPn-iz(tl- 17

+ ir) + pnx(t1

- 17),

i=l

z(t2and x(t 1

-17

17

+ ir) ~ z(t1- 17 + ir),

+ nr) > 0, x(t 2 -

17

n

i

= 1,2, ... ,n,

= 0, 1, ... , we see

+ nr) ~ p"(x(t 2 - 17)- x(t 1 - 17))

= Ap".

Similar to the proof of Corollary 4.4.1, we can show that there is an interval [t2, t3] such that

x(T-

17

+ nt) ~

Apn

forTE [t2, t 3 ] and all n. From ( 4.4.21) we get

z(to)

+ f(A)

t lta+ir i=O

(u- t2)q(u)j(pn)du

: 1, respectively, we assume (4.4.4) holds. Then Eq. (4.4.1) is oscillatory. Proof: If not, let x(t) be an eventually positive solution of Eq. (4.4.1). Then we have z'(t) > 0, z(t) < 0 eventually, where z(t)

= x(t)- px(t- r).

The following is similar to the proof of Theorem 4.4.1. We omit it here.

0

239

Second Order Equation& with Deviating Arguments 4.4.4. Further discussion for the linear case Now we pay attention to the following linear equation

(x(t)

+ px(t- r))" + q(t)x(t- u) = 0.

(4.4.24)

Assume that p;::: 0 and q E C(R+,R+)· Let x(t) be an Lemma 4.4.5. of ( 4.4.24). Set solution positive eventually y(t) = x(t) + px(t- r), and

z(t)

= y(t) + py(t- r).

(4.4.25)

Then z(t) > 0, z'(t) > 0 and z"(t) $ 0 and

z"(t) + rit; z(t- u) $ 0

(4.4.26)

eventually, where q*(t) =min {q(t),q(t- r)}. Proof: Suppose x(t) > 0 fort 2:: t 0 • Then y(t) > 0 fort;::: to+ r, and y"(t) fort 2:: t1 =to +max {u, T }. Therefore y'(t) > 0 fort ;::: t 1. Then

0, q E C(R+, R+)· Then every solution of Eq. (4.4.24) is oscillatory iffor some a E (0, 1)

1

00

t"'q(t)dt =co.

(4.4.32)


241

Then we turn to the equation with variable p as follows:

(x(t)

+ p(t)x(t- r))" + q(t)x(t- u) =

Theorem 4.4.4.

0,

t 2:: to.

(4.4.33)

Assume that

(i) T > 0, 0' > 0; (ii) q E C(R+, R+) and q(t) 2:: qo > 0; (iii) p E C(R+, R) and there exist constants Pl and P2 such that Pl :5 p(t) :5 P2, and p(t) is not eventually negative. Then every solution of Eq. (4.4.33) is oscillatory. Proof: If not, let x(t) be an eventually positive solution. Set

z(t) = x(t) + p(t)x(t- r). It is not difficult to show that z(t)

( 4.4.34)

< 0 eventually, which contradicts (iii).

0


(x(t)+

(~+sint)x(t-211'))" + (~+sint)x(t-411')=0,

t;:=:O. (4.4.35)

It is easy to see that the assumptions of Theorem 4.4.4 are satisfied here. Therefore, every solution of Eq. ( 4.4.35) oscillates. For example, x( t) = l!~:i! 1 is such 2 a solution.

The following example shows that, if the hypothesis (ii) of Theorem 4.4.4 is violated, the result may be wrong.

Example 4.4.2. Consider the equation (x(t) + (t -1)_ 1 , 2 x(t -1))" +

~c

3 12 (t-

2)_ 1 , 2 x(t- 2) = 0,

t;::: 2. (4.4.36)

All assumptions of Theorem 4.4.4, except (ii) are satisfied. Note, however, that x(t) = t 112 is a nonoscillatory solution.

Chapter 4

242

4.5. Classification of Nonoscillatory Solutions We consider the second order nonlinear neutral differential equations of the form

(

m x(t)- t;p;(t)x(tr;) ) " n

+ Lfi(t,x(gji(t)), ... ,x(gjt(t))) =0,

t?_t 0 •

(4.5.1)

i=l

In this section we assume that (i)

T;

> 0, p;

E C([t 0 ,oo),R+), i = 1, ... ,m and there exists 8 E (0,1) such

m

that 'L:p;(t) $ 1- 8, t?. to; i=l

(ii) 9i• E C([to, oo ), R), lim 9is(t) = oo, j t-oo

= 1, ... , n,

s = 1, ... , I!;

(iii) fiE C([ta,oo) x R 1,R), Ytfi(t,yt, ... ,yt) > 0 for YIYi > 0, i = 1, ... ,£, j = 1, ... , n. Moreover

whenever Set

IY;i

$ lx;i and x;y; > 0, i

= 1, ... ,l, j = 1, ... , n.

m

y(t) = x(t)- LP;(t)x(t- r;).

(4.5.2)

i=l

We shall give the classification of nonoscillatory solutions of Eq. ( 4.5.1). First we show some lemmas which will be useful for the main results. Lemma 4.5.1. Let x(t) be an eventually positive (or negative) solution of Eq. (4.5.1). If lim x(t) = 0, then y(t) is eventually negative (or positive) and t-oo

lim y(t) = 0. Otherwise, y(t) is eventually positive (or negative).

t-oo

Proof: Let x(t) be an eventually positive solution of Eq. (4.5.1). From (4.5.1), y"(t) < 0 eventually. Thus y'(t) is decreasing and y'(t) > 0 or y'(t) < 0 eventually. Also y(t) > 0 or y(t) < 0 eventually. If lim x(t) = 0, from (4.5.2) we have lim y(t) t-+oo

plies that y'(t)

= 0.

t-oo

Since y(t) is monotonic, so lim y'(t) t-+oo

=

0 which im-

> 0. Therefore, y(t) < 0 eventually. If lim x(t) = 0 fail, then t-oo

243


limsup x(t) > 0. We show that y(t) > 0 eventually. If not, then y(t) < 0 t-+oo

eventually. If x(t) is unbounded, then there exists a sequence {tn} such that lim tn = oo, x(tn) = max x(t), and lim x(tn) = oo. From (4.5.2), we have n-oo

t 0 ::;_t~tn

n--..oo

y(tn) = x(tn)- tp;(tn)x(tn- r;)

~ x(tn)(1- tp;(tn)).

i=l

(4.5.3)

•=1

Thus lim y(tn) = oo, a contradiction. If x(t) is bounded, then there exists a n-+oo sequence {tn} such that lim tn = oo, and lim x(tn) = limsup x(t). Since n-oo

n--+oo

t-oo

sequences {p;(tn)} and {x(tn- r;)} are bounded, there exist convergent subsequences. Without loss of generality we may assume that lim x(tn - r;) and n-+oo lim p;(tn), i = 1, ... , m, exist. Hence n-+oo

0

~ n-oo lim y(tn) =

lim (x(tn)-

n-oo

~ p;(tn)x(tn- r;)) L-J i=l

a contradiction again. Therefore, y(t) > 0 eventually. A similar proof can be given if x(t) < 0 eventually. D m

Lemma 4.5.2.

Assume that lim LP;(t) = p E (0, 1), and x(t) is an event-ex:> i=l

tually positive (or negative) solution of Eq. (4.5.1). If lim y(t) = a E R, then t-+oo lim x(t) = 1 ~P. If lim y(t) = oo (or - oo), then lim x(t) = oo (or - oo). t--+oo t-+oo t-oo

Proof: Let x(t) be an eventually positive solution ofEq. (4.5.1), then x(t) ~ y(t) eventually. If lim y(t) = oo, then lim x(t) = oo. Now we consider the case that t--+oo

t--+oo

lim y(t) = a E R. Thus y(t) is bounded which implies that x(t) is bounded t-+oo (see (4.5.3)). Therefore, there exists a sequence {tn} such that lim tn = oo n-+oo and limn-+oo x(tn) =lim sup x(t). As before, without loss of generality, we may t-+oo

Chapter

244

4

assume that lim p;(tn) and lim x(tn - T;), i = 1, ... , n, exist. Hence n-+oo

n-+oo

m

lim p;(tn) lim x(tn- T;) lim x(tn)-'""' n-co = n-+oo n-+oo L..J i=l

~lim sup

x(t)(1- p)

t-oo

i.e. -1 a ~lim sup x(t). t-oo - P

(4.5.4)

lim x(t~) = liminf x(t). On the other hand, there exists {t~} such that R-t>OO t-400 lim p;(t~) and lim x(t~- T;), '= Without loss of generality, we assume that n-+oo n-+oo 1, ... , m, exist. Hence a= lim y(t~) n-oo

m

lim x(t~)-'""' = n-+oo LJ i=l

lim p;(t~) lim x(t~- t;) n-+oo

n-+oo

:$liminf x(t)(1- p) t-+oo

or -1 a :$liminf x(t). t-+oo - p

(4.5.5)

lim x(t) = 1 ~ • A similar proof can be Combining (4.5.4) and (4.5.5) we obtain t-+oo P 0 given if x(t) < 0. We are now ready to prove the following results. m

Theorem 4.5.1.

Assume that lim L:P;(t) = p E [0, 1). Let x(t) be a t-+00 i=l

nonoscillatozy solution of Eq. (4.5.1). Let S denote the set of all nonoscillatory solutions of Eq. (4.5.1), and define lim y(t) = 0, lim y'(t) = 0}, S(O,O,O) ={xES: lim x(t) = 0, t-+oo t--+oo t-+oo

245

Second Order Equation& with Deviating Arguments

S(b,a,O) ={xES: lim x(t) = b :=_a_, lim y(t) =a, lim y'(t) = 0}, 1- p

t--+oo

y--+oo

t-+oo

lim y(t) = oo, lim y'(t) = 0}, S(oo,oo,O) ={xES: lim x(t) = oo, t-oo t-+oo t-+oo

lim y(t) = oo, lim y'(t) = d S(oo,oo,d) ={xES: lim x(t) = oo, t-+oo t-+oo t-+oo

:f. 0}.

Then S = S(O, 0, 0)

u S( b, a, 0) U S( oo, oo, 0) U S( oo, oo, d).

Proof: Without loss of generality, let x(t) be an eventually positive solution. lim y'(t) = 0, i.e., x E If lim x(t) = 0, by Lemma 4.5.1, lim y(t) = 0 and t--+oo t__,.oo t_..oo S(O,O,O). If limx(t) = 0, fails, by Lemma 4.5.1, y(t) > 0 eventually, it is easy t-+00

to see that y'(t) > 0, y 11 (t) < 0 eventually. If lim y(t) = a > 0 exists. Then t-+oo limx(t) = 1 ~ P = b, i.e., x E S(b,a,O). limy'(t) = 0, by Lemma 4.5.2, we have t-+oo t--+oo If lim y(t) = oo, by Lemma 4.5.2, lim x(t) = oo. Since y"(t) < 0 and y'(t) > 0, t-+oo

t-+oo

we have limy'(t) = d, where d = 0 or d t--+oo x E S(oo, oo, d).

> 0. Then either s E S(oo,oo,O), or 0

In the following we shall show some existence results for each kind of nonoscillatory solutions of Eq. (4.5.1).

Assume there exist two constants k 1 > k2 > 0 such that

Theorem 4.5.2.

m

m

LP;(t)exp(k1r;) > 1 ~ LP;(t)exp(k 2 r;), ~1

(4.5.6)

~1

and

... , exp( -k2gjt(u)))du eventually. Then Eq. (4.5.1) has a solution x E S(O,O,O).

(4.5.7)

Chapter

246

4

Proof: Let BC be the space of all bounded continuous functions on [to, oo) with the sup norm 1\xl\ = suplx(t)l. Define t~to

n=

{x E BC: exp(-k 1 t) $ x(t) $ exp(-k2t) and

lx(t2)-x(ti)I$Lit2-tii,

tht22::to},

where L 2:: k1 • Then f! is a nonempty, closed convex bounded subset of BC. For the sake of convenience, denote n

f(u,x(g(u))) = Lfi(u,x(gji(u)), ... ,x(gjt(u))), j=l (4.5.8) n

f( u,exp( -k2g(u))) =

L

fi( u, exp( -k29jl (u)), ... , exp( -k29jt( u))).

j=l Define a mapping T on

n as follows:

EP;(t)x(t- r;)- j 100 (u- t)f(u, x(g(u)))du, (Tx)(t) = {

( 4.5.9)

i=l

exp(- K(x)t), where K(x) = to,

ln(T;)(T) ,

t 0 $ t < T,

T is a sufficiently large number so that t - r; 2::

9i•(t)2::to, i=1, ... ,m, j=1, ... ,n, s=1, ... ,1!,fort2::T.

Condition (4.5.7) implies that

£

00

f(u,exp(-k2g(u)))du < oo.

Assumption (i) at the beginning of this section implies that for given a E (1-h, 1)

(4.5.10) Therefore, T can be chosen so large that for t 2:: T

(4.5.11)

247


and m

a+ l::exp(- k2 (t- r;)) ~ 1. i=l

Hence by (4.5.6) and (4.5.7) m

('Tx)(t) ~ LP;(t)x(t- r;) i=l

m

~ LP;(t)exp(- k2(t- r;)) i=l

= ( ~

~p;(t)exp(k2r1 ))exp(-k2t)

exp( -k2t),

for

t 2:: T,

and

('T x)(t) 2::

L p;(t)exp(- k (t- r;)) - loo (u- t)f(u,exp( -k g(u)))du m

1

2

~1

t

= exp( -k1 t) + ( ~p;(t)exp(k 1 r;) -1 )exp( -ktt)

-1

00

(u- t)/(u,exp( -k2 g(u)))du

2:: exp( -k 1 t) for t 2:: T, i.e., exp(-k 1 t)

~

('Tx)(t)

~

By the definition of K(x), and exp(-k 1 T) that k2 ~ K(x) ~ k1. Hence

exp( -k1 t)

~

(Tx)(t)

~

t 2:: T.

exp(-k2t), ~

('Tx)(T)

exp( -k2 t),

t0

~

~

exp(-k 2 T), we know

t < T.

Chapter

248

4

Now we show that (4.5.12)

Without loss of generality assume t2 and (4.5.12), we have

~

tt

~to.

~

For t2

tt

~

T, using (4.5.11)

I(Tx)(tt)- (Tx)(t2)l m

:5 L lp;(tt)x(tt - r;)- p;(t2)x(t2- r;)l i=t

m

:5 L (p;(t2)lx(t2- r;)- x(t1- r;)l + IP;(t2)- p;(tl)lx(tl- r;)) i=l

+

11:

2

(u- tt)f(u,x(g(u)))du +

1~ (t

2 -

tt)f(u,x(g(u)))dul

:5 [ ~ (p;(t2) + exp( -k2(t- r;)))L

+ 1.0 /(u,exp(-k2g(u)))du]lt2 -td :5 { [ tp;(t2) +

t

exp(- k2(t1 - r;)}]

+ (a-~ p;(t2)) }Lit2 -

= [~ exp(- k2(h- r;)) +a] Llt2- td :5 Llt2 -ttl· For to :5 tt :5 t2 :5 T, we have I(Tx)(t2)- (Tx)(tt)l = jexp(- K(x)(t2))- exp(- K(x)(tt))j

:5 IK(x)(t2)- K(x)(tt)l :5 Llt2 -ttl·

ttl

Second Order

Equation~

249

with Deviating Argv.menu

For to < t1 :::; T :::; t2, we have I(Tx)(t2)- (Tx)(ti)I :5 I(Tx)(t2)- (Tx)(T)I + I(Tx)(T)- (Tx)(t2)l :::; Llt2 - Tl +LIT - til = Llt2- t1l·

We have proved that (4.5.12) holds for all t 0 :::; t 1 :5 t2. Therefore, Tfl ~fl. Clearly, T is continuous. Since Tfl ~ fl, Tfl is uniformly bounded. Set X En, we see that I(Tx)(t)l :5 exp( -k2t) I(Tx)(t2)-(Tx)(t1)l :5Lit2-tll,

to :5tl :::;t2.

For given e: > 0 there exists aT'> to such that exp(-k2t) 0, a> 0.

250

Chapter

4

Using the notation (4.5.8), from (4.5.1) and (4.5.2) we have

y"(t)

= -f(t,x(g(t)).

Integrating it from s to oo for s 2: to we have

1

00

y'(s) =

f(u,x(g(u)))du.

Integrating it from T to t for T sufficiently large, we have

y(t) = y(T) + J;cu- T)f(u,x(g(u)))du

+ Since lim x(gjh(u)) u-oo

= b > 0,

j

ioo

(t- T)f(u,x(g(u)))du.

= 1, ... , n,

h

( 4.5.16)

= 1, ... ,£,there exists aT 2: t 0

such that x(gjh(u)) 2:! for u 2: T. Hence from (4.5.16) we have

which implies that (4.5.15) holds. Sufficiency. Set b1 > 0 and A > 0 so that A < (1- p )b 1. From ( 4.5.15) there exists a sufficiently large T so that fort 2: T we have t- r; 2: t 0 , i = 1, ... , m, and gjh(t) 2: t 0 , j = 1, ... , n, h = 1, ... , £, and

( 4.5.17) Define n to be the set of all continuous functions x( t) on [to, oo) such that 0 :5 x(t):::; b1 , t 2: t 0 , and define a mapping Tin as follows:

n

m

A+ ;;p;(t)x(t- r;)

t

+ JTuf(u,x(g(u)))du

{

(Tx)(t)

=

+ f1

00

(Tx)(T),

tf(u,x(g(u)))du, t 0 $t 0 fort?. t 0 • It is easy to check that x( t) = y(t)e-t is a solution of Eq. (4.5.27). D Corollary 4.5.1. Assume that 0 q*, r > r such that eventually

 0, and there exist constants

g(t)?.t-r.

If the "majorant" equation

(x(t)- px(t- r))"

+ q*x(t- r) =

0,

t?. t 0

has a positive solution, then Eq. (4.5.27) also has a positive solution. Proof: ( 4.5.30) has a positive solution if and only if

has a real root a. Clearly a must be negative. Let p.

or

= -a > 0, then

(4.5.30)

255

Second Order Equations with Deviating Arguments Hence

1 +-11"" -e-pr p

p

(s- t- r)q(s)exp[Jl(t- g(s))]ds

t+r

< ~e-pr + Lep(r-r) PJl2

- p

=1

'

i.e., (4.5.28) holds. Then, by Theorem 4.5.5, Eq. (4.5.27) has a positive solution.

4.6. Unstable Type Equations 4.6.5. Equations with constant p

Consider the second order linear neutral differential equation of the form (x(t)- px(t-

r))"

= q(t)x(g(t)),

t ~to,

(4.6.1)

limg(t) = oo, r > 0. where p E R,q E C([t 0 ,oo),R+), g E C([t 0 ,oo),R), t-+oo We will show later that Eq. ( 4.6.1) always has an unbounded, nonoscillatory solution. Therefore, we only need to find conditions for all bounded solutions of Eq. (4.6.1) to be oscillatory. Theorem 4.6.1. Assume that (i) 0 0 are constants; (ii) g(t) :5 t and g is nondecreasing fort~ t 0 ; (iii)

lim sup t-+oo

f91(t) (s- g(t))q(s)ds > 1.

( 4.6.2)

Then every bounded solution of Eq. (4.6.1) is oscillatozy. Proof: Assume the contrary, and let x(t) be an eventually positive bounded solution of Eq. (4.6.1). Define z(t) = x(t)- px(t- r).

(4.6.3)

lim z(t) = oo, > 0 fort~ T1 > T, then t-+oo which contradicts the boundedness of x. Therefore, z'(t) :5 0 fort~ T.

We have z"(t)

> 0 fort~ T

~ t 0 • If z'(t)

There are two possibilities for z( t) :

256

Chapter 4

(a) z(t) > 0

fort~

T; (b) z(t) < 0 fort~ T2

~

T.

In case (a), integrating (4.6.1) from 8 tot we have

z'(t)- z 1(8)

=it

q(u)x(g(u))du.

(4.6.4)

Integrating (4.6.4) in 8 from g(t) tot, we have

z'(t)(t- g(t))- z(t) + z(g(t))

=

r 1.t q(u)x(g(u))du d8

}g(t)

=

r

(8- g(t))q(s)x(g(8))ds

}g(t)

> [' (s- g(t))q(s)z(g(8))ds }g(t)

~ z(g(t))

r

(s- g(t))q(s)ds.

}g(t)

Hence for t

~

T

z(t) + z(g(t)) (

r

}g(t)

(s- g(t))q(s)ds -1) :50

which contradicts the positivity of z(t) and (4.6.2). In case (b), we have

x(t) < px(t- r) < p2 x(t- 2r) < · · · < pnx(t- nr) fort ~ T2 + nr, which implies that lim x(t) = 0. Consequently, lim z(t) = 0, a t-oo t--+oo contradiction . 0

Remark 4.6.1. Theorem 4.6.1 is also true for p = 0. Theorem 4.6.2.

Assume that

(i) p < 0 and q(t) > 0, t ~to; (ii) g(t) = t- u, where u is a constant, u > r;

Second Order

Equation~

with Deviating

Argument~

257

(iii) There exists o: > 0 such that (4.6.5)

lim sup q(t)fq(t- r) = o: t-+oo

and (s- (t- (u- r)))q(s)ds > 1- o:p.

limsupit t-+oo

(4.6.6)

t-(u-r)

Then every bounded solution of (4.6.1) is oscillatory. Proof: Assume the contrary, and let x(t) be a bounded, eventually positive solution of Eq. (4.6.1) and z(t) be defined as in (4.6.3). As shown before, z"(t) > 0, z'(t) < 0, and z(t) > 0 eventually, where z(t) is defined by (4.6.3). From (4.6.5) and (4.6.6), there exists a constant k > 1 such that

lim sup t-+oo

it

(s- (t- (u- r)))q(s)ds > 1- ko:p,

(4.6.7)

t-(u-r)

q(t)fq(t- r) < ko:,

t

~ t1

(4.6.8)

where t1 is a sufficiently large number. We rewrite Eq. (4.6.1) in the form z"(t)- p (q(t) )z"(t- r) = q(t)z(t- u). q t- T

(4.6.9)

Substituting ( 4.6.8) into (4.6.9) we have z"(t)- ko:pz"(t- r) ~ q(t)z(t- u),

t ~ t 1•

(4.6.10)

Set w(t) = z(t)- ko:pz(t- r).

(4.6.11)

Then w"(t) ~ q(t)z(t- u) > 0,

t ~ t 1•

(4.6.12)

By the boundedness of x(t), it is easy to see that w(t) > 0, t2 ~ t 1 • Since z( t) is decreasing, w(t) = z(t)- ko:pz(t- r) ~

(1- ko:p)z(t- r),

w'(t) ~ 0,

t ~ t 2.

t ~

(4.6.13)

Chapter

258

4

Combining (4.6.12) and (4.6.13) we have 1

w"(t) ~ 1 k q(t)w(t- (u- r)). - ap Integrating (4.6.14) from

8

tot

fort~ 8 ~

t2 we have

f.'

w'(t)- w1(8) ~ 1 _ 1kap • q(u)w(u- (u- r))du. Integrating (4.6.15) in

8

(4.6.14)

(4.6.15)

from t- (u- r) tot we have

w'(t)(u- r)- w(t) + w(t- (u- r)) ~

=

1-

1-

1k

1' f.' 1' 1'

ap t-(a-T) ~

1k

(u- (t- (u- r)))q(u)w(u- (u- r))du

ap t-(a-T)

r)) k > w(t- (u1-

ap

q(u)w(u- (u- r))duds

(u- (t- (u- r)))q(u)du,

t ~ t2•

t-(a-r)

Thus

w(t) + w(t- (u- r)) [

1-

1'

1k

ap t-(a-r)

(u- (t- (u- r)))q(u)du- 1]

~0 D

which contradicts (4.6. 7). Theorem 4.6.3. Assume that (i) p = 1, r > 0; (ii) g(t) ~ t, and g is nondecreasing fort~ t 0 ; (iii) either

1

00

tq(t)dt

=

00

(4.6.16)

to

or lim t

t-+oo

1

00

t

q( s )ds = oo.

(4.6.17)

259

Second Order Equations with Deviating Arguments Then every bounded solution of Eq. (4.6.1) is oscillatory.

Proof: Assume the contrary and let x(t) be a bounded eventually positive solution of Eq. ( 4.6.1) and z(t) be defined as in ( 4.6.3). There are two possibilities: (a) z"(t) ~ 0, z'(t) ~ 0, z(t) < 0 fort~ t 1 ~ t 0 , (b) z"(t) ~ 0, z'(t) ~ 0 and z(t) > 0, t ~ t 1 ~to. In case (a), there exists a finite number a > 0 such that lim z(t) t-oo Thus there exists t2 ~ t1 such that -a < z(t) < - ~ , t ~ t2, i.e.,

-a< x(t)- x(t- T) < -

2a ,

t

= -a:.

~ t2.

Hence x(t- T) > ~. t ~ t2, then there exists t 3 ~ t2 such that x(g(t)) > ~. ~ t 3 • From Eq. (4.6.1), we have

t

z"(t) ~ ~ q(t),

t ~ t3.

x(t)>x(t-T),

t~t1

(4.6.18)

In case (b), we have

then there exists M > 0 such that x(t)

~

M,

z"(t) ~ Mq(t),

t

~ t1•

Hence

t ~ t3•

(4.6.19)

In both cases we are lead to the same inequality (4.6.19). (4.6.19) from t toT forT> t ~ t 3 we have

z'(T)- z'(t)

~MiT q(s)ds,

t3 $

t < T.

Hence -z'(t)

which implies that

~MiT q(s)ds,

t3 $ t < T,

ft'; q( s )ds < oo and so -z'(t)

~M

1""

q(s)ds.

Integrating

260

Chapter

4

Integrating the above inequality from t to T for T > t we have

z(t)

~ z(T) +MiT J.oo q(u)duds = z(T)

+M [

lT(u-

t)q(u)du + (T- t)

!roo q(u)du],

t

~ tz

which leads to a contradiction to the boundedness of z in either case of (4.6.16) and (4.6.17). 0 Example 4.5.1. Consider

211" ( x(t)- x(t- 211") ) , == --x(t11"). t -11"

(4.6.20)

It is easy to see that all assumptions of Theorem 4.6.3 are satisfied. Therefore, every bounded solution of Eq. ( 4.6.20) is oscillatory. Eq. ( 4.6.1) may have unbounded oscillatory solutions. For example, (4.6.20) has a solution x( t) = t sin t.

Theorem 4.6.4. Assume that p > 1. Then Eq. ( 4.6.1) has a bounded positive solution if and only if

roo tq(t)dt < oo.

(4.6.21)

ito

Proof: Necessity. Let x(t) be a bounded positive solution of Eq. (4.6.1) and z(t) be defined by (4.6.3). Then Eq. (4.6.1) becomes

z"(t) = q(t)x(g(t)). There are only two possibilities for z(t) :

(a) z"(t)

~

(b) z"(t)

~

0, 0,

z'(t) :::; 0, z'(t) :::; 0,

z(t) < 0, z(t) > 0,

t t

~ ~

t1 t1

~to; ~to.

As in the proof of Theorem 4.6.3, in either case we are lead to the inequality

z 11 (t) ~ Mq(t),

for

t

~

tz,

261


where M > 0 is a constant and t 2 is a sufficiently large number. Integrating it twice we have

~MiT J.T q(u)duds

z(t)- z(T)

T >t

= M iT(u- t)q(u)du,

~ t2.

Letting T-+ oo, we see that (4.6.21) holds. The sufficiency part of Theorem 4.6.4 follows from the following more gen0 eral result. Theorem 4.6.5.

Assume p > 1 and q E C([to, oo), R) such that

1

00

sjq(s)jds < oo.

(4.6.22)

to

Then Eq. (4.6.1) bas a bounded positive solution. Proof: Let T ~ to be sufficiently large so that t t ~ T, and

1

00

p-1

sjq(s)jds $ - - , 4 t+r

t

+r ~

~to,

g(t

+ r)

T.

~ t0

for

(4.6.23)

Consider the Banach space BC of all continuous bounded functions defined on [to, oo) with the sup norm. Set

n={xeBC:

~$x(t)$2p,t~t 0 }.

Clearly, n is a bounded closed convex subset of BC. Define a map T: BC as follows:

(Tx)(t) =

{

n-+

p-1+~x(t+r)

- ~ ft";r(s- t- r)q(s)x(g(s))ds,

(Tx)(T),

t

~

T

to$ t $ T.

(4.6.24)

262

Chapter

For any

X

4

E n, from (4.6.23) we have

(Tx)(t)

~ p

1100

(s- t- r)jq(s)jjx(g(s))jds < 2p,

for

t :2: T,

12--1100

(s- t- r)jq(s)llx(g(s))jds :2: ~,

for

t :2: T.

+ 1 +p

t+.-

and (Tx)(t) :2: p-

p

t+.-

Therefore, Tn ~ n. We shall show that T is a contraction on we have

n.

In fact, for any

XJ' X2

E

n,

1 p

I(Txt)(t)- (Tx2)(t)j ~- lxt(t + r)- x2(t + r)l

11

+-

00

p t+.-

(s- t- r)jq(s)llxt (g(x))- x2(g(s))jds

~ llx1-x211~(1+P~ 1 )

=~(1+~)11xl-x211, t~T which implies that

!)

Since !(1 + < 1, it follows that Tis a contraction. Hence there exists a fixed point X En. Then x(t)

1100

1 = p -1 + -x(t + r)--

p

p t+.-

(s- t- r)q(s)x(g(s))ds,

t ~ T.

Second Order

Equation~

263

with Deviating Arguments

Differentiating it twice, we have

(x(t + r)- x(t))"

= q(t + r)x(g(t + r)),

t ~ T,

i.e., x(t) is a bounded positive solution of Eq. (4.6.1).

D

Remark 4.6.2. By a similar proof we can show that Theorem 4.6.5 is also true for p E (0, 1). The following result is about the existence of asymptotically decaying positive solutions of Eq. (4.6.1).

Theorem 4.6.6. Assume that 0 0 such that eventually

pe 01 r

+

1

00

(s- t)q(s )exp [a(t- g(s))]ds

~ 1.

(4.6.25)

Then Eq. (4.6.1) has a positive solution x(t) satisfying x(t)-+ 0 as t-+ oo. Proof: It is easy to see that if the equality in ( 4.6.25) holds eventually, then Eq. ( 4.6.17) has a positive solution x( t) = e-at. In the rest of the proof we may assume that there exists a number T > t 0 such that t- r ~ t 0 , g(t) ~ t 0 for t ~ T and (3 := pe 01 r

+ ~00 (s- T)q(s)exp[a(T- g(s))]ds < 1,

(4.6.26)

and condition ( 4.6.25) holds fort ~ T. Let BC denote the Banach space of all continuous bounded functions defined on [t 0 ,oo) with the sup norm. Let n be the subset of BC defined by

n=

{y E BC: 0

~

y(t)

~ 1,

t

~to}.

Define a map T : n -+ BC as follows:

(Ty)(t) = (1i.y)(t)

+ (12y)(t)

Chapter

264

4

where

pe"' .. y(t- r),

(1iy)(t) = {

(1iy)(T)

t?. T

+ exp[e(T- t))- 1,

(4.6.27)

to $ t $ T

and

(12y)(t) = {

j,""'(s- t)q(s)exp[x(t- g(s))]y(g(s))ds, t?. T

(4.6.28)

to $ t $ T,

('J2y)(T),

where e = ln(2- [3)/(T- t 0 ). It is easy to see that the integral in (4.6.28) is defined whenever y E il. Clearly, the set n is closed, bounded and convex in BC. We shall show that for every pair X' y E n

1ix + 'J2y E il. In fact, for any x,y

(1ix)(t)

+ (12y)(t) =

En, we have

pe"' .. x(t- r) +

$ pe"' .. $ 1,

(4.6.29)

+

for

1

00

1

00

(s- t)q(s)exp[a(t- g(s))]y(g(s))ds

(s- t)q(s)exp[a(t- g(s))]ds

t ?. T,

and

(1ix)(t)

+ (12y)(t) =

(1ix)(T)

+ (12y)(T) + exp[e(T- t)]-1

= {3 + exp[e(T- t)] - 1

$ {3 + exp[e{T- t 0 ))-1

=1

for

t 0 $ t $ T.

Obviously, (1ix)(t) + (12y)(t)?. 0 fort?. t 0 • Thus, (4.6.29) is true. From (4.6.26), we know that pe"' .. < 1, which implies that 1i is a contraction. We now shall show that 12 is completely continuous. In fact, from (4.6.25), there exists a positive constant M such that

1

00

q(s)exp[a(t- g(s))]ds $ M,

for

t?. T.

265


Thus we have

I:t(12y)(t)i = il"' q(s)exp[a(t- g(s))]y(g(s))ds +a ~

j

00

(s- t)q(s)exp[a(t- g(s))]y(g(s))dsi

M +a,

for

t > T,

and d

dt(12y)(t) = 0,

for

t 0 ~ t < T,

which shows the equicontinuity of the family 72!1. On the other hand, it is easy to see that 72 is continuous and the family of 72 n is uniformly bounded. Therefore 72 is completely continuous. By Krasnoselskii's fixed point theorem, T has a fixed pointy E !1. That is,

pe"'Ty(t- T) y(t) =

{

+ Jt,(s- t)q(s)exp[a(t- g(s))]y(g(s))ds,

t;::: T t0

y(T) + exp[c(T- t)]- 1,

~

t

(4.6.30) ~

T.

Since y(t) ;::: exp[c(T- t)]- 1 > 0 for to ~ t < T, it follows that y(t) > 0 for t;::: to. Set x(t) = y(t)e-"'t. Then (4.6.30) becomes

x(t)=px(t-T)+

j

00

(s-t)q(s)x(g(s))ds,

t;:::T.

Thus x(t) is a positive solution of Eq. (4.6.1) and x(t)-+ 0 as t-+ oo. Remark 4.6.3. For the case p 4.6.6 still holds. Corollary 4.6.1. a > 0 such that

= 0, if g(t) < t,

Assume tl1at 0

. = is a real root of ( 4.6.38), and hence (4.6.37) has a bounded positive solution. By Corollary 4.6.1, Eq. ( 4.6.36) has a positive solution x(t) satisfying x(t)-+ 0 as t-+ oo.

4.6.6.

Equations with variable p

We now consider the second order neutral differential equation

(x(t)- p(t)x(t- r))" where r, u E (0, oo ),

= q(t)x(t- u),

t;::: t 0

(4.6.39)

p, q E C([t 0 , oo ), R).

Theorem 4.6.7. Assume that (i) 0::; p(t) ::; 1, t ;::: to; (ii) 0 < k 1 ::; q(t) ::; k2, t ;::: to; (iii) for any >. > 0

liminf{p(t- u) t-+oo

q

/(t) {\r t- T

+ ,12

q(t)e>.o-}

A

> 1.

(4.6.40)

Then every bounded solution of Eq. ( 4.6.39) is oscillatory. Proof: Let x(t) be a bounded, eventually positive solution, say x(t- r) > 0, x(t-u)>O for t;:::t 1 ;:::t 0 .Set

z(t) = x(t)- p(t)x(t- r). It is not difficult to show that z"(t) > 0, z'(t) < 0, z(t) large enough and lim z(t) = lim z'(t) = 0. Then from (ii) t-+oo

(4.6.41)

> 0 fort;:::

t 1 , if t 1 is

t-+oo

( 4.6.42) and (4.6.43) Define a set A as follows: A={>.> 0:

z"(t) > >. 2 z(t) eventually}.

(4.6.44)

Chapter

268

4

It is easy to see that ~ E A, i.e., A is nonempty. We shall show that A is bounded above. In fact, ( 4.6.42) implies that

z"(t) Integrating from t to t

+i

~

t

k1 z(t- u),

~ t1

+ u.

we have

r+i u z(s- u)ds z'(t + 2")- z'(t) ~ k1 },

> kt

i z(t- i),

t ~ t 1 + u,

and then

u u z(t)-z(t+-) >k1 2

4

u 11+'\ z(s--)ds 2

t

This implies that (4.6.45) where a

= k1 ( ~2

).

Applying (4.6.45) four times we obtain that

z(t) > a 4 z(t- u),

t ~ t 1 + 2u.

(4.6.46)

In view of the boundedness of x(t) it is not difficult to see that liminfx(t) = 0. t-oo Choose a sequence {sn} such that sn ~ t 1 +2u, n = 1,2, ... , lim Sn = oo, and ft-+00

x(sn- u) = min{x(s): t1 :5 s :5 Sn- u},

n = 1,2, ....

Integrating (4.6.42) twice we have

z(t-u)>kt

r

f'x(u-u)duds,

lt-.r 1.

t~tl+u


269

and hence

z(sn- u) > k1

J. •n J.Sn x(u- u)duds •n-u

•

i.e.,

x(sn- u) < f3z(sn- u),

n = 1, 2, ...

where f3 = 2/k 1 u 2 • Then from (4.6.43), (4.6.46) we obtain

which implies that Jo:- 4 f3k 2 E A, i.e., A is bounded above. Set ), 0 =sup A. Then .>. 0 E (0, Jo: 4 f3k 2 ]. For any a E (0, 1) we have that for sufficiently large t (4.6.47) Set z(t) = z'(t) + a>. 0 z(t). Then

z'(t)- o:>.oz(t) = z"(t)- (o:>. 0 ) 2 z(t) ~ 0 eventually. It implies that z(t)e->-ot is nondecreasing. Since z(t) --+ 0, z'(t) --+ 0 as t --+ oo so z(t) --+ 0 as t --+ oo. Thus z(t) < 0, i.e., z'(t) + o:>. 0 z(t) :::; 0 eventually. Set w(t) = z(t)e>-ot. Then w'(t) = (z'(t)

+ o:>. 0 z(t)]e">.ot:::; 0.

We can rewrite (4.6.39) in the form

z"(t)

= p(t- u)

( q(t) ) z"(t- r) q t- T

+ q(t)z(t- u).

Then by ( 4.6.47) we have

z"(t)

~ (o:>. 0 ?p(t- u)

( q(t) ) z(t- r) qt-r

+ q(t)z(t- u)

(4.6.48)

Chapter

270

= (a>.o?p(t- u) ~

q(t) w(t- r)e-crAo(t-r) q(t- r)

[(a>.o)2p(t- u)

+ q(t)w(t- u)e-crAo(t- 1.

=

=

Remark 4.6.2. In the case p(t) p, q(t) q are constants ( 4.6.40) is also a necessary condition for the bounded oscillation of Eq. (4.6.39). Theorem 4.6.8.

Assume that p(t) $ 0, limsup{-p(t-u) t-+oo

tt)

q

t-

q(t) > 0,

T

u > r,

)}=aE(O,oo)

(4.6.49)

and

lim sup [' t-+oo

(s- t + (u- r))q(s)ds > 1- a.

(4.6.50)

lt-( 0, of z(t) we have

w'(t) $ 0,

t;::: t2,

(4.6.53)

t ;::: t2. By the monotone property

w(t) = z(t)- kaz(t- r) $ (1- ka)z(t- r),

t;::: t 2 ,

or 1

z(t);::: 1 _kaw(t+r),

t;:::t 2 •

Substituting this into ( 4.6.53) we obtain

w"(t);::: __2_k q(t)w(t- (u- r)), 1- a

t;::: t 2 + u.

Integrating from s to t for t ;::: s we have

w'(t)- w'(s);::: __2_k 1- a

j' q(u)w(u- (u- r))du, ~

Integrating this inequality in s from t - ( u - r) to t we have

w'(t)(u- r)- w(t)

+ w(t- (u- r))

s;::: t 2 •

Chapter 4

272

;::: -11- ka

it

(u- t +a- r)q(u)w(u- a+ r)du

t-11+r

;::: w( t - a + r) 1- ka

it

(u- t +a- r)q(u)du,

t;::: t2.

t-11+r

Thus

w(t) +w(t- (a- r)) [ -1 + - 1-

1 - ka

[' lt- 11+r

(u- t +a- r)q(u)du] 50

which implies that

('

(u-t+a-r)q(u)du 0. Then for each t• ;::: t 0 + r, there

Second Order Equation1 with Deviating Argument1

273

exists a set

A= {tl t* :5 t :5 t*

+ 2T, y(t- T)

~

.B.(t)}

with the measure roes (A)~ T, where ,B.(t) =min { ,B(t;r), ~ }.

Proof: For any fixed t*

~

B = {

t 0 + T we define a set

*

E

[t*,t*

+ T], y(t) > .a;t) }·

If B = 0(empty set), then py(t-T) ~¥,fortE [t*,t*+T] i.e., A= [t*,t* +T]. Now we consider the case that B =f 0, then roes (B)= a E (0, T). Let B denote the closure of B. In view of the continuity of y, we have y(t) ~ t E B. Define a set B + T = {t, t- T E B}. Then y(t- T) ~ ,B(t;r> fortE (B + T). Set

¥,

A = {[t*' t*

then roes A= T and y(t- T)

~

+ T]\B} u (B + T)

,B.(T) on A.

Theorem 4.7.1. Assume that (H1 )-(H3) hold. .FUrther assume that nondecreasing in y and

0

f is

for every closed set E whose intersection with every segment of the fonn t ~ t 0 + T, has a measure not smaller than T. Then every solution of Eq. (4.7.1) is oscillatory.

[t- T, t + T],

Proof: If not, without loss of generality, let x(t) be an eventually positive solution, say x(t) > 0 fort ~to. Set

z(t) = x(t) + px(t- T), Then (z(t) + r(t))" < 0 fort ~ t 0 + T. It is easy to see that (z(t)- r(t))' > 0

Chapter 4

274 eventually, which implies that 00 {

ito

f(t,x(t- a))dt < oo.

(4.7.3)

On the other hand, it is easy to see that (z(t)- r(t)) > 0 eventually. Then we have

z(t)

= x(t) + px(t- r) ~ r+(t),

t ~ t0

+ r.

By Lemma 4.7.1, for every t* ~to +2r there exists a set A= {tl t* :5 t :5 t* + 2r, x(t- r) ~ rt(t)} with mes (A) ~ r. Let us consider the set A- (r- a) = {tl t + (r- a) E A}. It is obvious that mes (A- (r- a)) ~ T and x(t- a) ~ rt(t + (r- a)), t E (A- (r- a)). From (4.7.3) we have

which contradicts the assumption (4.7.2).

0


(x(t)

+ px(t- r))" + t"lx(t)l" sign x(t) =

t 6 sin t,

t ~to

(4.7.4)

where p, r, v > 0, a, 6 E R. If ct + v6 > -1, then condition (4. 7.2) holds. From Theorem 4.7.1 every solution of (4.7.4) oscillates. For instance, every solution of the equation

(x(t) + x(t -1r))" + x(t) = sint

(4.7.5)

is oscillatory. In fact, x(t) = sin t is such a solution. The following result is for equations of the more general form

(x(t) + p(t)x(t- r))" + f(t,x(g(t)),x'(a(t)))

Theorem 4.7.2. Assume that (i) p, g, a and Rare continuous on t

~ t0 ;

= R(t).

(4.7.6)

275

Second Order Equation.5 with Deviating Argument.5

(ii) p(t) ~ 0, g is nondecreasing and lim g(t) !->ex>

(iii) f E C([to,oo) x R 2 ,R) and f(t,u,v)u (iv) for any T

= oo;

> 0 as u =I 0;

~to

lim sup J; R( s )ds= oo,

liminf J;R(s)ds= -oo, !->ex>

!->ex>

liminf J; J;R(u)duds= -oo, !->ex>

(4.7.7)

lim sup J; J; R( u )duds= oo. !->ex>

Then every solution of Eq. (4. 7.6) is oscillatory.

Proof: Assume the contrary. Then without loss of generality we assume there is an eventually positive solution x(t). Set z(t) = x(t) + p(t)x(t- r). Then z(t) > 0, t ~ T ~to. From (4.7.6) we have z"(t) < R(t). Thus

z'(t)- z'(T) < By (iv), there exists a sufficiently large T* T by T* in ( 4. 7.8) we have

J;

(4.7.8)

R(s)ds.

~to

such that z'(T*) < 0. Replacing

z' ( t) < [' R( s )ds

lr·

and

z(t)- z(T*) < [' (" R(u)duds.

jT• jT•

Therefore, liminf z(t) = -oo, which contradicts the positivity of z(t). t-->oo

0

Example 4. 7 .2. Consider

(x(t)

+ x(t- 1r))" + tx(t- 27r) =

t sin t.

(4.7.9)

It is easy to see that all assumptions of Theorem 4.7.2 are satisfied. Therefore, every solution of (4.7.9) is oscillatory. In fact, x(t) = sint is such a solution.

Chapter 4

276

As an application we give sufficient conditions for oscillation of the nonlinear hyperbolic equations of neutral type

uu(x,t)+puu(x,t-T) {

- ~u(x, t) + q(s, t, u) ~:(x,t)

= f(x, t),

= g(x,t),

(x,t)eG

(4.7.10)

(x,t) E u

where p and T are positive constants, ~ is the Laplacian in Rn. D is a bounded domain in Rn with the smooth boundary oD, the cylindrical domain G = D X (0, 00 ), u = oD X (0, 00 ), n is the vector of the external normal to u, G 01 = D x (a,oo), a E R. Assume that (A1)

q(x,t,u) E C(G x R,R),

f(x,t)

e C(G,R),

and g(x,t) E C(u,R);

(A2) q(x, t, {) ~ .8(t) p

for

f3 =a,

( 4.8.3)

lt-(cr-r)

where p E (0, 1) for a= 1, p E (O,oo) for a E (0, 1); (iii) every solution of the second order ordinary differential equation z"(t)

+ >.q(t)

t-u)fJ fJ (t - z (t) = 0

(4.8.4)

is oscillatory, where 0 < >. < 1 is a constant. Then every solution of Eq. (4.8.1) is oscillatory. Proof: Without loss of generality, let x(t) be an eventually positive solution of Eq. (4.8.1) and define

z(t)

= x(t)- px"'(t- r).

(4.8.5)

From (4.8.1), we know that z"(t) ::; 0. If z'(t) < 0 eventually, then lim z(t) = t-oo lim en= oo -oo. Thus lim x(t) = oo and there exists a sequence {en} such that n-+oo t-+oo and x(en) = max x(t)-+ oo as n-+ oo. Hence to~t~en

z(en) = x(en)- px"'(en- r) ~ x(en)- px"'(en) = x(en)[1- px"'- 1 (en)]

a contradiction. Therefore, z'(t)

-+

oo,

as

n-+ oo,

> 0. If z(t) < 0, then z(t) > -px"'(t- r). Then

x(t- r)

z(t)) 1/a >( - p

(4.8.6)

Substituting ( 4.8.6) into ( 4.8.1) we have

(4.8.7) As in the proof of Theorem 4.3.1, ( 4.8. 7) has no negative solution under the assumptions. This contradiction shows that z(t) > 0. By Lemma 4.3.1, for each

Chapter

280 k E (0, 1), there is

at~.:

4

2::: t 0 such that

t-0' z(t- 0') 2::: k -t- z(t),

for

t 2::: t,.,.

(4.8.8)

Substituting ( 4.8.8) into (4.8.1) we have (4.8.9) which implies that Eq. (4.8.4) has a nonoscillatory solution, contradicting the assumptions (iii). D Example 4.8.1. We consider

(4.8.10) We see that 9

i

t

911"2

(u- (t- 1r)) sin2 udu = 4

> 6.

t-Il'

That is, (4.8.3) holds. It is known (see [110]) that every solution of (4.8.11) is oscillatory for 0 < .X < 1. Then, by Theorem 4.8.1, every solution of Eq. (4.8.10) oscillates. In fact, x(t) = sin 3 t is such a solution. Example 4.8.2. We consider

(4.8.12) It is easy to see that all assumptions of Theorem 4.8.1 hold. Therefore, by Theorem 4.8.1, every solution of ( 4.8.12) oscillates. In fact, x( t) = sin 3 t is such a solution.

We now consider ( 4.8.1) of the unstable type. For the sake of convenience, we put Q(t) =: -q(t) 2:::0, t 2::: t 0 •

Second Order

Equation~

with Deviating

Argument~

Theorem 4.8.2. Assume that (i) p, r, u > 0, and f3 E (0, 1], Q(t) ;::: 0, t;::: t 0 ; (ii) lim sup JL 1.

281

(4.8.13)

t-+oo

Then every bounded solution of ( 4.8.1) is oscillatory.

Proof: Assume the contrary, and let x(t) be a bounded eventually positive solution of (4.8.1). Then z"(t);::: 0. By the boundedness of z, we have z'(t) < 0 eventually. If z(t) > 0 eventually, integrating (4.8.1) twice we have

i~ 0. Hence -px"'(t- r)

(~) {Jfa Q(t).

~

-d,

(4.8.15)

We note from (4.8.13) that

£

00

tQ( t )dt

Hence (4.8.15) implies that lim z(t) t-+oo

= 00.

= oo, we reach a contradiction.

(4.8.16) D

Chapter 4

282 Example 4.8.3. We consider

(4.8.17)

It is easy to see that the conditions of Theorem 4.8.2 hold for ( 4.8.17). Hence every bounded solution of (4.8.17) oscillates. In fact, x(t) = sint is such a

solution.

Assume that (i) p, r > 0, a ~ 0, a ~ 1, (3 > 0, Q(t) ~ 0, t (ii) There exists a constant A> 0 such that

Theorem 4.8.3.

~to;

ap exp{Aar + .At(1 - a)} :5 L < 1

(4.8.18)

and

p exp{.Aar + .At(1- a)}+ ["'<s- t)Q(t) exp{A(t- fi(s- a))}ds :51 (4.8.19) hold eventually. Then Eq.(4.8.1) has a positive solution x(t) which tends to zero as t--+ oo.

Proof: If the equality in (4.8.19) holds eventually. Then x(t) = exp( -.At) is a

positive solution of Eq. (4.8.1). Therefore, we may assume that there exists a T >to such that t- T ~to, t- a ~to fort~ T,

r :== p exp{.Aar + .AT(1- a)}+

loo

(s- T)Q(s)exp{A(T- fi(s- a))}ds < 1,

and (4.8.19) holds for all t ~ T. As before, let BC denote the Banach space of all bounded continuous functions defined on {t 0 , oo) with the sup norm. Let n be the subset of BC defined by f! == {y E BC: 0 :5 y(t) :51,

for

t

~ t 0 }.

283

Second Order Equation$ with Deviating Arguments

It is easy to see that n is closed, convex and bounded in BC. Define a map T : n __. BC by follows

(Ty)(t) = ('Jiy)(t)

+ (12y)(t)

where

('Jiy)(t) = {

p exp{..\aT + ,\t(1- a)}y(t- T), (Tiy)(T)

(12y)(t) = {

+ exp{e(T- t)} -1,

t ~T to ~ t ~ T,

J,""(s- t)Q(s)exp{..\(t- fj(s- u))}y.B(s- u)ds, t ~ T to ~ t ~ T

(12y)(T), and e

= ln 2/(T- to).

'Jix + 'J2y E fl. And 1i is a contraction and 12 is completely continuous. By Krasnoselskii 's fixed point theorem T has a fixed point y E fl. That is, It is easy to see that for every pair x,y En,

p exp{..\aT + ,\t(1- a)}y 0 (t- T) y(t) =

{

+ f,""(s- t)Q(s)exp{,\(t- f3(s- u))}y.B(s- u)ds, y(T) + exp{e(T- t)}- 1,

t ~T t 0 ~ t ~ T.

Since y(t) > 0 for t 0 ~ t < T, it follows that y(t) > 0 for t ~ t 0 • Set x(t) = y(t)exp( -,\t). Then

x(t) = px 0 (t- T)

+ l""(s- t)Q(s)x.B(s- u)ds,

t

~ T,

which implies that x( t) is a positive solution of (4.8.1 ). It is obvious that x( t) __. 0 D as t __. oo. Example 4.8.4. We consider ( x(t)- 21e x 3 (t- 1))" = Q(t)x(t),

t > 10

(4.8.20)

3t- 33).

(4.8.21)

where

Q(t) = 1 -

1 t2 - te

21

(9t 3

-

45t 2

-

Chapter 4

284

It is not difficult to see that all assumptions of Theorem 4.8.3 are satisfied. Therefore, ( 4.8.20) has a bounded solution x( t) which tends to zero as t -> oo. In fact, x(t) = te-t is such a solution of (4.8.20).

4.9.

Advanced Type Equations

We consider the second order advanced type differential equations of the form

(r(t)x'(t))'

+ q(t)x(h(t)) = 0

Jt r1ij =

where r E C([to,oo),(O,oo)), hE C([to,oo),(O,oo)) and h(t) Theorem 4.9.1.

~

(4.9.1)

oo, q E C([to,oo),R+), q 1= 0,

t.

Assume that

1

00

q(t)dt =

00.

(4.9.2)

to

Then every solution of Eq. (4.9.1) is oscillatory. Proof: Assume the contrary, and let x(t) be an eventually positive solution. It is easy to see that r(t)x'(t) > 0 fort~ T ~ t 0 • Then

£

00

q(t)x(h(t))dt

< oo

( 4.9.3)

0

which contradicts (4.9.2).

In the following we want to derive some oscillation criteria for (4.9.1) when

1

00

q(t)dt

< 00.

(4.9.4)

to

Lemma 4.9.1.

Let x(t)

> 0, t w(t)

~

th be a solution of Eq. (4.9.1). Set

= r(t)x'(t)/x(t).

(4.9.5)

285

Second Order Equations with Deviating Arguments Then w(t) > 0,

lim w(t) = 0,

1-+oo

(4.9.6)

and

w(t) =

ioo :(s)

2( )

ds

1

+

loo r

f.h(•)

q(s) exp( •

w(u)

r(u) du)ds,

t :2:

t2

:2:

t 1•

(4.9.7)

Proof: From (4.9.1) we have

(x(t)w(t))'

+ q(t)x(h(t)) = 0,

since

x(h(t))

_(_)_ = exp X

t

w2(t)

w'(t) + r(t)

ih(t) I

+ q(t)exp

w(s)

-( ) ds, r S

ih(t) 1

w(s) r(s) ds = 0.

(4.9.8)

Integrating it from t to T for T :2: t :2: t 1 , we have

w(T)-w(t)+

iT 1

w2(s) r(s) ds+

iT 1

q(s)exp

(J.h(•) w(u) ) • r(u)du ds=O.

(4.9.9)

Because r(t)x'(t) > 0, so w(t) > 0. We shall show that lim w(t) = 0. In fact, if

t-+oo

lim r(t)x'(t) = c > 0, then there exists a t 2 :2: t 1 such that fort :2: t 2

1-+oo

and hence lim w(t) = 0. If lim r(t)x'(t) = 0, then lim w(t) = 0 also. Letting t-+oo

t-+oo

T __... oo in ( 4.9.9) we obtain ( 4.9. 7).

Lemma 4.9.2.

t-+oo

D

Eq. (4.9.1) has a nonoscillatory solution if and only if there

Chapter

286

4

exists a positive differentiable function (t)y(t) ;::: 0. Then either y(k)(t)y(t);::: 0, k = 0, 1, ... ,nor there exists an integer£, 0::::; .e::::; n-2, which is even when n is even and odd when n is odd, such that y(t)y(t);::: 0,

k

( -1)"+ky(k)(t)y(t) ;::: 0,

= 0, 1, ... ,.e k=

.e + 1, ... , n.

Lemma 5.1.3. Assume that tbe hypotheses of Lemma 5.1.1 bold and

y(t)y. E (0, 1), there exists an M > 0 such that Jy(.Xt)J ;::: Mt"- 1Jy(n-l)(t)J for all large t.

Chapter 5

290

Lemma 5.1.4. Assume that (5.1.2) holds and that 0 :5 p(t) :5 1. Let x(t) be an eventually positive solution of the inequality

(x(t)- p(t)x(t- r))(n)

+ q(t)x(t- cr(t)) :50

(5.1.4)

and set

y(t)

= x(t)- p(t)x(t- r).

(5.1.5)

Then y(t) > 0 eventually. Proof: From (5.1.4) and (5.1.5), y(t) :5 -q(t)x(t- cr(t)) :5 0. Since q(t) 'f= 0, y 0 eventually, by Lemma 5.1.2, y'(t) < 0 eventually. Therefore, there exist i > 0 and t 1 ~ to such that y(t) :5 -i, t ~ t1. Thus

x(t) :5 -i + p(t)x(t- r) :5 -i + x(t- r),

t ~ t 1•

In particular,

x(t1

+ ir) :5 -(i + 1)i + x(t1 -

r)-+ -oo as t-+ oo

which is impossible. Therefore, y(t) must be eventually positive.

0

Lemma 5.1.5. Assume that (5.1.2) holds, and either

p(t) + q(t)cr(t) > 0

for t ~ T,

(5.1.6)

or

cr(t) > 0

and q(s)

'f= 0

for s E [t,T*]

and t ~ T

(5.1. 7)

where T* satisfies that T*- cr(T*) = t. Let b = max{r, T-min{t-cr(t)} }, and assume that the integral inequality t;::T

z(t) ~ p(t)z(t- r)

+ (n _1 1)!

100 (s- tt- 1q(s)z(s- cr(s)) ds, t

t ~ T (5.1.8)

291

O&cillation of Higher Order Neutral Equation& has a continuous positive solution y : [T-b, oo) integral equation 1

x(t) = p(t)x(t- r) + (n _ 1)!

-+

(0, oo ). Then the corresponding

roo lt (s- tt- 1 q(s)x(s- a(s))ds,

has a continuous positive solution x : [T- b, oo)

-+

t ~ T (5.1.9)

(0, oo ).

Proof: Define a set of functions n and a mapping T as follows:

n ={wE C([T- b, oo), R+): 0 $

w(t) :51, t

~

T- b},

1 1 [ p(t)w(t- r)z(t- r) + (n _ 1)! z(t)

(Tw)(t) =

1

00

(s - t)n-lq(s )w(s - a(s ))z(s - a(s )) ds],

t>T (5.1.10)

t- ~ + b(Tw)(T)

+ 1- t- ~ + b,

It is easy to see from (5.1.8) that TO C nand (Tw)(t)

T- b $ t $ T.

> 0 on [T- b, T) for any

wen.

Define a sequence {wk(t)} in n as follows: wo(t) = 1, and WkH(t) (Twk)(t), k = O, 1, 2, ... , t ~ T- b. From (5.1.8), by induction, we have 0$ WkH(t) $ Wk(t) $1,

t ~ T- b, k = 0, 1,2, ....

t ~ T- b, exists, and

Then w(t) = limk-oo Wk(t),

w(t) =

=

z~t) [p(t)w(t- r)z(t- r) + (n ~ 1)!

1

00

(s- tt- 1 q(s)w(s- a(s))z(s- a(s)) ds],

t

~ T,

and

w(t)

w(T) + 1 b = t-T+b

t-T+b > 0 for T- b :5 t < T. b

Set x(t) = w(t)z(t). Then x(t) satisfies (5.1.9) and x(t) > 0 for t E [T- b, T). Clearly, x(t) is continuous on [T- b, T]. Then, in view of (5.1.9), we see that x(t) is continuous on [T - b, oo ).

Chapter 5

292

Finally, it remains to show that x(t) > 0 fort;:?: T- b. Assume that there exists t* ;:?: T such that x(t) > 0 for T- b :5 t < t* and x(t*) = 0. Thus, by (5.1.9), we have

0

= x(t*) = p(t*)x(t*- T) + (n _1 1)! },./00 (s- t•t- 1q(s)x(s- O'(s))ds

which implies that p(t*) = 0 and q(s)x(s- O'(s)) = 0 for all s 2:: t* which 0 contradict (5.1.6) and (5.1.7). Therefore, x(t) > 0 on [T- b,oo). Lemma 5.1.6. Assume (5.1.2) holds, k E {1, 2, ... , n- 1}, c > 0, and the integral inequality

z(t) 2::

+£

c+ p(t)z(t- T)

(t- u)"- 1 x

1

00

(s- u)n-k-1 q(s)z(s- O'(s)) ds,

t ;:?: T, (5.1.11)

has a continuous positive solution z : (T - b, oo) -+ (0, oo) where b is defined as in Lemma 5.1.5. Then the corresponding integral equation

x(t)

= c + p(t)x(t- T) + £(t-u)"- 1

1

00

(s-ut_,._ 1 q(s)x(s-O"(s))ds,

also has a continuous positive solution x: (T- b, oo)

t 2:: T, (5.1.12)

-+

(0, oo ).

The proof of Lemma 5.1.6 is similar to the proof of Lemma 5.1.5 and hence we omit it here. Theorem 5.1.1. Let 0 :5 p(t) :5 1. Assume that (5.1.2) and the assumptions of Lemma 5.1.5 hold. Then every solution of Eq. (5.1.1) is oscillatory if and only if the corresponding differential inequality (5.1.4) has no eventually positive solutions. Proof: The sufficiency is obvious. To prove the necessity, we assume that (5.1.4) has an eventually positive solution x(t). Set y(t) = x(t) - p(t)x(t- T).

O~cillation

293

of Higher Order N eu.tral Equation~

By Lemma 5.1.5, y(t) > 0 eventually. According to Lemma 5.1.1, there exists an even number k such that 0 ~ k ~ n - 1, and

= k + 1, ... , n- 1,

( -1)iy(i)(t)

>0

for i

y(i)(t)

>0

for i = 0, 1, ... , k.

(5.1.13)

H k = 0, integrating (5.1.4) from t to oo we have y(t)?: (

1 )' n -1.

1

00

(s- tt- 1 q(s)x(s- u(s)) ds.

1

That is, x(t)?: p(t)x(t- r)

1

+ (n _

roo (s- tt- q(s)x(s- u(s))ds. 1

1)! },

By Lemma 5.1.5, the corresponding integral equation z(t) = p(t)z(t- r)

1

+ (n _

1)!

roo (s- t)n-

lt

1 q(s)z(s-

u(s)) ds

also has a positive solution z(t). Clearly, z(t) is an eventually positive solution of (5.1.1), contradicting the assumption. If 2 ~ k ~ n -1, then integrating (5.1.4) from t to oo n- k times, we have 1

y(t)?: (n _ k _ 1)!

roo (s- tt-A:- q(s)x(s- u(s)) ds.

lt

(5.1.14)

1

Then integrating (5.1.4) from T tot k times we have

y(t)?: y(T)

+ (k -1)! (~- k -1)! X

1

00

(t- u)A:-1

(s -u.)n-A:- 1q(s)x(s -u(s))dsdu,

where T is sufficiently large such that y(T)

x(t)?: y(T)

£

t?: T,

> 0. Thus we have, for t ?: T,

+ p(t)x(t- r) 1

t

roo (s- ut-A:-tq(s)x(s- u(s)) ds du.

+ (k- 1)! (n- k- 1)! Jr (t- u)A:-t lu

Chapter 5

294 By Lemma 5.1.6, the corresponding integral equation

z(t) = y(T) X

+ p(t)z(t- r) + (k _ 1)! (~ _ k _

1)!

J;

Lxo(s-ut-k-lq(s)z(s-a(s))dsdu,

(t- u)k-l t'2;.T,

has a positive solution z(t). Clearly, z(t) is a positive solution of Eq. (5.1.1 ). This contradiction completes the proof. D We now show some applications of Theorem 5.1.1. We compare Eq. (5.1.1) with the equation

(x(t)- p*(t)x(t- r))(n)

+ q*(t)x(t- a(t)) = 0

(5.1.15)

where p*, q* E C([to, oo ), R+ ). Theorem 5.1.2. Assume that the assumptions of Theorem 5.1.1 hold and

q(t) :5 q*(t),

p(t) ::::; p*(t) ::::; 1,

for t '2:. t 0 •

(5.1.16)

Then every solution of Eq. (5.1.1) is oscillatory implies the same for Eq. (5.1.15). Proof: Assume the contrary, and let x(t) be an eventually positive solution of (5.1.15). Let y(t) = x(t)- p*(t)x(t- r). As in the proof of Theorem 5.1.1 we see that y(t) > 0, y'(t) < 0, for large t and (5.1.13) holds. If k 0, integrating (5.1.15) from t to oo n times we have

=

1 ('" y(t) = y( oo) + (n _ 1)! lt (s- tt- 1 q*(s )x(s- a(s)) ds,

and so

x(t)

= y(oo) + p*(t)x(t- T) + (n ~ p(t)x(t- T)

1

1 ("" _ 1)! lt (s- t)n- 1 q*(s)x(s- a(s)) ds

(""

+ (n _ 1)! lt

(s- tt- 1q(s)x(s- q(s))ds

295

Oscillation of Higher Order Neutral Equations

where y( oo) = limt-+oo y(t) ~ 0. Using a similar method as in the proof of Theorem 5.1.1, one can see that Eq. (5.1.1) also has an eventually positive solution which contradicts the assumption. H k > 0, as shown in the proof of Theorem 5.1.1, we obtain that for t ~ T

y(t)

~ y(T) + (k- 1)! (~- k- 1)! x

Hence for t

x(t)

~

1

00

J;

(t-

u)k-1

(s- ut-"- 1 q*(s)x(s- u(s))dsdu.

T

~ y(T) + p*(t)x(t- T) + (k _ 1)! (~ _ k _ 1)! x [(t) + lq(t) :5 0, t 2: T. From (5.1.23), we have

It contradicts (5.1.13). Therefore, .e = 0. We show that x(t) is bounded. Otherwise, there exists a sequence {tn} such that limn-ootn = oo, x(tn) = maxx(s), •:9n

and limn-co x(tn) = oo. Then

y(tn) = x(tn)- p(tn)x(tn- r) 2: {1- p)x(tn)-+

00

as n-+ oo

which is impossible. Let {n -+ oo be such that a= limsupx(t) = lim x({n)· t~oo

n--+oo

Since

we get 0 :5 a :5 pa. This implies that a= 0 and hence x(t)-+ 0 as t-+ oo.

0

Using Lemma 5.1.7 we can compare the odd order delay differential equation

x(t) + q(t)x(h(t))

Chapter 5

300

Theorem 5.1.5. Assume tha.t p(t) ::; q(t), h(t) :5 g(t) < t,

roo q(t)h(tt-

lto

2

dt

= oo,

for n

~ 3,

(5.1.31)

or

roo q(t)dt = oo,

leo

for n = 1.

(5.1.32)

Then every solution of Eq. (5.1.29) is oscillatory implies the same for Eq. (5.1.30). Proof: Assume the contrary, and let x(t) be an eventually positive solution of (5.1.30). By Lemma 5.1.7, x'(t) < 0 and lim1- 00 x(t) = 0. We see that 0 = x(n)(t)

+ q(t)x(h(t)) ~ xoo

Theorem 5.2.1. Assume that (i) p E [0, 1), g is nondecreasing, g(t) :5 t, and (5.1.23) or (5.1.24) holds; (ii) either

limsup t->oo

{1

t

g(t)

(g(s)- g(t))n-lq(s)ds

+p

1g(t)

(g(s)- g(t))n-Iq(s)ds

}

g(t)-T

> (n- 1)! (1- p),

(5.2.2)

303

Oscillation of Higher Order Neutral Equations or lim sup {

t

(s- g(t))n- 1 q(s)ds

l

+ p lg(t)

}

g(t)-r

g(t)

t-oo

(s- g(t) + rt- 1 q(s) ds

> (n- 1)! (1- p).

(5.2.3)

Then every solution of Eq. (5.2.1) is oscillatory. Proof: Assume the contrary, and let x(t) be an eventually positive solution of (5.2.1). Set y(t) = x(t) - px(t- r). By Lemma 5.1.4, we know that

( -1)iy(il(t) > 0 and

lim y (n -1)!.

}g(t)-r

(5.2. 7)


t;

N+l

y to such that t- r ~ t 0 and g(t) ~to fort~ T, 1

f3 =pear+ (n _ 1)!

Jr("" (s- T)n- 1 q(s)exp[o:(T- g(s))] ds

< 1,

and (5.2.15) holds for all t ~ T. Let Y be the set of all continuous functions y defined on [to, oo) satisfying 0 ~ y(t) ~ 1 for t ~ t 0 • Let Y be endowed with the usual pointwise ordering: YI ~ Y2 YI(t) ~ Y2(t) for all t ~to. It is easy to see that for any subset A of Y there exist inf A and sup A. Define a mapping T on Y as follows:

pe

(Ty)(t) =

ar ( ) 1 y t-r + (n-l)! X

1"" (s- tt- q(s)exp[o:(t- g(s)))y(g(s)) ds,

(Ty)(T)

1

+ exp[e(T- t)]-1,

where q = ln(2- {3)/(T- to). For any y E Y, (Ty)(t)

(Ty)(t) ~pear+ (n _1 1)! (Ty)(t)

~

0

fort~ t 0 ,

1""(s- ttt

t0

~

t

~

t

~T

T,

and

1 q(s)exp[o:(t-

g(s))] ds ~ 1,

= (Ty)(T) + exp[e(T- t)]-1 ~ f3 + exp(T- t 0 ) -1 =

1,

t

~

T,

to ~ t ~ T.

Chapter 5

306

This means that TY C Y. Moreover, T is a nondecreasing mapping. By Knaster's fixed point theorem there exists y E Y such that Ty = y. That is, pe

y(t)

=

err (

y t- r X

1

00

)

+ (n _1 1)! t ;:::-: T

(s- t)"- 1 q(s)exp[a(t- g(s))}y(g(s)) ds,

y(T) + exp[e(T - t)] - 1,

(5 ·2 ·16 )

t 0 :5 t :5 T.

Since y(t) > 0 for t 0 :5 t < T, it follows that y(t) > 0 for all t ~ to. Set x(t) = y(t)exp( -at). Then we get

(x(t)- px(t- r))(n)

+ q(t)x(g(t)) = 0,

t ~ T,

i.e., x(t) is a positive solution of Eq. (5.2.1) and limt-oo x(t)

= 0.

0

Corollary 5.2.1. Assume there exist constants q E (0, 1) and u ~ 0 such that 0 :5 q(t) :5 q,

g(t)

~

(5.2.17)

t -u.

If the "majorant" equation

{x(t)- px(t- r))(n)

+ qx(t- u) =

0,

t ~to

(5.2.18)

has a positive solution, then Eq. (5.2.1) has a positive solution which tends to zero as t -+ oo. Proof: It is easy to see that (5.2.18) has a positive solution implies that the characteristic equation has a negative real root. Hence there exists a p > 0 such that 1 = pel't + qel'tr / p". Then we can show that (5.2.15) is satisfied for a = p. The conclusion of the corollary follows from Theorem 5.2.2. The following result depicts the asymptotic behavior of decaying nonoscillatory solutions of Eq. (5.2.1), where a nonoscillatory solution x is said to be decaying if x -+ 0 as t-+

00.

Theorem 5.2.3. Assume the conditions of Theorem 5.2.2 hold. Then the decaying nonoscillatory solutions of Eq. (5.2.1) are of the following two types:

307


(i) Solution x satisfies •

x(t) = w(t)p"

.L + o(p~)

as t--+ oo

(5.2.19)

where w is a r-periodic continuous function such that

x(t)w(t) > 0 for all large t.

(5.2.20)

(ii) Solution x satisfies

lx(~)l

---+

oo

as t--+ oo.

(5.2.21)

P" Furthermore, for any continuous positive and r-periodic function w, Eq. (5.2.1) has a decaying nonoscillatory solution x with the asymptotic property (5.2.19) if and only if

roo q(s

lto

)p(g(•)-•)/r

ds

< oo.

(5.2.22)

We omit the proof here. For the existence of bounded nonoscillatory solutions of Eq. (5.2.1) we first present a general result where q changes sign, then we employ it to obtain a necessary and sufficient condition for the case that q is nonnegative. Theorem 5.2.4. Assume that p > 0, p =F 1, q E C([t 0 , oo ), R) and (5.2.23)

Then Eq. (5.2.1) has a bounded positive solution. Proof: We only prove it for the case p > 1. The proof for the case 0 < p < 1 is in a similar way. Choose T ~ t 0 such that g( t + r) ~ t 0 for t ~ T and 1

loo

(n + 1)! t+r s

n-11 q(s )I ds :::; -4-, p- 1

t

~T.

(5.2.24)

Chapter 5

308

Consider the Banach space BC of all continuously bounded functions defined on [t 0 ,oo) with the sup norm IJxiJ = sup{lx(t)l: t ~to}. Set

n=

{x E BC: ~p :S x(t) :S 2p fort~ to}.

Then n is a bounded closed convex subset of BC. Let the mapping T: be defined as follows

n .-. BC

1

p-1+-x(t+r) p

(Tx)(t) =

- (

Pn

(T x )(T), Since for any

X

E

00 (s-t-rt- 1 q(s)x(g(s))ds, t~T ~ 1 ) .1 1t+r t0

:S t :S T.

n

(Tx)(t) :S p -1 + 2 +

( _2 1) 1

1

00

. t+r

n

p-1

(s- t- r)n-llq(s)lds

:S P + 1 + - 2 - < 2p for t 1 2

(Tx)(t)~p-1+--(

~

T,

2 )' 100 (s-t-r)n-llq(s)lds n -1 . t+r

1

p-1

p

2

2

2

> p - - - - - = - fort>T -

'

~ ~ (Tx)(t) = (Tx)(T) :S 2p,

t0

-

and

:S t :ST.

It follows that Tn c n. We now prove that T is a contraction mapping. In fact, for any Xl' X2 E we have that for t ~ T

n


1

1 p

309

p-1

:5 -llxl- x2ll + -llxl- x2114p =

~ (1+~) llx1-x2ll

which implies that 11Tx 1 - Tx2ll =sup I(Tx1)(t)- (Tx2)(t)l =sup I(Txl)(t)- (Tx2)(t)l (~;to

~

t~T

~) llx1 -

< (1 + p -4 Since t{1 + ~)

x2ll·

< 1, Tis a contraction on n. Hence there is an

X

E

n such that

Tx = x. It is easy to see that x(t) is a bounded positive solution of Eq. (5.2.1). 0

Theorem 5.2.5. Assume that p > 1, and q E C([to,oo),R+)· Then Eq. (5.2.1) has a bounded positive solution if and only if

roo sn-lq(s) ds
0; (ii) (-1)iz(il(t) > 0, i = 1,2, ... ,n, and z(t) < 0. H (i) holds, since p > 1 and x(t) > px(t- r), we conclude that x(t)--+ oo as t--+ oo, which contradicts the boundedness of x(t). If (ii) holds, then there is a d > 0 such that limt-oo z(t) = -d. Hence -d :5 z(t) :5 -~ eventually, and px(g(t)) ;::: ~ eventually. Substituting this into Eq. (5.2.1) we have pz(t) :5 -~q(t). Integrating it from t to T n times for T > t sufficiently large and noting the signs of z(i)(T), i = 0, 1, ... , n- 1, we obtain -pz(T);:::

( d )' 2n-1.

1T t

(s-

tt-

1 q(s)

ds.

Chapter 5

310 Letting T

--+

D

oo we get (5.2.25).

We now consider the equation m

+L

(x(t)- x(t- r))(n)

q;(t)x(t- u;) = 0

(5.2.26)

i=l

where n ;:: 1 is an odd integer, r

> 0, u; ;:: 0,

q; E C([to, oo), R+ ), i = 1, 2, ... , m.

Theorem 5.2.6. Assume that for some T ;:: to

roo

}'1 tn-l T

L q;(t) dt = m

00.

(5.2.27)

i=l

Then every solution of Eq. (5.2.26) is oscillatory. Proof: If not, let x(t) be an eventually positive solution of (5.2.26). By Lemma 5.1.4

y(t)

= x(t)- x(t- r) > 0

(5.2.28)

eventually. Then according to Lemma 5.1.1 there exists a k E {0, 2, ... , n- 1} such that

y(t)>O, ( -1)i+ly(i}(t) > 0,

i=O, ... ,k, i = k + 1, ... , n- 1.

If k = 0, then there exist M > 0 and T;:: t 0 such that x(t) ;:: M, (5.2.26) and (5.2.29) we have

(5.2.29)

t ;:: T. From

m

y(n)(t) :::;

-ME q;(t).

(5.2.30)

i=l

In view of (5.2.27), we have

(5.2.31)

311


On the other hand from (5.2.29) (5.2.32) where n-1 F(t) = tn-ly(n-l)(t) +

L( -1)i(i + 1) · · · (n -1)tiyi(t) > 0.

(5.2.33)

i=O

It is easy to see that (5.2.32) and (5.2.33) contradict (5.2.31).

If k > 0, from (5.2.29), there exist M 1 > 0 and T1

~

to such that (5.2.34)

We claim that there exists Mz

> 0 such that (5.2.35)

In fact, from (5.2.34) (5.2.36) Let L = min{x(t): T1 :5 t :5 T1 +r}. Choose o 1 such that 0 r )k, MI/2r k} and

< o 1 :5 min{L/(T1 +

k

G(t) = (M1- ko1r)tk-l + o1 L(-1)iCiritk-i ~ 0,

t ~ T1

(5.2.37)

i=l provided T1 is sufficiently large. Since x(t) ~ L for T 1 :5 t :5 T 1 + r, it follows that x(t) ~ o 1 tk, T1 + r. From (5.2.36) and (5.2.37), for T1 + r :5 t :5 T1 + 2r

By induction we have

x(t) ~ o 1tk

for T1

+ ir :5 t :5 T1 + (i +

l)r, i

= 0, 1,2, ....

T :5 t :5

Chapter 5

312

Therefore, (5.2.35) is true for M2 = a1. Substituting (5.2.35) into (5.2.26) we have m

y 0 is a constant. (5.2.38) and (5.2.27) lead to (5.2.39) On the other hand, from (5.2.29)

1;

sn-k-ly(n)(s)ds = Fn-k(t)- Fn-k(T)

(5.2.40)

where

Fn-k(t) = tn-k-ly(n-l)(t)- (n- k -1)tn-k-2Y(n-2)(t)

+ .. · + (n- k -1)!y(kl(t) > 0.

(5.2.41)

(5.2.40) with (5.2.41) contradicts (5.2.39). The proof is complete.

0

Theorem 5.2.7. Eq. (5.2.26) has a bounded positive solution if and only if 00

roo

m

L: lt.to+•r. tn-1 L: q;(t) dt < oo. i=O

Proof: Sufficiency.

(5.2.42)

i=1

Choose T

~ t0

such that (5.2.43)

Set

1

00

K(t) =

sn- 1

(t- T 0,

~q;(s)ds,

+ r)K(T)/r,

t

~T

T-r:5t:5T

(5.2.44)

01cilla.tion of Higher Order Neutra.l Equa.tion1

313

and y(t) = E:o K(t-ir), t ~ T. Clearly, K E C(R,R+), y E C([T,oo), (0, 1]) and y(t) = y(t - r) + K(t). Define a set of functions as follows:

X= {x E C([T,oo),R): 0 :5 x(t) :5 y(t), t ~ T}.

(5.2.45)

In X, x 1 :5 x 2 means that x 1(t) :5 x2(t) for all t ~ T. It is easy to see that for any subset A of X, there exist in£ A and sup A. Define a mapping T on X as follows:

(Tx)(t) =

1

oo (

{

x(t-r)+

t

t)n-1 m

~;_ 1 )! ~q;(s)x(s-u;)ds,

t

~T+r

•- 1

ty(t) ( t ) (Tx)(T+r)(T+r)y(T+r) +y(t) 1- T+r ' t E [T,T+r] (5.2.46) where r = max1~i~m{r,u;}. From (5.2.43) and noting that y(t) :51 we have 0 :5 (Tx)(t) :5 y(t- r) + K(t) = y(t),

t

~

T

+r

and 0 :5 (T x )(t) :5 y(t),

t E [T, T

+ r].

Therefore, T X C X. Clearly, T is nondecreasing. By Knaster's fixed point theorem, there is an x E X such that T x = x. That is,

{

x(t- r) +

x~)=

1

oo

t

(s _ t)n-1 m (n _ 1)! ~ q;(s)x(s- u;) ds,

t

~

T+r

-1

ty(t) ( t ) x(T+r)(T+r)y(T+r) +y(t) 1 - T+r '

t E [T,T+r].

(5.2.47) It is easy to see that x(t) > 0 fortE [T, T + r) and hence x(t) > 0 for all t ~ T. Therefore, x is a bounded positive solution of Eq. (5.2.26). Necessity. Let x(t) be a bounded positive solution of (5.2.26). Set z(t) = x(t)-x(t-r). Then z(t) :50, ( -1); z(i)(t) > 0, i = 0, 1,2, ... , n-1, eventually. Hence there exist a > 0, t 1 ~ t 0 + r, such that x(t) ~ a, t ~ t 1 - r. Integrating

314

Chapter 5

Eq. (5.2.26) from t to oo for t

z(t)~

~ t1

l oo (

8 _

n times we have

t)n-1 m

( _ )' Lq;(8)x(8-u;)d8, n 1 . i=1

t

(5.2.48)

and hence for l = 0, 1, 2, ... ,

From the boundedness of x the above inequality implies that ~roo (8-(tl+ir))n-1~ ()d LJ Jt (n - 1)! ~ q; 8 8 < oo. i=1 t, +i7" •=1

Then 00

L

roo

m

lt . 8n-1 Lq;(8)d8 < oo.

i=1 ,, +•..

i=l

0 From Theorem 3.2.5 we obtain the following result. Theorem 5.2.8. Eq. (5.2.26) has a bounded positive solution if and only if

roo

j,l tn T

m

L q;(t) dt
1,

(5.2.54)

i=1

then every solution of (5.2.53) is oscillatory. We now consider the mixed type equation m

(x(t)- px(t -r))(n)

+ Lq;x(t- u;) =

0

(5.2.55)

i=1

where r, u; and q; are constant, p E (0, 1), r, q; E (0, oo ), and u; E R, i = 1, ... , m. Let m; be the least nonnegative integer such that 'F; = m;r + u; > 0, i = 1, 2, ... , m. Let £ be a positive integer such that £ ;::: max{ m;, i = 1,2, ... ,m}.Set q.~ "

= q 1pm 1+(i-1), _

qt-m 1 +1+i - q2p

;• -_

n m1 1, ... ,('.-

m2+(i-1)

,

+ 1'

z· = 1, · · ·, fn- -

m2

+ 1,

Chapter 5

316

_- qm pmm+(i-1) ' q• m-t (m-1)(l+1)-( E m; )+i

· 1, ... ,(t) + L qiz(t- n6;) ~ 0.

(5.2.58)

i=l

By Theorem 5.4.1, (5.2.58) implies that the equation L

z 1. t-+00

Proof: Let x(t) be an eventually positive solution. Set w(t) = y

t·

ds 2: KQo(t)


Proof: For case (i) we have

Ql(t) = where K1

1oo t

2

h(s)Qo(s) ds

+ Qo(t) 2::: K

2

1"" t

dH(s) H2(s)

K

+ H(t)

_ Kt - H(t)

= K 2 + K. Q2 (t) =

J ~' it has no real roots. Therefore, lim Kn = oo. Hence for fixed t* 2::: to n-+oo

lim Qn(t*) 2::: lim Kn/ H(t*) = R--f'OO

00.

R--f'OO

By Theorem 5.3.2 (b), every solution of (5.3.1) is oscillatory. For case (ii) we have that

Q 0 (t)=

dH(s) 1 oo q(s)(1-p(cr(s)))ds2:::K 1oo t H (s) = 1

i.e., (ii) implies (i). For case (iii) we have that

2

K H(t)'

Chapter 5

324

1 h(s)Q~(s)ds ~ 00

Q 1 (t) = Q 2 (t)

=

1 h(s)Q~(s) 00

ds

+ Qo(t) ~ (K + 1)Qo(t) =

1 h(s)Q~(s)ds + 00

~ (K~ K

KQo(t),

Qo(t)

~ K~

K1Qo(t),

1 h(s)Q~(s)ds + 00

Qo(t)

+ 1)Q0 (t) = K2Qo(t).

In general, Qn(t) ~ KnQo(t) where Kn = K~_ 1 K +1. Then {Kn} is an increasing sequence and lim Kn = oo. Since Qo(t) ¢. 0, there exists a t* ~ to such that n ..... oo

Qo(t*) > 0. Then

By Theorem 5.3.2 (b) every solution of (5.3.1) is oscillatory.

5.4. Classification of Nonoscillatory Solutions Consider the higher order nonlinear neutral delay differential equation

(x(t)

+ px(t- r))(n) + f(t,x(t- u1(t)), ... ,x(t- um(t)))

= 0

(5.4.1)

where n ~ 2, p E R, T > 0; u; E C([t 0 ,oo),R.t), 0 ~ u;(t) ~ h < oo, i = 1, ... ,m; f(t,xl, ... ,xm) E C([to,oo) X Rm,R) and xd(t,xl,···•xm) > 0 ift ~ t 0 , x 1 x; > 0, i = 1,2, ... ,m. If there are continuous functions q;( t) ~ 0, i = 1, ... , m, such that m

I: q;(t) > 0 and i=l

(5.4.2) for y; ~ x; > 0 (or y; (or sublinear).

~ x;

< 0), i = 1, 2, ... , m, then f is said to be superlinear

From the definitions of superlinearity and sublinearity it is easy to obtain the following lemma.


325

= 1, 2, ... , m.

Lemma 5.4.1. Suppose that 0 0, t ~ t 0 • Denote

Then there exist sequences {tn} and {sn} such that lim tn = oo, lim Sn = oo, lim x(tn)/t~ = Q and lim x(sn)/s~

n--+oo

n--+oo

. z(tn) b = 1Im -.n--+oo

t~

n-+oo

= q. If p E [0, 1), then

n-+oo

. x(tn) + px(tn- r) Q 1Im = n--+oo . > + pq t~ -

and

. x(sn) b = lt'm z(s_n) = 1Im n-+oo

Hence q + pQ ~ Q q=Q.

s~

n_.,oo

+ px(snr) . s~

+ pq or (1- p)q ~ (1- p)Q,

5 q + pQ .

i.e., q 5 Q, which implies that

Chapter 5

326

Similarly, if p

> 1, we have

. z(tn+r) b = hm ( ) . ;::: q + pQ, n-+oo tn + T 1

. z(sn+r) b = hm ( )"1 :5 Q + pq. n-+oo

Sn

+T

Then Q + pq ;::: q + pQ or (p -1)q ;::: (p- 1)Q, i.e., q ;::: Q, which implies that q = Q. Therefore, b = lim z(~) = lim x(t) t-+oo

or q = b/(1

+ p).

t1

+ px(t- r) t1

t-+oo

= q + pq,

Hence lim x(t)jti = b/(1 + p). t--+00

Theorem 5.4.1. Assume that p ;::: 0, p =f. 1, x(t) is a nonosci11atory solution of Eq. (5.4.1). (a) If n is even, then x(t) belongs to one of the following types:

.

A2j-1 (oo, a )-type : A2j-1 ( oo, 0)-type:

x(t)

hm - 2 . 2 =oo, t ;-

1.... 00

.

1--+oo

x(t) t ;-

.

x(t)

hmsup2'2 = oo,

A2j-1 (a, 0)-type:

. 2 hm t- 2 )-

t .... oo

= a =f. 0,

.

x(t)

. 1 hm -t 2 J-

1--+oo

.

=a =f. 0,

x(t) t }- 1 =0,

hm - 2 .

1--+oo

lim x~t) t2J-1

t-+oo

=0

where j = {1, 2, ... , ¥} and a is a constant. (b) Ifn is odd, then x(t) belongs to one of the following types:

.

A2j(oo,a)-type: A2j( oo, 0)-type: A2j( a, 0)-type:

Ao(a)-type: Ao(O)-type:

where j E {1, 2, ... ,

n;- 1 }

x(t)

hm -t 2;. 1 =oo,

t--+00

.

x(t)

hmsup~ t-+oo t ;-

.

x(t)

=oo, a

=f. 0,

lim x(t) =a

=f. 0,

hm - . = t-+oo t 2J- 1 t-+oo

lim x(t) = 0

1-+oo

and a is a constant.

.

x(t) t 2-J . = a

hm -

t-+oo

.

x(t) - . =0, t 2J

hm -

t-+oo

.

x(t)

hm - 2-. =0, t 1

1:-+oo

=f. 0,

0Jcillation of Higher Order Neutral Equations

327

Proof: Without loss of generality we assume that x(t) is an eventually positive solution of (5.4.1). Then z(t) > 0 and z(nl(t) < 0 fort~ T1 ~to. If n is even, by Lemma 5.1.1, there exist Tz ~ T1 and R = 2j- 1, where j E {1,2, ... , ~},such that

z(k)(t) > 0 for t ~ T 2 , k = 0, 1, ... ,R -1 and

Especially, z< 2i- 2l(t)

> 0,

z( 2i- 1l(t)

> 0,

z( 2il(t) < 0,

t ~ Tz. Denote

Then 0 ::5 Rzj-I < oo, 0 < Rzj-2 ::5 oo. If .R.zj-1 > 0, by !'Hospital's rule, we have lim z~t) = lim t-+oo t2J-1

z'(t)

t-+oo

Since 0 :::; x(t)/t 2i - 1 5.5.1, we have

:::;

(2j -1)t2i-2

= ... = lim z(2i-1l(t) t-+oo (2j -1)!

=

.e2j-I (2j- 1)! ·

z(t)/t 2i - I , x(t)/t 2i-I is bounded on [to, oo ). By Lemma

r

x(t) -

t.:.~ t2i-1

.e2j-1 - (2j- 1)! (1

f.

+ p)

0.

It follows that lim x(t)/t 2i- 2 = oo. Hence x(t) is an A 2j_ 1(oo,a)-type solution. t-+oo

If l2j-I = 0 and l2j-2 = oo, then by !'Hospital's rule, it is easy to see that

lim z~t) = 0 t2J-l

t-+oo

and

.

z(t)

hm -t 2 ]. 2 =oo.

t-+oo

By Lemma 5.4.2 we have

.

x(t) . x(t) sup~ = oo. t ]- 1 =0 and hm t-+oo t ]-

hm - 2 .

t-+oo

Hence, x(t) is an A2j-1(oo,O)-type solution. If .R. 2 j-I = 0 and 0 < £2j_ 2 < oo, then, by !'Hospital's rule, we have lim z~t) = l2j-2 t2J-2 (2j - 2)!

t-+oo

f.

O.

Chapter 5

328

By Lemma 5.4.1 we have lim x(t) t-+oo t2i-2

=

f.2j-2

(2j - 2)!(1

'I O.

+ p)

It follows that lim x(t)/t 2i-l = 0. Hence x(t) is an A2j_ 1(a,O)-type solution.

t-+oo

For the case that n is odd, the proof is similar.

0

Theorem 5.4.2. Suppose that n is even, p ~ 0, p -::J 1, and f is superlinear or sublinear. Then Eq. (5.4.1) has an A2 j_ 1 (oo,a)-type nonoscillatory solution if and only if there is a k -::J 0 such that

j oo

sn- 2 j if(s, k(s- o-1(s)) 2 j-l, ... , k(s- O"m(s)) 2 j-l )Ids< oo

(5.4.6)

to

wherej E {1,2, ... ,~}. Proof: Necessity. Let x(t) be an eventually positive A2 j-l ( oo, a)-type solution. Then z(t) > 0 fort ~ T1 ~ t 0 • Since z(t) < 0, z(i)(t) is eventually monotonic, i = 1, 2, ... , n- 1. Since lim x(t)jt2i-l =a> 0, there exists T2 ~ T1 such that t-+00

~at2i-l < x(t) -2 < ~at2i-l , t >_ T-2. 2 -

(5.4.7)

= (2j -1)! (1 + p)a.

(5.4.8)

We show that lim z< 2i-l)(t)

t-+oo

In fact, by !'Hospital's rule

.

z(t)

bm 2 ._ 1 t-+oo t J For the case that j 2j

(t) (21. - 1 )'.

= (1 + p)a.

(5.4.9)

z(i)(t) is eventually monotonic for i = 2j,

+ 1, ... , n- 1, from (5.4.8) we have lim z< 2i)(t) = lim z< 2i+ 1)(t) = · · · = lim z

t-+oo

t~oo

(5.4.10)

329

0Jcillation of Higher Order Neutral EquationJ

Integrating (5.4.1) n- 2j times and using (5.4.10), we have

z< 2i>(t) = ( -1t- 2i

{oo J.oo lt

.. · J.oo G(s)ds ds1 · · · dsn-2i-1

Bn-2j-l

(5.4.11)

St

where G(s) = f(s, x(s - 0'1 (s )), ... , x(s - O'm(s ))). Integrating (5.4.11) from T2 to oo and noting (5.4.8) we have

This together with (5.4.3), (5.4.4) and (5.4.7) implies that

!.

{00 00 ... J.oo Fj,k(s)dsds1 ... dsn-2j < oo

}T2

"n-2j

(5.4.12)

Bt

t

if f is where Fj,k(s) = f(s, k(s - u 1 (s))li-I, ... , k(s - O'm(s))li- 1 ), k = superlinear, k = 324 iff is sublinear. This shows that (5.4.6) holds. Sufficiency. Assume that (5.4.6) holds. We may assume that k > 0. Let d = k/2 iff is superlinear and d =kif f is sublinear. Set R(t) = t 2 i- 1 • If p E (0, 1), we have limp

t-+oo

R(t)

= p

Choose p 1 E (p, 1) and T ~ to

.

and

R(t- T - h)

hm t-+oo

R(t-r) R( t )

= 1

>1-

1-p -4p -.

+ T + h such that

and

R(t-r) R(t)

1-p

> 1- ~ fort ~ T,

(5.4.13)

and

~T00 1.00

•n-2j

···

1.00 Fj,k( s) ds ds1 · · · dsn-2j < (1 -

p )d

-'---7--''---

•t

8

(5.4.14)

where Fj,k(s) = f(s, k(s- 0'1 (s)) 2i-I, ... , k(s- O'm(s)) 2i- 1 ). LetT* = T- r- h. Then pR(T)/R(T*) < P1· Introduce the Banach space:

CR[T*,oo) = { x: x E C((T*,oo),R),

t~'f. ~;g~

< oo}

330

Chapter 5

with the norm llxlln

=

sup lx(t)I/R2(t). Define a subset t~T·

n of Cn[T*,oo)

as

follows:

n=

{x:

E Cn[T*' 00 ), dR(t)

X

:5 x(t) :5 2dR(t)}.

Then n is a bounded, convex and closed subset of C n[T*, oo ). Define two maps 7i and T2 : n --+ R [T*' 00) as follows:

c

3d(l + p) R(t)2

px~;:) r) R(t),

T* :5 t < T,

(1ix)(t) = { 3d(l + p) R(t)- px(t- r), 2

t

~ T,

and

T* :5 t < T,

0,

(T2x)(t)

=

rt r•n-1 ... Jr Jr

r•n-2j+2

lr

···100

100

...

•n-2i+t

G(s) ds ds1 · · · dsn-b

t

~ T,

•t

where G(s) = f(s,x(s- a 1 (s)), ... ,x(s- am(s))). It is not difficult to see that foranyx,yED, 1ix+T2yED. We show that 7i is a contraction. Let x, y ED. Then, forT* :5 t < T 1(7ix)(t)- (7iy)(t)l lx(T- r)- y(T- r)l R2(t) = p R(t)R(T)

and

fort~

= P

lx(T- r)- y(T- r)l R 2(T- r) R2(T- r) R(t)R(T)

:5 p

lx(T- r)- y(T- r)l R(T) R 2(T- r) R(T*) :5 PI

t~u.f.

ix(t)- y(t)l R2(t) '

T

I('Iix)(t)- (1iy)(t)l lx(T- r)- y(T- r)l lx(t- r)- y(t- r)l R 2(t) = p R2(t) :5 P R2(t- r)

1, we can prove that Eq. (5.4.1) has an A2j- 1 (a,O)-type solution. 0

Chapter 5

336

Theorem 5.4.4. Suppose that n is odd, p ;::: 0, p =/= 1, f is superlinear or sublinear. Then Eq. (5.4.1) has an Az;(oo,a)-type nonoscillatory solution if and only if there is a k =/= 0 such that

1

00

sn-Zj- 1 lf(s, k(s -

0'1 (s )) 2 j, ... ,

k(s - O'm(s )?j-l )Ids < oo

to

where j E {1, 2, ... ,

n;- 1 }.

Proof: The proof is similar to that of Theorem 5.4.2. We need only note that R(t) and 72 there are replaced by R(t) = t 2 i and 0,

(Tzx)(t)

=

{

T*

~

lt 13n-1 T

t 1, respectively.

D

Theorem 5.4.5. Suppose that n is odd, p ;::: 0, p =/= 1, f is superlinear or sublinear. Then Eq. (5.4.1) has an A2;(a,O)-type nonoscillatory solution if and only if there is a k =/= 0 such that

1

00

sn- 2 ilf(s, k(s -

O't (s )) 2 i-l, ... ,

k( s -

O'm)) 2 j-l )Ids

< oo

to

where j E {1, 2, ... , n;- 1 }. Proof: The proof is similar to that of Theorems 5.4.3 and 5.4.4.

D


337

Theorem 5.4.6. Suppose that n is odd, p 2:: 0, p =f. 1, f is superlinear or sublinear. Then Eq. (5.4.1) has an Ao(a)-type nonoscillatory solution if and only if there is a k =f 0 such that roo sn-llf(s,k, ... ,k)!ds

ito

< 00.

Proof: The proof is similar to that of of Theorem 5.4.2. We need only note that R(t) and 72 there are replaced by R(t) = 1 and roo J.oo .. · J.oo G( s) ds ds 1 .. · dsn-b

}T (Tix)(t) = {

•n-1

100 !.00 .. ·!.00 t

Bn-1

T* '5: t

0 such that

roo

ito

ap91(t)fr' ... 'ap9N(t)fr) dt

tn-1 fL-tfr F(t,

0

such that pc(1

+ w±)f(p- J.L) :5 a,

To= min{T- T, inf 9I(t), ... , inf 9N(t)} > max{to, T }, t~T

t~T

-

(5.5.7)

and (5.5.8) Let X be the set of functions defined by X= {x E C[To,oo): lx(t)l :5 CJ.Ltfr fort~ T0 }

which is a closed convex subset of Frechet space C(T0 , oo) of continuous functions on [To, oo) with the usual metric topology. With each x E X we associate the

342

Chapter 5

functions X±(t): [T0 ,oo)-+ R defined by ...1!!:._ p 11rw±(t)- f(=F1)ip-ix (t

X±(t) = { p -~-~

+ ir), t;:: T- r, (5.5.9)

i=1

X±(t)(T- r),

To

~

t

~

T- r.

It is easy to see that, for each x E X, X±(t) are well defined and continuous on [To, oo) and satisfy $±(t)±p$± (t-r)=x(t) , t;::T.

(5.5.10)

x E X implies that

(5.5.11) Substituting (5.5.11) into (5.5.9) we obtain

IX±(t)l ~ pew± pt/r + _!!!:.__ 1-'t/r ~ pc(w± + 1) ptfr, P-1-'

P-1-'

P-1-'

t;:: T- r,

and hence we have

l$±(g;(t))l

~

pc(w± + 1) pfl;(t)/r p-1-'

~ apg;(t)/r, t;:: T, i = 1, ... , N.

(5.5.12)

Now we define the mappings T±: X-+ C[To,oo) as follows:

T±x(t)

={

(-1)n- 1

l

eo

(s _ t)n-1 (

T±x(T),

_

n

1

To

~

t

~

1) 1 f(s,$±(g(s)) )ds, ·

t;::T (5.5.13)

T

where f(s,$±(g(s)) = j(s,$±(g1(s) ), ... ,$±(gN(s))) . It needs to show that T± are continuous and map X into compact subsets of X, so that Schauder-Tychonoff fixed point theorem is applicable to T±. In fact, if x E X, from (5.5.13), (5.5.3), (5.5.12) and (5.5.8) we obtain

343


~

p.t/r /, 00 sn- 1 p.-•/r F(s, Jf±(g1(s))J, ... , 1f±(gN(s))1) ds

which implies that T±.x E X. Thus T±.(X) C X. Let {xk} be a sequence in X converging to an x E X in the topology of C[To, oo ). Since 00

00

2)±1)ip-ixk(t + ir) ..._. 'L)±1)ip-ix(t + ir)

ask~

00

i=1

i=1

uniformly on every compact subinterval of [To, oo ), the Lebesgue dominated convergence theorem shows that T±.xk(t) ~ T±.x(t) uniformly on compact subintervals of [To, oo) as k ~ oo. This shows the continuity of T±.· Finally, since the inequalities

J(T±.x)'(t)J ~ F(t,apgt(t)fr, ... ,ap9N(t)fr),

t;?: T, for n = 1

and

hold for all x E W, we conclude that the sets T±.(X) have compact closure in C[To, oo) by the Ascoli-Arzela Theorem. Therefore, applying the SchauderTychonoff Theorem we see that there exist ~±. E X satisfying

~±.(t)

roo (s- t)n-1

• (n _ 1)! f(s,~±.(g(s)))ds,

= (-1t- 1 lt

t ;?: T.

From (5.5.10) we have •

•

~±.(t)±P~±.(t-r) = (-1t-

1

/,

t

t)n-1 • (n _ 1)! f(s,~±.(g(s)))ds,

oo ( s-

t;?: T. (5.5.14)

This concludes that t±.(t) are solutions ofEq. (5.5.1) and Eq. (5.5.2) on [T0 ,oo), respectively.

Chapter 5

344

From (5.5.9) and (5.5.11) we have

I

{::J:(t)-

~ ptfrw::l:(t)\

p-J.'

:5

~ 1-'t/r,

~ T- T

t

p-J.'

(5.5.15)

which shows that {::~:(t) satisfy the relations (5.5.6) and (5.5.5), respectively. It 0 is obvious that e::~:(t) are oscillatory. Theorem 5.5.2. that

Suppose that p > 1 and there exists a constant a > 0 such

r>a tn-1F(t,apgt(t)fr, ... ,apgN(t)fr)dt < oo.

(5.5.16)

lto

Then (i) for any continuous periodic oscillatory function w_(t) with period r, Eq. (5.5.2) has an unbounded oscillatory solution X-(t) such that x_(t) = k1ptfrw_(t)

+ o(1)

as t-+ oo;

(5.5.17)

(ii) for any continuous oscillatory function w+(t) such that w+(t+r) = -w+(t) for all t, Eq. (5.5.1) has an unbounded oscillatory solution x+(t) such that

(5.5.18) where k1 and k2 are constants.

Proof: The proof is similar to that of Theorem 5.5.1. Choose c > 0 and T such that, pc(1 +w±)/(p-1) :5 a, (5.5.7) holds, and !roo tn-1 F(t, apgl (t)/r' ... 'apgN(t)fr) dt

Define a set Y C C[To, oo) and mappings T::1: : Y

-+

:5 c.

~ t0

(5.5.19)

C[To, oo) as follows:

Y = {y E C[To,oo): jy(t)l :5 c, t ~To}

T±y(t)

={

(-1)n- 1

T±y(T),

1

00

t

(s _

t)n-1 _

(

n

1) 1 f(s,fi::J:(g(s)))ds, .

To :5 t :5 T,

t~T (5.5.20)

345

Oscillation of Higher Order Neutral Equations where Y± : [To, oo)

--+

R are defined by 00

Y±(t) = ~ ptfrw±(t)- l:C=t=l)ip-iy(t + ir), p- 1 i=l

t

2:: T- r,

00

Y±(t) = ~ ptfr w±(t)- l:C=t=l)ip-iy(T + (i- 1)r), p -l i=l

To ~ t ~ T- r.

It is easy to see that y E Y implies that Y± E C[To, oo) and Y±(t) ±w±(t- r)

= y(t),

t;::: T.

Furthermore, since

~ (t)l < pew± tfr + _c_ < pc(w± + 1) ptfr I Y± - p- 1 p 11 ' p-

p-

t

>_ T.o.

And hence

IY'±(o;(t))\ ~ ap9 '(t)fr,

t 2:: T, i E IN = {1, ... , N}.

Proceeding as in Theorem 5.5.1, by the Schauder-Tychonofffixed point theorem, T± have fixed points 77± E Y and 7J±(t) satisfy Eq. (5.5.1) and (5.5.2), respectively. Since 77±(t) --+ 0 as t --+ oo, the solutions 7J±(t) satisfy (5.5.17) and (5.5.18), respectively. 0 With a slightly different discussion we can show the following result. Theorem 5.5.3. Suppose that p a > 0 such that

l

> 1, and there exist constants J.l E (1,p) and

oo J.l-tfrF(t,apg~(t)fr, ... ,apgN(t)fr)dt 0

where oi(t) = max{g;(t), t}, i E IN. Then

< oo,

(5.5.21)

Chapter 5

346

(i) for any continuous periodic oscillatory function w_(t) with period r, Eq. (5.5.2) has an unbounded oscillatory solution x_(t) such that (5.5.22)

(ii) foranycontinuousoscillatoryfu nctionw+(t) such thatw+(t+r) = -w+(t) for all t, Eq. (5.5.1) has an unbounded oscillatory solution x+(t) such that

where kt and k2 are constants.

Remark 5.5.1. For m = 1, 2, ... the functions acos(2mrrt/r) + bsin(2mrrt/r)

(5.5.23)

acos((2m- l)rrt/r) + bsin((2m- 1)rrt/r)

(5.5.24)

and

where a and bare constants with a 2 + b2 > 0, can be used as w_(t) and w+(t), respectively, in the above theorems. Example 5.5.1. Consider the equation (x(t)

± e- 1 x(t -1))(n) +

(1

± e- 2 )e(l-vB)tlx(Bt)l"sgnx(Bt) =

0

(5.5.25)±

where v and 8 are positive constants with vB > 1. Here F(t, v) = (1+e- 2 )e 0 such that for each e solution of the equation

(y(t)- (p- e)y(t- T))(n)

+ (q- e)y(t- u) =0, t?: to

(5.6.2)

e

[0, e0 ] every

(5.6.3)

is oscillatory. Proof: If not, for any e0 > 0 there exists e E [O,e 0 ] such that (5.6.3) has a positive solution. Then the characteristic equation of (5.6.3) (5.6.4)

Chapter 5

348

has a negative real root .X •. On the other hand, by the assumption the characteristic equation of (5.6.2)

has no real roots. F(oo)

= oo and F(-oo) = oo implies that

F(.X)

~

m

> 0. Set

Since G( -oo) = oo, there exists a > 0 such that G(p.) > 0 for Jl < -a. Choose co E (0, such that p- co ~ p/2, q- co ~ q/2, and co(anear + e"'") $ m/2. For c::; co, if p. < -a, then

t)

if 0

> p.

~

-a, then

m

m

> F(u)can ear- ce"'" > m -2 - => 0. ,.. 2 That is, there is an co > 0 such that for every c E [O,co] (5.6.4) has no real roots. 0 We reach a contradiction. Theorem 5.6.1. Assume that (i) T > 0, u ~ 0, p, q E C([to,oo),R+), g, hE C(R,R); (ii) g(u)u ~ 0 as u =f. 0, lim g(u)/u = 1, and lg(u)l $ Llul for L > 0; u-+0

(iii) uh( u) > 0 for u =f. 0, lim h( u )/u = 1, and lh( u )I ~ h0 > 0 for lui sufficiently u-+0 large; (iv) 0 0, liminf q(t)fq(t- r) = M > 0, and p 1 M < 1; t---.oo

t-.oo

(vi) every solution of the linear equation

(y(t)- P1My(t- r))(n)

+ qoy(t- u) =

is oscillatory. Then every solution of (5.6.1) is oscillatory.

0,

t ~to

(5.6.5)

349


Proof: Assume the contrary, and let x(t) be an eventually positive solution of (5.6.1). Set z(t) = x(t)-p(t)g(x(t-r)). Thenz(t) :50. If lim z(n-l)(t) = -oo, t ..... oo

then lim z(i)(t) t-oo

= -oo,

i

= 0, 1, 2, ... , n- 1.

(5.6.6)

Thus x(t) is an unbounded solution and there exists a sequence {tk} such that tk-+ oo and limk_, 00 x(tk) = oo where x(tk) = maxto::569k x(s) fork= 1,2, .... Hence

z(tk) = x(tk)- p(tk)g(x(tk- r)) = x(tk)- p(tk) g(xt"- ri) x(tk- r)

X tk- T

which contradicts (5.6.6). Therefore, lim z(n-l)(t) = £ E R. Then integrating t-+oo

(5.6.1) from t 1 to oo we find that

1

00

q(s)h(x(s- O'))ds < oo

(5.6.7)

t,

which implies that liminfx(t) = 0. Let {tk} be a sequence such that t-+oo

lim

k-+oo

x(t~:)

t~:-+

oo and

= liminf x(t) = 0. Then lim z(t~:) :50. On the other hand, t--+oo k-+oo

Since z(t) is monotonic, we have lim z(t) = 0. Then it is easy to see that t-+oo

lim z(i)(t)=O,

t-+oo

i=0,1,2, ... ,n-1,

(5.6.8)

and lim x(t) = 0. We rewrite Eq. (5.6.1) in the form t ...... oo

(x(t)- P(t)x(t- r))(n)

+ Q(t)x(t- 0') = 0

(5.6.9)

where

P(t) = p(t)g(x(t- r)) x(t- r)

and

Q() t

= ( )h(x(t- r)) qt

( ) . xt-r

(5.6.10)

Chapter 5

350 Hence z(t)

= x(t)- P(t)x(t- r), and (5.6.9) becomes

z(t)- P(t- u)

Q(t) z(t- r) Q(t- r)

+ Q(t)z(t- u) = 0.

(5.6.11)

For any small c > 0, Q(t) ~ qo(1- c), and P(t- u)Q(t)/Q(t- r) > p 1 M(1- c) for all large t. Then from (5.6.11) we have

z(t)- p 1 M(1- c)z(n)(t- r) + qo(1- c)z(t- u) :::; 0.

(5.6.12)

Integrating (5.6.12) n times, we have

qo(1- c) z(t) ~ p 1 M(1- c)z(t- r) + (n _ 1)!

joo (s- tt

-1

z(s- u) ds

(5.6.13)

joo(s-tt _z(s-u)ds

(5.6.14)

t

for all large t. By Lemma 5.1.2 the equation

qo(1- c) u(t)=p 1 M(1-c)u(t-r)+ (n- 1)!

1

t

has an eventually positive solution u(t). Hence the equation ( u(t)-

p 1 M(1- c)u(t- r) )

(n)

+ qo(1- c)u(t- u) = 0

has an eventually positive solution. This contradicts (vi) by Lemma 5.6.1.

0

We consider the nonlinear neutral differential equation

(x(t)- p(t)g(x(r(t))))(n)

+ q(t)h(x(u(t))) = 0,

t ~to

(5.6.15)

where n is an odd integer. The following is an existence result of nonoscillatory solutions of Eq. ( 5. 6.15).

Theorem 5.6.2. Assume (i) p, q E C([to,oo),R+)i (ii) g, hE C(R,R), xg(x) > 0, xh(x) > 0 as x such that Jg(x)- g(y)J :::; Llx- yJ,

=f. 0,

and there exists an L

x, y E [0, 1],

>0

351


and h is a nondecreasing function; (iii) r, u E C([to, oo ), R), 0:::; t- r(t) :::; M, t lim u(t) = oo;

~to,

where M is a constant, and

t-oo

(iv) there exist a> 0 and c E (0, 1) such that Lp(t)ea(t-r(t)) :::; c < 1 and

Lp(t)ea(t-r(t))

ioo + (n eat -1)! t (s- tt-lq(s)h(e-acr(s})ds:::; 1,

t

~to.

Then Eq. (5.6.15) has an eventually positive solution x(t) which tends to zero exponentially as t -+ oo.

Proof: Denote T =min { inf r(t), inf u(t)}. Let BC denote the Banach space

t;;::to

t2:to

of all bounded continuous functions defined on [T, oo) with the sup norm. Let be the subset of BC defined by Q = {y E BC:

0:::; y(t):::; 1,

t0

:::;

n

t < oo}.

Clearly n is a bounded, closed and convex subset of BC. Define operators ~ and 72 on n as follows:

p(t)eatg(y(r(t))e-ar(t>),

(~y)(t)=

{

t to

-(~y)(t 0 )

,f

+

(

t)

1-- ,

to

t

~to

T:::;

t:::; t 0 ,

00 eat (s- tt- 1 q(s)h(y(u(s))e-acr(•)) ds, (T2y)(t)= { (n-1). t t t 0 (72y)(to), T:::;t:=;to.

t

~to

By (iv), for every pair x, y E Q we have ~x + 72y E Q. Conditions (ii) and (iv) imply that ~ is a contraction on n. It is easy to see that

where M 1 is a positive constant. Thus 72 is completely continuous. By Krasnoselskii's fixed point theorem there exists a y E Q such that (~ + 72)y = y.

Chapter 5

352 That is,

p(t)e"'tg(y( r(t))e-arr(tl) y(t) =

+ (n e~t1)! tto y(to)

ioo

+ ( 1-

It is easy to see that y(t)

x(t) = p(t)g(x(r(t)))

(s -tt-l q(s )h(y( a(s ))e -ar.,.(•l) ds,

t:) ,

T

~to

t

~to.

:5 t :5 t 0 •

> 0 fort~ T. Set x(t)

+ (n -1. 1 )'

t

1

00

= y(t)e-"' 1 • Then

(s- tt- 1 q(s)h(x(a(s)))ds,

1

This implies that x(t) is a desired solution of (5.6.15).

0

Example 5.6.1. Consider

(5.6.16) where q(t) = e-it - fe 3 e-~ 1 > 0 for all large t. In our notations p(t) = 1/4, g(x) = x 3 , h(x) = x!, and L = 3. Obviously, the hypotheses of Theorem 5.6.2 are satisfied. Therefore, Eq. (5.6.16) has a solution x(t) which tends to zero exponentially as t-+ oo. In fact, x(t) = e-t is such a solution of (5.6.16). We now consider the equation

(x(t)- g(x(t- r)))(n)

+ h(t,x(t- a))= 0,

t ~to

(5.6.17)

where n is an odd integer. Theorem 5.6.3. Assume that (i) T > 0, a~ 0, g E C(R,R), hE C([to,oo) x R,R); (ii) g is nondecreasing, xg(x) ~ 0 for x E R, and there exists a a E (0, 1] such

that

.

Jg(x)l

hmsup-- = t-oo JxJ"' lxl-oo

{ ME [0, oo)

c E (0 ' 1)

for a E (0, 1) for a= 1;

353

Oscillation of Higher Order Neutral Equations (iii) his nondecreasing in x, h(t,x)x 2::: 0 for (t,x) E [to,oo) nonzero constant f3

1

00

X

R, and for any

sn-I h(s, {3) ds = oo ·sign {3;

to

(iv) there exists a positive integer N such that

y(t) + h(t, y(t- a)+ g(y(t- a- r)

+ g(y(t- a- 2r) + · · · + g(y(t- a- Nr)) · · · )) :S 0 (5.6.18) has no eventually positive solutions, and

y(t) + h(t, y(t- a)+ g(y(t- a- r)

+ g(y(t- a- 2r) + · · · + g(y(t- a- Nr)) · · ·))

2::: 0

(5.6.19) has no eventually negative solutions. Then every solution of Eq. (5.6.17) is oscillatory. Proof: We prove it for the case that a = 1 in condition ii). The case that a E (0, 1) can be similarly proved. Assume the contrary, and let x(t) be an eventually positive solution of Eq. (5.6.17). Set

z(t) = x(t)- g(x(t- r)).

(5.6.20)

z(t) = -h(t, x(t- a)) :S 0.

(5.6.21)

Then eventually

Therefore, z(il(t) are eventually monotonic, i = 0, 1, ... , n- 1. Now let us consider the following two possibilities: z(t) < 0 and z(t) > 0 eventually. Assume z(t) < 0 eventually. Since n is an odd integer, it follows from Eq.(5.6.17) that z'(t) < 0 eventually. Then either lim z(t) = -oo,

t-+oo

(5.6.22)

Chapter 5

354

or lim z( t) = -l E ( -oo, 0).

(5.6.23)

t-oo

If (5.6.22) holds, we have from (5.6.20) that lim g(x(t - r)) = oo, and so t-oo x(t) -+ oo as t -+ oo. This implies that there exists a sequence {ek} such that lim x(ek) = oo where x(ek) = max{x(t): to :5 t :5 ek}. Hence from condition ii) k-oo

which contradicts (5.6.22) and so (5.6.23) holds. Thus we have (-1)iz(i)(t) > 0, i = 1, 2, ... , n- 1, eventually. Then eventually

z(t) = x(t)- g(x(t- r)) < Hence there exist t 1 ~to and f3 > 0 such that x(t) this into Eq. (5.6.17) we obtain

~

z(t) + h(t,/3) :50 fort~ t1

l 2.

f3

fort~

t 1. Substituting

+ u.

(5.6.24)

Multiplying (5.6.24) by tn-l, then integrating it from t 1 + u to t, and letting t -+ oo we obtain from condition iii) that (5.6.25) On the other hand,

i

t

sn-lz(n)(s)ds = F(t)- F(t1

+ u)

(5.6.26)

tt+D'

where n-2

F(t)

= tn-lz(n-l)(t) + L( -1)i(i + 1)(i + 2) · · · (n -1)tiz(i)(t) i=O

> (n -1)z(t).

(5.6.27)

Combining (5.6.25)-(5.6.27) we obtain that lim z(t) = -oo, contradicting (5.6.23). t-oo

355

Oscillation of Higher Order Neutral Equations Assume z(t) > 0 eventually. Then

x(t) = z(t) ~

+ g(x(t- r)) =

z(t) + g(z(t- r)

z(t)

+ g(z(t- r) + g(x(t- 2r)))

+ g(z(t- 2r) + · · · + g(z(t- Nr)) · · · )).

Substituting this into Eq. (5.6.17) we find that

z(t) + h(t, z(t- u)

+ g(z(t- u- r)) + g(z(t- u- 2r)) + · · · + g(z(t- u- Nr)) · · · )) $

0

0

which contradicts condition (iv).

Theorem 5.6.4. In addition to condition (i) of Theorem 5.6.3 we assume that (i) g is nondecreasing, xg(x) ~ 0 for x E R, and there exists ad> 0 such that g(d) < d; (ii) his nondecreasing in x, h(t,x)x ~ 0 for (t,x) E [to,oo) X R, and

1

00

sn-tlh(s, c)l ds < oo

for any c :f; 0.

(5.6.28)

to

Then Eq. (5.6.17) has an eventually positive solution. Proof: Let (3 > 0 be such that (3 T ~ t 0 such that

£

00

t

~

+ g(d)
0 fori= 0, 1, ... , n and lim y(t) = 0. It follows that lim Q(t) =co. By Theorem 5.3.3 in [4] (5.6.34) t~oo t-+oo has no eventually positive solutions. This contradiction shows that condition (iv) is satisfied. Then all the conditions of Theorem 5.6.3 are satisfied, and so every D solution of (5.6.32) is oscillatory. Similarly, we have the following result. Theorem 5.6.6. Assume p = 0, are oscillatory.

f3 E (0, 1). Then all solutions of Eq. (5.6.32)

5.7. Unstable Type Equations We consider the existence of positive solutions of higher order neutral delay differential equations of the unstable type

(x(t)- p(t)x(t- r))(n) = q(t)x(t- q),

t ~ t0

(5.7.1)

where

n is an even integer,

T

> 0,

q

~

0, p, q E C([t 0 , co), R+ ).

(5.7.2)

Theorem 5.7.1. Eq. (5.7.1) has a positive solution on [t 0 ,co) which tends to infinity as t -+ co.

Chapter 5

358

Proof: Set

Pt(t) = {

p(t),

t

~to

(5.7.3)

p(to), and

qt(t) = {

q(t), (5. 7.4)

q( to), Then p 1 and q1 are nonnegative and continuous on R. Let (5. 7.5) where

(5. 7.6) Clearly,

p1 is continuous and nondecreasing on R. Now define

y(t)=exp[

ltf'

exp((

0 -2T

Then for t

~

n

~ 2 ) ·1 ltt

(s-u)"- 2 H(u)du)ds].

(5.7.7)

0 -2T

to

Pt(t) y(t(t) r) =pt(t)exp [ Y

lt ( t-T

exp ( _1 2)1 n ·

:::; Pt(t)exp [- rexpc;rPt(t))] :::; Pt(t)exp (

-~Pt(t)) :::; ~·

1•

t 0 -2T

)]

(s-u)"- 2 H(u)du ds


359

That is, 1

4y(t)

~

~to.

(5.7.8)

i = 0, 1, 2, ... , n,

(5.7.9)

p1(t)y(t- T),

t

From (5.7.7) we find that

y(t) > 0, and for t

~

to - 2T

y(t) > H(t)exp[( It follows that for t

n

~ 2)I.

1'

(t- u)n- 2 H(u)du]y(t)

to-2r

~ H(t)y(t).

to - 2T

~

q1(t)y(t- u) < q1(t)y(t- u) < q1(t) < ! y(t) H(t)y(t) - H(t) - 4. That is, (5.7.10) Integrating (5.7.10) from to- 2T tot we obtain 1 y(t) ~ ( _1 1 ) 1 -4

n

·

1'

fort~ t 0 -

2T n times and using (5.7.9),

(t- st- 1q1(s)y(s- u)ds,

t

~to-

(5.7.11)

2T.

to-2r

Combining (5.7.8) and (5.7.11), we have

-1 2 y(t) ~ PI(t)y(t- T)

+ ( _1 1) 1 n

"

1'

(t- st- 1q1 (s)y(s- u) ds,

t

~to.

to-2T

(5.7.12) Let BC denote the Banach space of all bounded and continuous functions defined on R with the sup norm. Set

n=

{z E BC: 0 :5 z(t) :51, -oo < t < oo}.

360

Then

Chapter 5

n is a bounded, closed and convex subset of BC.

Define a mapping T :

n-+ BC as

y~t) [Pl(t)y(t- r)z(t- r) + (n =1)!(t- to+ r)n-l (Tz)(t) =

+(

n

~ 1)'. j'to-2r (t- st- 1q1(s)y(s- u)z(s- u)ds],

(Tz)(to),

t;::: to

t :5 to. (5.7.13)

In view of (5.7.5) and (5.7.7) we have fort;::: t 0 y(t);::=:exp[1'

to-2r

;::: e 1'

to-2r

;::: ( e2 _ )' n 1.

exp((

exp ((

J'

t-r

n

n

~ 1 ).1(s-to+2rt- 1)ds]

~ 1)'(sto+ 2rt.

1)

ds

(s-to+2rt- 1 ds (5.7.14)

By (5.7.12)- (5.7.14), we see that 0 :5 (Tz)(t) :51, t E R, which shows that T maps n into itself. Next we will show that T is a contraction on fl. In fact, for any ZI, Z2 E n and t ;::: to i(Tzi)(t)- (Tz2)(t)i :5 ytt) [PI(t)y(t- r) iz1(t- r)- z2(t- r)i

+(

n

~ 1) 1.

1'

to-2r

(t- st- 1PI(s)y(s- u)

x iz1(s- u)- z2(s- u)i ds] 1

:5 2llz1 - z2il· Hence


361

which shows that T is a contraction on n. Then by the Banach contraction principle, T has a fixed point z E n, that is,

ytt) [p1(t)y(t- r)z(t- r)

+ ( ~ )'

z(t) =

n

1.

jt

+

(n:

1)!(t- t 0

+ r)n- 1

(t- st- 1q1(s)y(s- u)z(s- u) ds],

t

to-2r

~to

t :5 to.

z(to),

Set x(t) = y(t)z(t). Then x(t) is a positive continuous function on Rand satisfies

x(t) = P1(t)x(t- r)

+ ( _1 )' n

1.

jt to-2r

(t- st- 1q1(s)x(s- u)ds

+ (n _T 1)! ( t- to+ r )n-1 , t

~

t0 •

It is easy to see that x(t) is a positive solution of Eq. (5.7.1) and satisfies that x(t) ~ oo as t ~ oo. D

Remark 5.7.1. Theorem 5.7.1 is also true ifn is an odd integer. In the following we present some results on existence of a bounded positive solution of Eq. (5.7.1).

Theorem 5. 7 .2. Assume there exists an o: E (0, oo) such that 0 < p(t)e"'r :5 p < 1,

t

~to,

(5.7.15)

and

p(t)e"'r

1

roo

+ (n _ 1)! lt

(s- t)n- 1q(s)exp[o:(t- s + u)] ds :51,

t

~ t0 •

(5.7.16)

Then Eq. (5.7.1) has a bounded positive solution on [t 0 ,oo) which converges to zero exponentially as t ~ oo.

Chapter 5

362

Proof: Let m = max{ r, u} and let BC denote the Banach space of all bounded and continuous functions defined on [to - m, oo) with the sup norm. Let Q be the subset of BC defined by Q=

{y E BC: 0 S y( t) S 1 for t

~

to - m}.

Clearly, Q is a bounded, closed and convex subset of BC. Define two mappings Tt and 72 on n as follows:

p(t)eary(t- r), (Tty)(t)=

{

~to

t

t+h ( t+h) to+ h (Tty)(to) + 1 - to+ h '

(5.7.17)

to-mS t S to

and 1 I {oo(s-tt-lp(s)y(s-u)ea(t-s+cr)ds, (Tiy)(t) = { (n -1). lt

(Tiy)(to),

to-mS t S to

(5.7.18) where his a constant satisfying t 0 + h > 0. In view of (5.7.15) and (5.7.16) for any Yt, Y2 E Q, we have 'TiYt + TiY2 E Q. Using (5.7.15) one can verify that Tt is a contraction on Q. On the other hand, it is easy to see that ift(Tiy)(t)i is uniformly bounded for y E Q, and hence 'J2 is completely continuous on Q. Therefore, by Krasnoselskii 's fixed point theorem, there exists a y E Q such that 'Jiy + Tiy = y. That is, fort ~to

y(t) = p(t)eary(t-r)+ (

1 )' n -1.

1

00

(s -tt- 1 q(s)y(s -u)ea(t-•+cr) ds (5.7.19)

1

and for to - m S t S to

t+h ( t+h) y(t) = to+ h y(to) + 1 - to+ h > O. It follows that y( t) > 0 for all t ~ t 0

x(t)

-

(5.7.20)

m. Set

= y(t)e-at.

(5.7.21)


x(t)

= p(t)x(t- r) + (n _1 1)!

1

00

1

(s- tt- 1 q(s)x(s- u) ds,

t

~to.


363

Consequently, x(t) defined by (5.7.21) is a positive solution of (5.7.1) on [to, oo), and x(t) converges to zero exponentially. 0 With Eq. (5.7.1) we associate the linear equation with constant coefficients

(x(t)- px(t- r))(n)

= qx(t- u).

(5.7.22)

We have the following corollary. Corollary 5.7.1. Assume that 0 < p(t) ~ p < 1, 0 ~ q(t) ~ q, t ~to, and the characteristic equation of (5. 7.22)

has a positive root. Then Eq. (5. 7.1) has a bounded positive solution on [to, oo) which converges to zero exponentially as t --+ oo. Theorem 5.7.3. Let p(t)

~

1. Assume

f'JO sn-lq(s)ds = 00.

},.

(5.7.23)

Then all bounded solutions of Eq. (5. 7.1) are oscillatory. Proof: Let x(t) be a bounded positive solution of Eq. (5.7.1) and set

z(t) = x(t)- x(t- r).

(5.7.24)

Then it is not difficult to see that eventually

(-1)iz(i)(t) > 0,

i=

o, ... ,n.

(5.7.25)

Hence x(t) > x(t- r) eventually. Then there exists M > 0 such that x(t) ~ M, t ~ t~, where t1 is a sufficiently large number. From (5.7.1)

(5.7.26)

Chapter 5

364

We note that (5.7.25) and (5.7.26) lead to

roo sn-lq(s)ds < oo.

(5.7.27)

lto

This contradicts {5.7.23), and hence concludes the theorem.

0

=

Theorem 5.7.4. Assume p(t) solution if and only if

1. Then Eq. (5.7.1) has a bounded positive

(5.7.28)

Proof: Necessity. Let x(t) be a bounded positive solution of Eq. {5.7.1) and z(t) be defined by {5.7.24). In view of (5.7.25) we have

z(t) 2:: (n _1 1)!

1

00

1

(s- t)n- 1 q(s)x(s- u)ds,

or equivalently,

x(t) 2:: x(t- r) + (n _1 1)!

1

00

1

(s- t)n- 1 q(s)x(s- u)ds.

(5.7.29)

As in the proof of Theorem 5.7.3 we have

1

00

sn- 1 q(s)ds < oo.

(5.7.30)

to

Set

Let t1 2:: to+ m be such that {5.7.29) holds fort 2:: t 1 - m. Since x(t) 2:: x(t- r) fort 2:: t1 - m, there exists an M > 0 such that x(t) 2:: M fort 2:: t 1 - m. Thus _ 1)! x(t) ~ x(t- r) + (n M = x( t - r)

+M

1

00

1

00

1

(s- t)n-lq(s)ds

Q( s) ds,

t

~ t1,

365

Oscillation of Higher Order Neutral Equations and then

Since x(t) is bounded, it follows that

f1

00

i=l

(5.7.31)

Q(s) ds < oo.

.

tt+•r

According to Theorem 3.3.3 we have

1

00

sQ(s) ds < oo

(5.7.32)

tt

which implies that (5.7.28) holds. Sufficiency. (5.7.28) implies {5.7.32). By Theorem 3.3.3 we have

Choose t1

~to

so large that

Define a function H as follows:

1 { =

00

H(t)

Q(s) ds,

(t- t1 0,

+ r)H(tl)/r, t :5 t1 - r. 00

Then His a nonnegative and continuous function on R. Set y(t) = ,E H(t- ir). Then y(t) is continuous on Rand satisfies 0 < y(t) :51

y(t)

= y(t- r) + H(t),

t E R.

i=O

fort 2::: t 1 , and

Chapter 5

366

n

Let be the set of all continuous functions z defined on [t 0 , oo) and satisfying that 0 $ z(t) $ 1 for every t ;,:: t 0 • The set n is considered to be endowed with usual pointwise ordering$: z 1 $ z2 {:==} z 1(t) $ z2(t) for all t;,:: to. It is easy to see that for any subset A of n there exist inf A and sup A. Define a mapping T on n as follows:

- 1- [y(t- r)z(t- r) y(t)

(Tz)(t)

=

where h is a constant satisfying t 1 + h > 0. For any z E and t ;,:: t 1 we have

n

0$ (Tz)(t) $ ytt) [y(t- r)

+ (n ~ 1)! lX>(s- tt- 1 q(s)ds]

1 = y(t) [y(t- r) + H(t)] = 1,

i.e., Tn c n. Moreover, Tis a nondecreasing mapping. Therefore, by Knaster's fixed point theorem there exists a z E such that T z = z. That is, for t ;,:: t 1

n

z(t)

= ytt) [y(t- r)z(t- r) + (n ~ 1)!

1

00

(s- tt- 1 q(s)y(s- a)z(s- a)ds], (5.7.33)

and for to $ t

< t1 - h > 0. - h z(ti) + ( 1- -t+h) z(t) = -t+h tl + tl +

It follows that z(t) > 0 on [to,oo). Set x(t) = y(t)z(t). Then we conclude from 0 (5.7.33) that x(t) is a bounded positive solution of Eq. (5.7.1).

Theorem 5.7.5. Assume there exist positive constants M 1 and M 2 such that 1 < Pl $ p(t) $ P2, t;,:: to. Then Eq. (5. 7.1) has a bounded positive solution if


367

and only if

j oo sn- q(s)ds < oo. 1

(5.7.34)

to

Proof: Necessity. Let x(t) be a bounded positive solution of Eq. (5.7.1). Then x(t) > 0 fort ~ t1 >to. Set

z(t) = x(t)- p(t)x(t- r).

(5.7.35)

Then z(nl(t) = q(t)x(t -a)~ 0,

t ~ t1 +a.

According to the boundedness of x(t), we have ( -l)iz(il(t) > 0 fort~ t1 +a, i = 0, 1, ... , n- 1. Since x(t) > x(t- r) fort ~ t 1 +a, there exists a constant M > 0 such that x(t) ~ M > 0 fort~ t 1 +a. From (5.7.1) we have z(nl(t) ~ Mq(t), t ~ t1 +a, which leads to (5.7.34). Sufficiency. Let t 1 ~ to + r +a and ao > max{1, 2 (M~-l)} be such that 1

1.

00

(n- 1)! t+,.

t . 1 -

Consider the Banach space BC of all continuous bounded functions x defined on [to, oo) with the sup norm. Let n denote the subset of BC defined by fl= { xEBC:

Pl-1 } ( )~x(t)~2p1,t~to. ao P2 -1

Then n is a bounded, closed and convex subset of BC. Define a mapping T: n-+ BC as follows ( 1 ) [Pl- 1 + x(t + r) pt+r

(Tx)(t) =

(n (Tx)(tJ),

~ 1)! t0

1:

(s- t-

~

t

~ t1•

rt-

1

q(s)x(s -a)ds],

t

~t

1

Chapter 5

368

It is not difficult to see that T maps n into n and T is a contraction on n. Therefore, T has a fixed point X E n, that is, T X = x. Clearly, x( t) is a bounded D positive solution of Eq. (5.7.1) on [t1,oo).

We now present some bounded oscillation criteria for the even order equation

(x(t)- p(t)x(t- r))(n)

t ~to

= q(t)x(g(t)),

(5.7.36)

where n is an even integer, r > 0, p, q, g E C([to,oo),R+), g is nondecreasing, g(t) :5 t, and lim g(t) = oo. t-+00

Theorem 5.7.6. Assume that 0

lim sup

:5 p(t):::; p < 1 and

r (g(t)- g(s)t- q(s) ds > o. 1

(5.7.37)

Jg(t)

t-+oo

Then every bounded solution of Eq. (5. 7.36) is either oscillatory or tending to zero as t --+ oo. Proof: Let x(t) be a bounded positive solution of (5.7.36), say, 0 < x(t) : 0, k = 1, ... ,n, t ~ t 1, and hence lim z(t) = l E (-oo,oo). t-+oo

If R. < 0, then z(t) < 0 eventually which leads to lim x(t) = 0 and hence t-+oo lim z(t) = 0. We reach a contradiction. t-+oo If R. ~ 0, by Taylor's formula, for v ~ u ~ t 1

z(u)- z(v) =

~ (-1); z(v)(v -u)i + (-1)n-I z (- 1)n-l z(n-l)(v)(v- u)n-l - (n-1)!

'

v >_ u >_ t1,

369

Oscillation of Higher Order Neutral Equations and then

where t2 is so large that g(t);?: t 1 fort;?: t2. We rewrite (5.7.38) in the form

Multiplying the above inequality by q(s) we have

z(nl(s) -x(g(s))

1"

z 1(g(u))g'(u)du;?:

1

(-1)n-1 z (n -1)! (1- ap).

limsupl' t-+oo

(5.7.44)

t-(a-T)

Then every bounded solution of Eq. (5. 7.1) is oscillatory.

Proof: Let x(t) be a bounded positive solution and z(t) be defined by (5.7.35). Then ( -1)iz(i)(t) > 0, i = 0, 1, 2, ... , n. We rewrite Eq. (5.7.1) in the form z(t) = q(t)z(t- u) + p

q(t) z(t) 2: q(t)z(t- u).

(5.7.48)

w(t) 5 (1- kap)z(t- r).

(5.7.49)

Since z'(t) < 0,

Combining (5.7.48) and (5.7.49), we have that eventually 1

w(t) 2: 1 k q(t)w(t- (u- r)). - ap

(5.7.50)

373

Oscillation of Higher Order Neutral Equations By the boundedness of w(t), we know that eventually ( -1)iw(il(t) = 0, 1, 2, ... , n. Integrating (5.7.50) n times we have

> 0,

i

-w(t) + w(t- (u- T))

>

1

- (n-1)!(1-kap)

x

it

(s- (t- (u- T)))"- 1 q(s)w(s- (u- T)) ds

t-(u-r)

~

w(t - (u - T)) 1 (n- 1). (1- kap)

1 1

(s-(t-(u-T)))"- 1q(s)ds,

t-(u-r)

or

0

!

~ w(t)+w(t-(u-T))[( n _ 1)1. 1 _ kap )

which contradicts (5.7.47).

1 1

t-(u-r)

(s-(t-(u-r)))"- 1 q(s)ds-1]

0

5.8. Notes Lemmas 5.1.1-5.1.3 are taken from monograph (110]. Theorem 5.1.1 was first obtained by Gopalamy, Lalli and Zhang (54], the present form is taken from Zhang, Yu and Wang (215]. Theorems 5.1.2 and 5.1.3 are from (54]. Theorem 5.1.4 is adopted from (215]. Theorem 5.1.5 is based on (54]. Theorem 5.1.6 is taken from Yan (174], some further comparison results can be seen from Yu, Wang and Zhang [183], see also Ladas, Qian and Yan (108] for some other results. Theorems 5.2.1, 5.2.2, 5.2.4 and 5.2.5 are taken from Zhang and Yu [211]. Theorem 5.2.3 is adopted from Naito [140]. Theorems 5.2.6-5.2.8 are adopted from Zhang and Gopalamy [204]. Theorem 5.2.9 is from [211]. The material in Section 5.3 is taken from Zhang [192], also see Lalli, Ruan and Zhang [112] for the nonlinear equation case. The materials in Section 5.4 are adopted from Chen [18], also see Jaros and Kusano [90]. Section 5.5 is from Jaros and Kusano [91]. Theorem 5.6.1 is taken from Ladas and Qian [105], Theorems 5.6.2 and 5.6.3 are taken from Zhang and Yu [211]. Section 5.7 is taken from Chen, Yu and Zhang [12], also see Yu and Zhang [185].

6 Oscillation of Systems of Neutral Differential Equations

6.0. Introduction and Preliminaries Oscillation of systems of n neutral differential equations is an interesting and hard problem. In this chapter we will present some recent contributions. Here we use the definition for weaker oscillation given in Chapter 1, i.e., a vector solution is said to be oscillatory if at least one of its nontrivial components has arbitrarily large zeros. For the linear autonomous neutral delay differential system in the form m dN Q;X(t- u;) = 0 d N [X(t)- PX(t- T)} + i=1 t

L:

(6.0.1)

=

=

1, ... , m, 1, ... , m, are given n x n constant matrices, r, O'j j where P, Qj, j are nonnegative numbers, the above definition is equivalent to a stronger one, i.e., a vector solution of Eq. (6.0.1) is oscillatory if and only if its every component

374

375

SystemJ of Neutral Equations

has arbitrarily large zeros. There is a basic result for the oscillation of system (6.0.1) based on its characteristic equation: Theorem 6.0.1. equation

System (6.0.1) is oscillatory if and only if its characteristic

m

det [.XN (I- Pe->.r)

+L

Qie->.u;]

=0

j=l

has no real roots. Although this is a fundamental result, it is not easy to apply. Therefore, in Section 6.1, we give some explicit conditions for oscillation of (6.0.1) using Lozenskii measures, or logarithmic norms, of matrices. The criteria are sharp in the sense that even for the scalar case they are still the best known up to now. In Section 6.2 we consider system (6.0.1) for the case that Qj, j = 1, ... , m, are variable matrices. Oscillation criteria for (6.0.1) are obtained by comparing (6.0.1) with some systems with constant matrix coefficients. Those results are extended to a kind of nonlinear system and are applied to some Lotka-Volterra models. The nonautonomous system considered in Section 6.2 is discussed further in Section 6.3. Some comparison results for oscillation of systems with scalar equations are obtained. Section 6.4 deals with nonlinear systems of the form

dN dtN [x;(t) ± a;(t)x; (h;(t))] =

n

L Pj(t)/;j (x i(Yii(t))). i=l

Some results on the existence of nonoscillatory solutions are given. For the criteria of oscillation in Section 6.1 we need the following notations and definitions. For any n x n real matrix A we denote by .X;( A), i = 1, ... , n, the eigenvalues of A satisfying

Re.Xt(A);:::: Re.X2(A);:::: · · ·;:: Re.Xn(A). We d efi ne

IIAII·,-

sup

- - -, -IIAxll;

zER",z¢0 11X 11 i

. -1,2, ... ,oo, where x =

t

(x 1 , ... ,xn)T ,

Chapter 6

376

llxll; = ( En lx;l'.)1/i , i < oo, and llxlloo = j=l

m!IJC

l~J~n

{lx;l}. For each s. = 1,2, ... , oo,

the Lozenskii measures p;(A) of A are defined as follows: ·(A) J.ll

-

li

h-+~+

III+ hAll; h

1 I

and v;(A) = -p;( -A), i = 1,2, ... , oo. In general, without specification, we denote by p(A) and v(A) any pair of p;(A) and v;(A), i = 1,2, ... , oo. It has been shown that p;(A) and v;(A), i = 1, 2, ... , oo, exist for any n x n matrix A and can be explicitly calculated for i = 1, 2, and i = oo :

p 1 (A)

= s~p {a;;+ I

EIa;; I}.

v1 (A) =

i~f {a;;-

j=l

J.loo(A)

v2(A) =An

Ela;;l}, i=l

J

I};

#i

= A1 (~(A+ AT)),

= s~p {a;;+

Ia;;

J=l

j¢i

J.12(A)

t

(~(A+ AT));

v (A) = iqf {ajj- Eja;jl}· 00

i=l

J

i¢j

i¢j

For any n x n matrices A and B and any Lozenskii measures we have i)

p(A +B)

:5 p(A) + p(B),

= -p(A),

ii)

v( -A)

iii)

p(aA) = ap(A),

iv)

p(A);;:: ReA1(A),

v(A +B);;:: v(A)

+ v(B);

J.&( -A)= -1.1(A); v(aA) = av(A),

v(A) :5 ReAn(A).

a >0;

(6.0.2)

In Section 6.1, we will obtain some criteria for oscillation by using Lozenskii measures. Since the criteria are given by the general form of Lozenskii measures J.l and v, we will actually have infinitely many different results corresponding to each criterion in the theorems. Moreover, three of them, which are given by J.li and v;, i = 1, 2, oo, can be expressed explicitly. But for the scalar case, where p(A) = v(A) =A, all of them coincide to give the same results.

377

Systems of Neutral Equation&

6.1. Systems with Constant Matrix Coefficients To simplify the discussion and proofs we first consider a simpler equation

dN [X(t)- PX(t- r)] + QX(t- u) = 0 dt

-N

(6.1.1)

where P, Q are n x n matrices, r, u ~ 0. The following conditions will be used to determine the oscillation of Eq. (6.1.1):

f[v(P)]I: v(Q)(kr + u)N

(AI)

~ ( ~) N,

k=O

L•[Jl(P)]-(k+l) v(Q)[(k + 1)r- u]N

(A 2 )

~ ( ~) N,

1:

(A 3 )

L)Jl(P)]-(k+I} v(Q)[-(k

+ 1)r + u]N

~ ( ~) N

1:

where

L:*

and

L:.

1:

(k

denote the sums over all the terms for k E Z+ such that

1:

+ 1)r- u > 0 and -(k + 1)r + u > 0, respectively.

Remark 6.1.1. The inequality in (A 3 ) will become strict if the sum has only one term. As we mentioned in Section 6.0, a solution X(t) of (6.1.1) is oscillatory, if every component of it has arbitrarily large zeros. Theorem 6.1.1. Assume N is odd and v(Q) is sufficient for (6.1.1) to be oscillatory:

> 0. Then each of the following

i) Jl(P) = v(P) = 1, ii) 0

< v(P)

~

Jl(P) ~ 1, and (AI) holds,

iii) 1 ~ v(P) ~ Jl(P), and (A 2 ) holds, iv) v(P)

< 1 < Jl(P), and (At), (A 2 ) hold.

Theorem 6.1.2. Assume N is even and v(Q) > 0. Then each of the following is sufficient for (6.1.1) to be oscillatory:

Chapter 6

378

i) 0 < p(P) ii) p(P)

~

1, and (Aa) holds,

> 1, and (A 2 ),

(A 3 ) hold.

Remark 6.1.2. The condition (A 3 ) in Theorem 6.1.2 is required in the sense that if the set {k E Z+ : -(k + 1)T + o > 0} is empty and v(P) > 0, then Eq. (6.1.1) must have a nonoscillatory solution. In fact, the above assumption implies that r ~ T. If (6.1.1) is oscillatory, then (6.1.2) has no real root. Let

Then

p(F(O)) = p(Q) > 0 implies that p(F(A)) > 0 for all A E R. But

as A -+ -oo) This contradiction shows that all solutions cannot be oscillatory. The following lemma will be needed in the proofs of the results. Lemma 6.1.1. Let A bean x n real matrix. If either v(A) > 0 or p(A) < 0, then det(A) =/: 0. Proof: From (6.0.2), if v(A) > 0, then Re An(A) > 0. Hence ReA,(A) > 0 fori= 1,2, ... ,n. Thus det(A)

= A1(A) · · · An(A) =/: 0.

The case that p(A) < 0 is similar. Proof of Theorem 6.1.1:

0

The characteristic equation of (6.1.1) is (6.1.2)

i) Assume p(P) = v(P) = 1. Let

379

System3 of Neutral Equations Then v(F(O))

= v(Q) > 0, and

For.\> 0

v(F(.\)) ~ .\N(1- p(Pe-.\r)) + v(Q)e-.\ 17

= .\N(l- e-.\r) + v(Q)e-.\

17

(6.1.3)

> 0.

For.\< 0

v(F(.\)) ~ l.\lNv(- I+ Pe-.\r) ~

+ v(Q)e-.\

17

I.\IN ( -1 + vPe-.\r) + v(Q)e-.\ > 0. 17

Thus v(F(.\)) > 0 for all .\ E R. By Lemma 6.1.1 det F(.\) (6.1.2) has no real root. ii) Assume 0 ~ v(P) ~ p(P) ~ 1. Clearly .\ .\ > 0, by (6.1.3)

Hence det(F(.\))

=f. 0 for .\ >

=f. 0 for

.\ E R, i.e.,

= 0 is not a root of (6.1.2).

For

0 by Lemma 5.2.1. Let .\ = -s, and denote (6.1.4)

Then.\< 0 is a root of (6.1.2) if and only if s > 0 is a root of det(G(s)) = 0. By Lemma 5.2.1, if det(G(s)) = 0 has a roots> 0, then v(G(s)) ~ 0. Since N is odd,

Chapter 6

380

which implies that 0 $ v(P)e 8 r

< 1. As a result

00

= L)v(P)]k v(Q)es(kr+u)

(6.1.5)

k=O

>

~[v(P)]"v(Q) [s(kr;u)e]N·

The equality cannot hold since es(kr+u) attains its minimal value at a different point s for different k. Thus

(N)N ,

oo t;[v(P)]" v(Q)(kr + u)N < -;

contradicting (A 1 ). Therefore, (6.1.2) has no real root. iii) Assume 1 $ v(P) $ ~-t(P). Similar to ii) we see that A $ 0 is not a root of (6.1.2). Assume A > 0 is a root of (6.1.2). By Lemma 6.1.1 we have

(6.1.6) which implies that ~-t(P)e->.r

> 1. Therefore,

00

= 2)~-t(P)j-(Hl) v( Q) e>..[(Hl)r-u) k=O

>

~·[~-t(P)j-(HI) v(Q) [ A((k + ~T- u)e] N'

that is,

~·[~-t(P)j-(Hl) v(Q)[(k + 1)r- u]N < ( ~) N, contradicting (A2).

381

SyJtemJ of Neutral Eqv.ationJ

iv) Clearly, A = 0 is not a root of (6.1.2). From the proof of ii) and iii) we see that (AI) and (A 2) imply that any A > 0 and A < 0 can not be a root of (6.1.2). 0 Proof of Theorem 6.1.2: i) Assume p.(P) :5 1. Similar to the proof of Theorem 6.1.1 ii), we see any A ~ 0 is not a root of (6.1.2). Assume A = -s < 0 is a root of (6.1.2). Then by Lemma 6.1.1 and from (6.1.2)

Since N is even,

which implies that p.(P)ear > 1. As a result

00

= L[p.(P)]-(Hl) v(Q) e•[-(l:+I)r+u] l:=O

> ~Jp.(P))-(l:+I) v(Q) [ s[-(k +~)r + u]e] N

1

that is,

~..[p.(P)]-(l:+l) v(Q)[-(k + 1)r + u]N < ( ~) N Note that the inequality may become an equality if the sum has only one term, and this contradicts (A 3 ). ii) Assume p.(P) > 1. If A is a real root of (6.1.2), then A =f. 0 and AN > 0. By the proof of i), (Aa) implies that A < 0 cannot be a root of (6.1.2). By the proof of Theorem 6.1.1 iii), (A2) implies that A > 0 cannot be a root of (6.1.2). 0

382

Chapter 6

The above oscillation criteria for Eq. (6.1.1) can be easily extended to the equation

dN d N [X(t)- PX(t- r)] t

m

+L

by using the following conditions where q =

Q;X(t- Uj) = 0

(6.1.7)

i=l

(IT v(Q;)) /m, 1

j=l

(6.1.8) (6.1.9)

where

I:*, I:. k

are defined the same as before;

k

I:*, I:. j,k

denote the sums

j,k

over all the terms for 1 ::::; j ::::; m, k ;::: 0 such that ( k + 1 )r - u i > 0 and -(k + 1)r + Uj > 0, respectively. The inequality of (B 3 ) will become strict if the sum has only one term. Theorem 6.1.3. Assume N is odd and v(Qi) ;::: 0 but not all zero, j = 1, ... , m. Then each one of the following is sufficient for (6.1. 7) to be oscillatory:

i) J.L(P) = v(P) = 1, ii) 0::::; v(P) ::::; J.L(P) ::::; 1, and (BI) holds, iii) 1 ::::; v(P) ::::; J.L(P), and (B 2 ) holds, iv) v(P)

< 1 < J.L(P), and (B 1 ), (B 2 ) hold.

383

SystemJJ of Neutral EquationJJ

Assume N is even and v(Q;) ;::: 0 but not all zero, j = Theorem 6.1.4. 1, ... , m. Then each one of the following is sufficient for (6.1. 7) to be oscillatory:

i) 0 < p(P) :5 1, and (B 3 ) holds, ii) p(P) > 1, and (B2), (Ba) hold. Proof of Theorem 6.1.3 and 6.1.4: Similar to those of Theorem 6.1.1 and 6.1.2. To show the difference we only give an outline of the proof of Theorem 6.1.3 ii) as an example.

Corresponding to (6.1.5) we now have m

oo

sN;::: LL)v(P)]"v(Q;)e•(kr+"J)

(6.1.10)

j=lk=O

that is,

f.; t.;[v(P)]" m

oo

v(Q;)[kr + u;]N

(N)N ,

< -;-

contradicting (6.1.8). From (6.1.10) we also have

00

= mq L[v(P)]" e•(kr+") k=O

> mq ~[v(P)]" [ s(kr;

u)e]

N

1

Chapter 6

384

that is, oo

mq :L:lv(P)]"(kr

(N)N ,

+ u)N < -;-

k=O

0

contradicting (6.1.9).

The idea in the proofs of the above theorems may also be applied to the case that v(Q;) are not all nonnegative. As an example we give a result for the equations of the form

dN d

N

t

m

[X(t)- PX(t- r)]

+ :L:[G;X(t- u;)- H;X(t- r;)] = 0 ~1

where P, G;, H; are n x n matrices, v(P) > 0, v(G;)- p.(H;) zero for j = 1, ... , m, r, u;, Tj ~ 0, j = 1, ... , m.

~

(6.1.11)

0 and not all

Theorem 6.1.5. Assume N is odd. Then each one of the following is sufficient for (6.1.11) to be oscillatory. ~ 1, u; < Tj < r, and (B 2 ) holds for the case where v(Q;) are replaced by v(G;)- p.(H;), j = 1, ... ,m . .FUrthennore,

i) v(P)

where

v(G·)} 1 a= min { - - ln - -1l~j~m

T j - Uj

p.(H;) '

bJ. = _1_ ln v(G;)(r- u;), p. (Hj )( r- Tj ) Tj - Uj

j

= 1, ... , m.

ii) v(P) > 0, p.(P) $1, u; > Tj > O,and(B 1 )holdsforthecasewherev(Q;) are replaced by v(G;)- p.(H;), j = 1, ... ,m . .FUrthennore, v(P- 1 G;) > 0, p.(P- 1 H;) > 0, j = 1, ... ,m and

- (~)N

+a*N [v(P)tl

+ 't[v(P-lG;)e-bj.,.; J=l

-p.(P-lH;)e-bjr;] > 0,

385

Systems of Neutral Equations wbere a*= min { -1- l n v(P-lGj)}• l~j~m

Uj- Tj

p.(P-lHj)

1 v(P- 1 G ·)u· b"! = - - ln ' ', J Uj-Tj p.(P- 1 Hj)Tj

j = 1, ... , m.

Proof: The characteristic equation of Eq. (6.1.11) is det

(AN(I-Pe-~r)+ t(Gje-~""i -Hje-~ri))

=0.

(6.1.13)

J=l Let m

F(A) =AN (I- Pe-~r) + L)Gie-~""i - Hie-~r1 ).

j=l Then A= 0 is not a root of (6.1.13) since the assumption before Theorem 6.1.5 implies that

v(F(O)) =

Hence det F(O)

v(~(Gi- Hi))~ ~(v(G;)- p.(Hi)) > 0.

f. 0.

i) Assume v(P) Lemma 6.1.1

~

1 and Uj < Tj < r. If A > 0 is a root of (6.1.13), then by m

0 ~ v(F(A)) ~ AN(1- p.(P)e-~r) + ~:)v(Gj)e-~"i- p.(Hj)e-~ri) j=l m

~ AN(l- p.(P)e-~r)

+ :E(v(G;)- p.(H;)) e-~ri. j=l

This is a similar inequality to (6.1.6) for Eq. (6.1.1). By a similar discussion we can get a contradiction to (B2) where v(Q;) are replaced by v(Gj)p.(Hi),

386

Chapter 6 j = 1, ... , m. If .X

< 0 is a root of (6.1.13), let .X= -s, and denote m

rp(s) = -sN(Ie-.r- P) + L(G;e-•(r-.,.;)- H;e-•(r-r;>), (6.1.14)

j=l

Then det( ¢>( s)) = 0, and since N is odd,

v(f3;(s));:: v(G;)e-•(r-.,.;) -p.(Hj)e-•(r-T;) ~l;(s),

j = 1, ... ,m.

v(/3i (s)) ;:: 0 if and only if s

Set a =

< - 1-

- Tj- l1j

ln v(Gj) ~ aj, p.(Hj)

j = 1, ... ,m.

m,in {a j}. We have v( if>( s)) > 0 for 0 < s ::; a. Consider the case

l~J~m

that s > a. Then

v(a(s));::-

(~)N +aN v(P),

and since

we see that the minimum of f i are attained at S . _ _1_ )

-

Tj- l1j

ln v(Gj)(r- l1j) _ b· p.(Hj)(r- Tj) - )>

and thus

j=1, ... ,m.

387

Systems of Neutral Equations Therefore by (6.1.14) and (6.1.12) m

+L

v((s)) ;::: v(a(s))

v(,B;(s))

j=1

;: : - (N

N )

m

+ aNv(P) + L

er

[v(G;)e-b;(r-tr;) -1-'(H;)e-b;(r-r;l]

i= 1

>0 contradicting that det( ( s)) = 0. ii) By (6.0.2) v(P) det ( -

> 0 implies that

).N (I-

p-1e>.r) +

t

p- 1 exists. (6.1.13) is equivalent to

[P-1G;e>.(r-tr;) - p-1 H;e>.(r-r; l]) = 0.

J=1

With >.

= -s, we have

det (sN(I- p-le-•r) +

t

[P- 1 G;e-•(r-tr;)- p-l H;e-•(r-r;l]) = 0.

J=1

(6.1.15) Then we have a duality between (6.1.13) and (6.1.15) as follows:

Using this duality and part i) we obtain the desired result.

0

Remark 6.1.3. A special case for Eq. (6.1.7) is that P and Qj, j = 1, ... , m, are symmetric matrices. Since for any symmetric matrix A, !-' 2 (A) = >. 1 (A), 112(A) = >.n(A) and 1-'(A);::: At(A), v(A) ~ An(A) for any Lozenskii measures, then if we use 1-' 2 and 112 in the previous theorems, they will give the best results among those using all the Lozenski measures. For another special case, we obtain a criterion for oscillation of the following

Chapter 6

388 delay differential system of first order m

X(t) + QoX(t) +

L Q;X(t- u;) = 0

(6.1.16)

j=l

where Q;,j = 0, ... ,m, are n x n matrices, and u; result of this section is given in the following.

~

O,j = 1, ... ,m. The main

m

Theorem 6.1.6.

Assume v(Q;);::: 0, j

= 0, 1, ... ,m, and Ev(Q;) > 0. Then j=l

either one of the following guarantees that (6.1.16) is oscillatory: (6.1.17)

(6.1.18)

The above inequalities become strict if m = 1. Proof: The characteristic equation of (6.1.16) is det

(>.! + Qo + t

(6.1.19)

Q;e->.tTi) = 0.

J=l m

= ..\! + Q0 + E Q;e->.tTi. Then det (F(..\)) j=l has a real root ..\ 0 < 0, then let ..\0 = -s0 , and let

Let F(..\)

=F 0 for ..\ > 0. If (6.1.19)

m

G(s) = -sl + Q0

+ L Q;e•ri. j=l

Then det ( G( so)) = 0 implies that v ( G( so)) $ 0. Hence m

0 ~ v(G(so)) ~-so+ v(Qo) +

L v(Q;)e

80

1Ti

j=l m

so - v( Qo) ;:::

L v( Q; i=l

)e"(Qo)tTj e(so-v(Qo))tTj.

(6.1.20)

389

Systems of Neutral Equations Obviously, s 0

v( Qo) > 0 and then

-

m

1~

L v( Qj

)e"(Qo)"i e 0, j reduce to the following

=

393

Systems of Neutral Equations Assume that the equation

Corollary 6.2.2.

m

x'(t) + q0 x(t) +

L i];x(t-

Oj) =

(6.2.10)

0

j=l

is oscillatory. Thus Eq. (6.2.9) is also oscillatory. In particular, assume m

> 1 or -

~q-·eiio";u·e

L...J

1

1

(f

* e> 1'

j=l

(IT q;_)1/m em m

!I!.tT•

m

where u* =

L u;, j=l

i=l

the inequalities become strict ifm = 1. Then Eq. (6.2.9) is oscillatory. Proof: The characteristic equation of (6.2.10) is m

F(A) =A+ i]o

+L

ijje->.";

= 0.

i=l

Eq. (6.2.10) is oscillatory implies that F(A) > 0 for all A E R. Then for any qj E [q;,q;], j = O, ... ,m, we see that m

A+ q~

m

+L

qje->.u; ~A+ i]o

j=l

+L

i];e->.u; > 0.

j=l

This means that the equation m

x'(t)

+ q~x(t) + L

qjx(t- u;)

=0

j=l

is oscillatory. By Theorem 6.2.1, Eq. (6.2.9) is oscillatory. The rest of the proof is immediate from the first part and Corollary 6.2.1. 0 Finally we show how our results may be extended to certain nonlinear systems of the form m

X'(t)

+ L Q;(t,X(t))X(t- u;) = 0 j=l

(6.2.11)

Chapter 6

394

where Tj > 0, j = 1, ... ,m. Let Qi(t,X) = ((qj)tk(t,X)), j = 0, ... ,m, ben X n matrices, where the (qj)tk(t,X(t)) are continuous, uniformly bounded, and do not change signs on R+ for all solutions X(t). Denote

for f., k = 1, ... , n, j = 1, ... , m, where the infima and suprema are taken over all solutions of the system (6.2.11) and for any (qj)tk E ((qj)tk,(qj)tk) define Qj = ((qj)tk), j = o, ... ,m. Theorem 6.2.2. i) If the conditions of Theorem 6.2.1 or Corollary 6.2.1 are satisfied, then system (6.2.11) is oscillatory. The proofs are essentially the same as Theorem 6.2.1, Corollary 6.2.1, and so we omit them here. We remark that if we can give a priori upper and lower bounds for all solutions, then with some general assumptions we can easily show the uniform boundedness of the functions (qj)tk(t,X(t)) in (6.2.11) which are required by Theorem 6.2.2. As an application of our results, we derive oscillation properties for LotkaVolterra models in the system (predator-prey and competition) cases. We consider the system of delay logistic equations m

N;(t)

= N;(t)[a;- L b;jNj(t- u)],

i

= 1, ... , m

(6.2.12)

j=l

with N;(t) = cp;(t), t E [-u, 0], where

ue(O,oo),

a;,b;jER for i,j=1, ... ,m, cp;(O)

> 0,

i

cp;EC([-u,O),R+)

and

= 1, ... , m.

We assume that (6.2.12) has an equilibrium N• = (N{, ... , N:,.)r with positive components. Set N;(t) = Niez;(t) for t 2: 0 and i = 1, ... , m, then the x;(t)

395

SyJtemJ of Neutral EquationJ

satisfy m

x;(t) + LPii(ez;(t-r) -1)

= 0,

i

= 1, ... ,m,

(6.2.13)

j=l

where Pii = b;jNJ, i,j = 1, ... , m. As shown in (see [76], Section 5.4) we see that every solution of (6.2.13) satisfies lim x(t)=O, t-oo if the matrix P

i=1, ... ,m,

(6.2.14)

= (Pii) satisfies m

v 00 (P)

= ~~~m {Nj(bjj- 3-

L

lb;jl)} > 0.

i=l,i#j

Now we are ready to derive criteria for oscillation. Theorem 6.2.3.

Assume v00 (P) > 0, and for some i = 1, 2, ... , or oo,

v;(P)ue > 1.

(6.2.15)

Then (6.2.13) is oscmatory, and hence every solution of (6.2.12) is oscillatory about N*. Proof: Rewrite (6.2.13) in the form m

x;(t) +

L Pi;(t)x;(t- u) = 0,

i

= 1, ... 'm

i,j

= 1, ... , m,

(6.2.16)

j=l

where 1

ez;(t-u) _

Pi;(t)

= Pij

Xj

(t

- 0'

)

for

and by (6.2.14), lim Pi1·(t) = p;;, i,j = 1, ... , m. Define the matrix P*(t) = t-oo (Pi;(t)). Then lim P*(t) = P. t-oo By (6.2.14) and the continuity of the Lozinskii measures we see that there is a To ~ 0 such that v;(P*(t))ue > 1, t ~ T0 • Then by Theorem 6.2.1, we obtain the desired result. 0

396

Chapter 6

6.3. Comparison with Scalar Equations Here we investigate the systems of first order neutral delay differential equations of the form

d

m

-(X(t)- PX(t- r)) dt

+L

Q;(t)X(t- u;(t)) = 0.

(6.3.1)

j=l

We assume that (i) P is an n x n diagonal matrix with diagonal elements Pl>P2, ... ,pn such that Pi ;::: 0 for i = 1, ... , n and T > 0; (ii) for each j = 1, ... , m, Q;(t) = ((q;)tk(t)) satisfy

(q;)tk(t) E C([to, oo), R+ ), (iii)

O"j

E C([t 0 ,oo),R+) and lim

t-+oo

Theorem 6.3.1. Assume that 0 j = 1, ... ,m, where

q;(t)

=

(t -u;(t)) ~Pi ~

1,

= oo;

k=l

j = 1, ... ,m.

i = 1, ... , n, and qi(t) ;::: 0,

min {(qj)tk(t)- tl(qj)tk(t)l},

l~l~n

£, k = 1, ... , n;

j

= 1, ... ,m.

(6.3.2)

k#

Further assume that there exists a nonnegative N such that every solution of the scalar equation m

N

u'(t)+ Lq;(t)Lp!u(t-hr-u;(t)) =0 j=l

(6.3.3)

h=O

is oscillatory. Then every solution of system (6.3.1) is oscillatory, where P• = min Pi· l~i~n

Proof: Assume the contrary. Then (6.3.1) has a nonoscillatory solution X(t) = (xl(t), ... ,xn(t))T, i.e., there exists t1;::: to such that X(t) ::j:. 0, and Xi(t) ::j:. 0 if it is nontrivial fort ;::: t 1 , i = 1, ... , n. Set Ci = sign xi(t), i.e.,

397

Systems of Neutral Equations

8;

= 1, if x;(t) 2:0 and 8; = -1 if x;(t) < 0. Then Y(t)= (Yl(t), ... ,yn(t)) T = (81x1(t), ... ,8nxn(t) )T

is a solution of the system m

r))' + 2: Qi(t)Y(t- o-i(t))

(Y(t)- PY(t-

(6.3.4)

= 0

j=l

where Qj(t) Set

= ((qj)tk), and (qj)tk =

t- (qj)tk,

£, k

= 1, ... , n,

j

= 1, ... , m.

zt(t) = Yt(t)- PtYt(t- r), n

v(t)

= L z;(t),

n

w(t)

= LY;(t). l=l

l=l

From (6.4.4) we have m

z~(t) +

n

2: "E(qj)tk(t)yk (t- o"j(t)) = 0,

f = 1, ... , n,

j=l k=l

and hence

v'(t)+

t;t; m

n

(

n

)

t;(qj)tk(t) Yk(t-o"j(t)) =0.

By the definition of qi(t) we find m

v'(t) +

L qj(t)w(t- o"j(t)) :5 0.

(6.3.5)

j=l

Since qi(t) 2: 0, w(t- O"j(t)) ? 0, (6.4.5) implies that v'(t) < 0 and thus v(t) is decreasing. Set lim v( t) = £. Then £ E [-oo, oo ). We claim that £ 2: 0. t->oo

Otherwise, there exist i E {1, ... , n} and a > 0 such that for t ? T? t 0

z; = y;(t)- p;y;(t- r) :5 -a.

(6.3.6)

398

Chapter 6

H Pi E (0, 1), then (6.3.6) leads to lim Yi(t) = 0 as before, consequently, t-oo lim Zi(t) = 0, a contradiction. If Pi = 1, then (6.3.6) implies that c-oo

Yi(T + nr) $ y(T)- no:-+ oo,

as

n-+ oo,

a contradiction again. The above discussion shows that v(t) > 0 for all large t. Since v(t) $ w(t)- p.w(t- r), we obtain w(t) ~ 2:~= 0 p~v(t- hr). Substitute it into (6.4.5) we have m

N

v'(t) + Lqi(t) LPZv(t- hr- u;(t)) $0. j=l h=O By Corollary 2.6.5 Eq. (6.3.3) has a positive solution, contradicting the assumption. 0 Remark 6.3.1. For the case that n = 1, Theorem 6.3.1leads to the fact that every solution of the neutral delay equation m

(x(t)- px(t- r))' + L q;(t)x(t- u;(t)) = 0

(6.3.7)

i=l

is oscillatory if every solution of the delay equation m

N

y'(t) + Lq;(t) LPhy(t- hT- 1 fori= 1, ... ,n, q;(t) ~ 0, where q;(t) are defined by (6.3.2), j = 1, ... , m. Assume there exists a j 0 E {1, ... , m} and a nonnegative integer N such that J;' qj0 (t)dt = oo, and every solution of the equation m

N

u'(t)- L q;(t + r) L(p*)-(h+I)u(t + (h j=l h=O

+ 1)r- u;(t + r)) =

0

(6.3.9)

399


is oscillatory. Then every solution of system (6.3.1) is oscillatory, where p* = max p;. t::;i=:;n

Proof: Assume X(t) = (x 1 (t), ... , Xn(t)) Tis a nonoscillatory solution of (6.3.1). As in the proof of Theorem 6.3.1.

is a nonoscillatory solution of the system (6.3.4). Thus n

m

(Yt(t)- PtYt(t- r))' + L L(q;)tk(t)yk (t- u;(t))

= 0,

l = 1, ... , n. (6.3.10)

j=l k=l

Rewrite (6.3.10) in the form 1

1

m

n

(Yt(t)- -yt(t + r))' = - L L(q;)tk(t + r)yk(t + T - Uj(t Pt i=t k=t Pt

+ r)), (6.3.11)

i= 1, ... ,n.

Set n

v(t) ==

L 1=1

n

1

(Yt(t)- -yt(t + r)), Pt

w(t) =

E Yt(t). l=l

Summing up both sides of (6.3.11) for l = 1, ... , n we obtain

(6.3.12) Hence t~~ v(t) = l E (-oo,oo]. We claim that l > 0. Otherwise, we get w(t) ~ p1• w( t +T) for all large t, where p. = min p;, which implies that lim w( t) = oo. lO (6.4.31)

Then there exists a positive solution of the system (6.4.1r) with the property lim x;(t) t-+oo

tk;

= 0,

lim x;(t) t-+oo tk;-1

= oo.

(6.4.32)

409


Proof: Let a;, b;, 1 :5 i :5 n, be positive constants. Then we choose 6; such that 0 < 26; =min {a;,b;}, 1 :5 i :5 n. We put 26;/(1- {3;) = S;, and

s = ( o:/2:151. · · · • (li;6.:!1)!) T • LetT~

max {-y(t 0 ), 1} be such that (6.4.13) holds and

G;(T,k;,pt,"S) < 6;(N- k; -1)!, --

G;(T,k; -1,p; ,6)

6;

< 2(N _ k;)!,

(6.4.33) 1 :5 i :5 n.

(6.4.34)

Let C[To, oo) be the space defined in the proof of Theorem 1. We consider the closed convex subset S of C[To, oo) defined by

S=

{

T

Tk;-1

Z=(z1 , ••• ,zn) eC[T0 ,oo): z;(t)=6; (k;- 1)! forte [To,T],

6·t";- 1 26·t" } 1)! :5 z;(t) :5 (k; ~ 1)! fort~ T, 1 :5 i :5 n .

2 (~; _

(6.4.35) With every (zl! ... , znf E S we associate the functions (x 11 ••• , Xn)T defined by the formula (6.4.25). From (6.4.25) in view of (6.4.26), (6.4.35), (6.4.2) and (6.4.3) we obtain

z;( t) :5 x;( t)

< 26; [ Q (h ( ))"; - (k;- 1)! t k·• + fo"i i t + ...

+ Pr;- 1 (H;(n;(t) S·t";

:5 (k; '_ 1)! , t ~ T,

1, t)) ";

+ Pr; (H;(n;(t)), t) ";]

1 :5 i :5 n.

(6.4.36)

410

Chapter 6

Define an operator T = (1i, ... , Tn)T : S 6;Tic;-1

6·tlc;-1 1

(k;- 1)!

C[To, oo) by

t E (To,T],

(k;- 1)! '

T;Z(t) =

-+

+ (- 1)n-lc; i'

(t- s)lc;-1 (k;- 1)!

T

ioo (u- s)N-/c;-1 T

(N- k;- 1)!

(6.4.37)

n

LP;;(u)f;;(x;(g;;(u )))duds,

X

t ~ T, 1:5 i :5 n.

j=1

Clearly, Sis a closed convex subset of C[To,oo). We show that T(S) C S. Let Z = (z1 , ••• , Zn)T E S. From (6.4.37) in view of (6.4.9) and (6.4.36) we get for t ~ T, 1 :5 i :5 n :

6;tk;-1 T;Z(t) :5 (k;- 1)! 6·tlc;- 1

6; 2

o·t"'- 1

t

I

- 2(k;- 1)! ,

If k;

T

n

k· _ 1) 1 L(P;;);,(u)f;;(x;(g;;(u)))du ds du

•

O;t"'- 1 - (k;- 1)!

>

)N-k 1 -1 I

i=1

•

(t- u)"'- 2 du (k;- 2)!

~

1~i

T,

~

n.

= 1, then from (6.4.37) in virtue of (6.4.39) we have T;Z(t)

~ 6;-

hJ.oo (~; ~

)N-2 2)!

t

T

•

n

~(p;;)!(u) J=l

x /;;(x;(g;;(u))) duds

1 6· - 2 "

>-

t

~

T,

1~i

~

n.

We have proved that T(S) C S. Proceeding similarly as in the proof of Theorem 6.4.1 we obtain that the operator T is continuous and T (S) has a compact closure. Therefore, by the Schauder-Tychonov fixed point theorem there exists a Z = (z1 , ••• , Zn )T E S such that (TiZ, ... ,TnZ)T = (z 11 ••• ,zn)T and the components (x 1 , ••• ,xn)T satisfy that for t ~ T the system

n

x LP;;(u)f;;(x;(g;;(u))) duds;

1 ~ i ~ n,

i=l

where z;(t) = x;(t)- a;(t)x;(h;(t)). Differentiating (6.4.40) (k;- 1)-times and k;-times, we get

z1"'-l)(t) = 6; + (-1)N-k, ('

J.oo

}T •

..:.,.(u_-----=s):._N_-_"'....,.-_1 (N- k;- 1)!

(6.4.40)

Chapter 6

412 n

x LPi;(u)fi;(x;(g;;(u))) duds,

t ~ T,

ik;)(t) = (-1)N-k;

l

t)N-k;-1 ..:.,.u-=----'-:-~

oo (

(N- k;- 1)!

t

I

1 $ i $ n,

(6.4.41)

j=1

n

x LPi;(u)/i;(x;(g;;(u)))du ds,

t ~ T,

1 $ i $ n,

(6.4.42)

j=1

respectively. Then (6.4.42) implies that

1.

liD t~oo

-(k;)(t) -z.1

0.

(6.4.43)

From (6.4.41) on the basis of (6.4.9), (6.4.39), (6.4.3), (6.4.36) and (6.4.35) we conclude

n

- LP;;(u)i;/;;(x;(g;;(u))) duds j=1

The last inequality together with (6.4.31) implies that li m -(k;-1)(t) z; -- oo.

t-oo

(6.4.44)

By L'Hospital's rule, (6.4.43) and (6.4.44)

z;(t) 0 1l. i D -=,

t-oo

tk;

. z(t) hm k 1 =oo. t ;-

t-oo

(6.4.45)

413

Systems of Neutral Equations Then from (6.4.45) in view of (6.4.2), (6.4.3) and (6.4.36) we get lim x;(t) t-+oo

tk;

= 0,

. x;(t) 11m --=oo

t-+00 tk;-1

respectively. The proof of Theorem 6.4.2 is complete.

6.5.

,

D

Notes

Theorem 6.0.1 is given by Orino and Gyori [5]. The content in Section 6.1 is from Erbe and Kong (36]. Section 6.2 is based on the work by Kong and Freedman (93]. Section 6.3 is extracted from Wang (164]. Section 6.4 is a modification of the results by Marusiak (131].

7 Boundary Value Problems for Second Order Functional Differential Equations

7 .0. Introduction In this chapter we shall discuss certain boundary value problems (BVPs) associated with second order functional differential equations. Although the study of the existence and location of zeros of the solutions of ordinary differential equations is fundamental in the study of BVPs for such equations, the relation between solutions to certain BVPs for FDEs and the oscillatory behavior of such solutions is less clear. In spite of this, we shall present a number of approaches to the study of such problems, which, in some sense, parallels the corresponding techniques for BVPs associated with ODEs. Although we do not claim in any sense to be complete, it is hoped that these techniques will give an idea of what may be obtained. In Sections 7.1 and 7.2 we establish existence via Lipschitz and Nagumo-type methods, respectively. In Sections 7.3 and 7.4 we employ topological methods and Section 7.5 deals with existence and uniqueness for a certain

414

Boundary Value Problem, for Second Order Functional Equations

415

type of singular problem.

7 .1. Lipschitz Type Conditions In this section, we shall consider the existence and uniqueness of solutions of the BVP

(p(t)x'(t))' = f(t,x,,x'(t)) xo

= cp,

x'(T)

(7.1.1)

= Tf

(7.1.2)

where

x,(u) the function

= x(t + u)

for u E J = [-r,O],

f is defined on the set [O,T]

X

D

X

R",

D

c C( J), T > 0,

and where C(J) is the space of all continuous bounded functions on J with values in R". We assume C(J) is endowed with the norm II·IIJ defined by the formula

llcpiiJ =sup lcp(u)l. uEJ

Moreover, (cp, Tf) belongs to D X R" and pis a positive continuous function defined on [O,T]. The following lemma will play an important role in the proof of our results.

Lemma 7.1.1. Let g E C([O,T],R"), {

e, Tf E R". Then the problem

(p(t)y'(t))' = g(t) y(O)

= e,

y'(T)

(7.1.3)

= 77

has a unique solution

y(t) = where ~(T)

e+ p(T) '7 ~(t)

= J: ~·

-1' ~(u)g(u)

du-

~(t)

lT

g(u) du

(7.1.4)

Chapter 7

416

Proof: By differentiating (7.1.4) we can easily verify that (7.1.4) is a solution of (7.1.3). From (7.1.4) we have

y'(t)

= p;~~T]-

ly(t)l < -

ptt)

iT

(7.1.5)

g(u)du

1(1 + I77IP(T)~(T) + 2~(T)T O$s$T max lg(s)l

(7.1.6)

and

I '(t)l $ p(T)IT7I y

V

+ '£V O$t$T max lg(t)l

0

where v = mino 0 on [0, T], then (7.1.1) and (7.1.2) has a unique solution. This result is best possible. Proof: By continuous dependence of solutions on parameters, there exists a sufficiently close to but less than 1 such that

(p(t)w'(t))'

1

+ -(azw'(t) + a 2 w(t)) = 0 a

(7.1.18)

Boundary Value Problems for Second Order Functional Equations

419

has a solution with w(O) > 0, w'(t) > 0 on (0, T]. Integrating (7.1.18) from t to T with t < T we have

11T

p(T)w'(T) +a

t

(a2w'(s) + a 1 w(s)) ds = p(t)w'(t)

or

11T

-

a

t

(a2w'(s)

+ a1w(s))ds :5 p(t)w'(t).

(7.1.19)

x

Now we consider the map defined by (7.1.13) and calculate 8(x1 , 2 ) according to the definition (7.1.16) and using the formula (7.1.4). It is easy to see that

!x1(t)- x2(t)l w(t)

:5

w~t)

1' ~(u) lf(u,xlu,x~(u))- f(u,x2u,x~(u))l

du

~(t)iT' f (u,xlu,xl'( u )) -f(u,x2u,x I2 (u))!du + w(t) 1 :5

_1_1' ~(u) w(t)

0

[w(u) adlxlu- X2ui!J + v(u) a2lxHu)- x~(u)l] du w(u) v(u)

fT( a1w(u )llxlu-X2uiiJ+ ( )lxHu)-x~(u)l)d w(u) a2v u v(u) u :5 8(x1,x2) w(t) [1' + a2v(u)} du ~(t) + w(t) },

0

+

~(u){a1w(u)

~(t) 1T(a1w(u) + a2v(u))du].

(7.1.20)

Similarly, from (7.1.5) we obtain (7.1.21) We require that

Chapter 7

420

and (7.1.23) If we choose v(t) = w'(t), then (7.1.23) follows from (7.1.19). Integrating (7.1.19) from 0 tot< T we can obtain (7.1.22). With this choice of a, w, v (7.1.22) and (7.1.23) hold. This implies that a:I [" lx'(t)ll(x'(t),x"(t))l dI - lu H(lx'(t)l) t rlx'(v)l ds I IJlx'(u)l H(s) . s2

=

By the Mean Value Theorem there exists u E [0, T] such that

and the continuity of x 1 implies the existence of v E [0, T] such that

lx'(v)l

= tE(O,T] max lx'(t)!.

Set w

=iT

lx'(tW dt.

Then (7.2.5) gives w

=

Jg(w) s2 ds > IJlx'(v)l s2 ds I· VwJT H(s) - VwJT H(s)

Hence

lx'(v)l $ g(w)

(7.2.6)

which implies that (7.2.4) holds. The Green's function associated with the boundary value problem

x"(t)

= 0,

x(O) = 0

= x(T)

Chapter 7

424

lS

G(t,s) =

{

(1-~)s,

O$s$t$T (7.2.7)

t (1- -f), 0 $ t $ s $ T.

Then the boundary value problem (7.2.1) is equivalent to the integral equation

x(t) =

1T G(t,s)f(s,x~,x'(s))ds.

(7.2.8)

Let B be the Banach space of all continuously differentiable functions x: [0, T] -+ Rn with the norm

1\xl\* =max{ sup Jx(t)J, sup lx'(t)l}. tE(O,T]

tE(O,T]

The main existence result is the following. Theorem 7.2.1. Assume that there exist t.p E C 2 ([0, T], R+ - {0}) and F E C([O,T) X R x R,R) satisfying the following condition: for any (t,u,v) with 1\ull~.p'(t),

one has

(i) (u(O),f(t,u,v)) $ ~.p(t)F(t,~.p(t),t.p'(t))

+ lvl 2 -

t E [0, T], llull = t.p(T), (u(O), v) =

(ii)

~.p"(t)

t.p'(t) 2

+ F(t,~.p(t),~.p'(t)) $0.

Assume moreover, that there exist numbers a E [0, 1) and for any (t,u,v) with t E [O,T), llull $ t.p(t) and vERn

(iii) (u(O),f(t,u,v)) $ aJvl 2

b;::: 0 such that,

+b

(iv) l(v,f(t,u,v))l $ H(Jvl)Jvl where H : R+ -+ R+ - {0} is an increasing continuous function which satisfies the condition (7.2.9)

Then the boundary value problem (7.2.1) has at least one solution x such that Jx(t)J $ ~.p(t) for all t E [0, T].

Boundary Value Problem" for Second Order Functional Equations

Proof: Define a mapping S : B

(Sy)(t)

=

-t

425

B by

1T G(t,s)f(s,y.,y'(s))ds.

By (iii) and (iv), f maps bounded subsets of [O,T] x C(J) subsets of Rn, so the map S defined by (7.2.10) is compact. Now define A: B - t Band the set by

(7.2.10) X

Rn into bounded

n

(Ax)(t) =

-1

T

a 2 G(t,s)x(s)ds,

(7.2.11)

and

f! 71 = {x E B, \It E [O,T], jx(t)l < cp(t), lx'(t)i < 77}

(7.2.12)

for a certain 77 > 0. Clearly, A is completely continuous, f! 71 is open and bounded in B. We will show that I - A is one-to-one and that if 77 and a are chosen large enough, the equation

x = >.Sx + (1 - >.)Ax,

>.e(O,l)

(7.2.13)

n

has no solution which belongs to an". Then s has a fixed point X E by Theorem 1.4.8. In order to prove the linear operator I- A is one-to-one, we must show that x = Ax implies x = 0. In fact, if x satisfies the equation x - Ax = 0, then it is a solution of the boundary value problem

x"(t) = a 2 x(t) (7.2.14)

x(O) = 0 = x(T). This problem has the unique solution x = 0. Therefore the operator I - A is one-to-one. Finally, we prove that if X E n'l is a solution of (7.2.13), then there exists no E (0, T) such that either jx(t)l2 - cp 2 (t) reaches the maximum value 0 at

t=

e eor lx'(e)l = TJ·

a

Assume that jx(t)l2 - cp 2 (t) reaches the maximum value 0 at Then

eE (0, T).

426

Chapter 7

(u(O),x'(e))- cp({)cp'({) = 0 (7.2.15) (7.2.16) Since xis a solution of (7.2.13), That is,

x"(t) + >.f(t,xt,x1(t)) = (1- >.)a 2 x(t).

(7.2.17)

For). E (0, 1), by our assumptions and (7.2.15) we have

J

=(u(O), ->.f(t, u, v)) + lvl = -(u(O), J(t, u, v ))

2 -

0 which contradicts (7.2.16). By Theorem 1.4.8, S defined by (7.2.10) has a fixed point y. Then

x(t) = {

h(t),

-r

:5 t :5 0 (7.2.18)

y(t), 0:5t:5T is a solution of the boundary value problem (7.2.1). Now we will show that for any solution x(t) of (7.2.13) lx(t)l :5 cp(t), lx'(t)l has an upper bound which does not depend on). E [0, 1]. Let x(t) be a solution of (7.2.13), multiply both sides of (7.2.17) by x(t), and integrate by parts over [0, T]. We have

(x(t), x"(t))

{T

+ >.(u(O), f(t, u, v)) = (1- >.)a2 (u(O), u(O)} {T

T

- Jo lx'(tWdt+>. Jo (u(O),v(t,u,v))dt=(1->.)a 2 1lx(tWdt


427

or

-1T

lx'(tW dt

+ >..bT +a>..

iT

~

t E [O,T].

lx'(tW dt;:::

o.

Hence

and by Lemma 7.2.1

lx'(t)l

g(K),

Choose "1 > g(K). The fixed pointy of S belongs ton. Thus proof is complete.

lx(t)l

~ rt>(t). The

0

Example 7.2.1. Consider the linear equation

x"(t) + q(t)L(xt) + m(t)x'(t)

=0

(7.2.19)

where q(t), m(t) are bounded functions with bounds Q and M respectively and L( 'f') is a linear bounded operator in C( J). We shall show that the assumptions of Theorem 7.2.1 are satisfied for (7.2.19). In fact, let IlLII be the norm of L. Set A = QIILII and define F by

F(t,x,y) =

(A+ ~2 ) llxll +MIYI·

(7.2.20)

Thus we have

(u(O), f(t, u, v)}

= q(t)

{L(u), u(O)} + m(t)u(O)v ~ Allull 2 + Mvu(O) (7.2.21)

0

< -

[M2 llull- l(u(O), v}l +I 1] llull v

2

~ ~ 2 llull 2 + l(u~~~~~:}l 2 + lvl 2 - Ml(u(O),v}l + Mllulllvl _ 2 1(u(O), v}ll

llull

I v

Chapter 7

428

< M211ull2 + l(u(O), v)l { l(u(O), v)l - 2lvl} -

!lull

!lull

4

+ lvl2- Ml(u(O),v)l + Mllulllvl $

~ llull2- l(u~~~~:)l 2 + lvl

2 -

Ml{u(O), v) I+ Mllu!llvl.

Thus

(u(O),J(t,u,v))

$

Allull 2 + ~2 llull 2 -

=!lull{ (A+

l(u~l~li:)l 2 + lvl 2+ Mllulllvl

~2 ) !lull+ Mlvl} + lvl

2 -

l(u~~~I;W (7.2.22)

That is, condition (i) is satisfied for tp may be easily verified. For (iii) we notice that

= llull 2 ;

condition (ii) is also satisfied, as

(u(O), J(t, u, v )) = q(t)(L( u), u(O)) + m(t)u(O)v $

Allull 2 + Mvlu(O)I $A*+ B*lvl (7.2.23)

where A* = AC, B* = MC, Cis the bound of tp. If lvl < 1, then (iii) is obvious from (7.2.23). If lvl > 1, then (iii) follows from the inequality for each B1 ;::: 0. Indeed, we have

Finally, it is easy to see that (iv) is satisfied. Hence (7.2.19) with

x(t) = h(t),

-r $

t $ 0, h(O) = 0, x(T)

=0


429

and

x(t)

~

cp(t)

has a solution.

7.3. Leray-Schauder Alternative We shall now consider the boundary value problem

t E [O,T]

x"(t) = f(t,x 1 , x'(t)), aoxo- a1x'(O) f3ox(T)

= cp E C([-r,O],Rn)

(7.3.1) (7.3.2)

+ f3tx'(T) =A ERn

where f E C([O,T] x C(J) X Rn,Rn), C(J) = C([-r,O]), ao, at, f3o, f3t are nonnegative real constants such that (7.3.3) We assume that a 0

> 0. In the case where ao = 0, (7.3.2) is replaced by xo = x(O) -atx'(O) = cp

(7.3.4)

f3o x(T) + (3 1 x'(T) =A We omit the discussion for this case here. The key tool is the Leray-Schauder alternative. This method reduces the problem of existence of solutions of a BVP to the establishment of suitable a priori bounds for solutions of these problems. The Green's function with respect to the BVP

x"(t) = 0,

t E [O,T]

aox(O)- a1x'(O) = 0

(7.3.5)

430

Chapter 7

IS

1 { (f3ot- /3oT- f3t)(aos +at), G(t,s)=f (aot + at)(f3os- f3oT- f3t),

0~s ~t ~T 0 ~ t ~ s ~ T.

(7.3.6)

The following is an existence result for BVP (7.3.1) and (7.3.2). Theorem 7.3.1. Assume that there exists a constant K such that llxll[-r,TJ ~ K

llx'll[o,TJ ~ K

and

for every solution x of the BVP for the equation x"(t)

= J..f(t,x 1 ,x'(t)),

(7.3.1)>.

t E (O,T]

with (7.3.2) and>.. E (0, 1). Then the BVP (7.3.1) and (7.3.2) has at least one solution. Proof: First we assume that rp(O) = 0. We let X denote the space C 1 [0, T] with the norm llxllt = max{llxll[o,TJ, llx'lho,T}}· Then, the set C = {x EX: aox(O)atx'(O) = 0} is a subspace of X. Define a mapping F: C-+ X as follows:

1 T

Fx(t) =

0

G(t, s)f(s, x., x'(s)) ds

1

+ l(at + a 0 t)A,

t

E (0, T]

(7.3.7)

where

x.(8)

={

8 '2:. -s

x(s+O), ;. [atx'(O)

+ rp(s + 8)],

8 < -s.

(7.3.8)

Clearly, F(C) c C. We shall show that F is completely continuous. Indeed, let B be a bounded subset of C. Then there exists b '2:. 0 such that llxllt ::::; b, x E B. Moreover, for any tt, tz in [0, T] and x E B we have

Therefore, B is an equicontinuous set.

431


Now we consider the subset

B of the space C(J) defined by

B = {xt:

x E B, t E [0, T]}

where Xt is defined by (7.3.8). We shall show that there exists a compact subset D of C( J) such that

jj ~D. In order to show that, it suffices to prove that the set B is uniformly bounded and equicontinuous. In fact, for any x E B and t E [0, T] sup lx,(9)1 8E[-r,O)

=

sup lx(t + 9)1 $ b, 8E[-r,O)

where

} ~b =max { b, la1l lao I (b + llc,oll[-r,oJ) and hence B is uniformly bounded. Moreover, for any x E B, t E (0, T] and 81. 92 in [-r, 0] we have

lx(t

+ 91)- x(t + 62)!,

Ic!. [c,o(t + 81) + a1x'(O)] -x(t + 82)!,

if t + 81 < 0, t + 92 :2:: 0

lc!.[c,o(t + 91)- c,o(t +92)]1, ift + 91 < 0,

t

+ 92 < 0

lx(t + 91)- c!. [c,o(t + 82)

+a1x'(O)]I,

if t

+ 91 > 0, t + 92

$ 0.

For any e > 0, if t + 81 :2:: 0, t + 82 2:: 0, by the equicontinuity of B, there exists 6 = o(e) such that for every X E B, t E (0, T], (}1' 92 E [-r, 0] we have

provided 191 -921 < o(e).

Chapter 7

432

If t + 81 < 0, t

+ 82

~

0, then

But, by the uniform continuity of the function cp on [-r, OJ, the equicontinuity of the set B and for the given c > 0, there exist o' (e) > 0 and o" (c) > 0 so that

and e

lx(st)- x(s2)l < 2' Hence, if

provided

181 - 8zl < min{h"'(e:), c5"(e:)}

=

isl- s21 < o"(c).

6, then

and so 1 1-ao [c,o(t

+ Ot)- c,o(O)] I + lx(t + Oz)- x(O)l < -e2 + -e2

=e.

The rest of the cases are similar. Therefore, for every e > 0 there exists 6 = h(e) so that

for every x E B and t E [0, T], which proves the equicontinuity of the set there exists a compact subset D of C( J) such that

fj

fi.

So,

c;, D.

Now we consider a bounded sequence {x.,} in C. Then, as is known, for every t E [0, T] the sequence {x 1.,} is bounded in C( J) and, moreover, there exists a compact subset Din C(J) such that x 111 ED for every v and t E [0, T]. Thus, if d is the bound of {x.,}, it is obvious that the set V = [0, T] x D x B(O, d) is compact in [0, T] x C( J) x Rn, where B(O, d) is the closed ball with center 0 and radius d in Rn. We set (J

= max{lf(t,u,v)l: (t,u,v) E V}


433

and

and set

Then we have

IIFxvl![o,T] :5

K,

li(Fxv)'li!o,T] :5 K.

Hence, for any it, t 2 in [0, T] and arbitrary v we have

jFxv(t!)- Fxv(t2)1 I(Fxv)'(t!)- (Fxv)'(t2)1

=I J:• =I J:•

(Fxv)'(s)dsl :5

K lt2- ttl

(Fxv)"(s)dsl :5 Bitt- t2l,

i.e., the sequences {Fx v}, {( Fx v)'} are equicontinuous. That is, F is completely continuous. Finally, we observe that by assumptions the set £(F) = { x E C: x = >. Fx for some >. E (0, 1)} is bounded. Hence the assumptions of Theorem 1.4.10 are satisfied and hence the operator F has a fixed point x E C. Then it is clear that the function

z(t) = {

x(t),

t E [O,T]

do [cp(t) + atx'(O)],

t E [-r, OJ

(7.3.9)

is a solution of the BVP (7.3.1) and (7.3.2). If cp(O) =f. 0, by the transformation cp(O)

y=x--ao

the BVP (7.3.1) and (7.3.2) is reduced to the BVP

y"(t) = f(t,yl

+ "'~~),

y'(t))

= [(t,yt,y'(t)),

t E [O,T]

Chapter 7

434

aoYo - a1y'(O) = cp- cp(O) f3oy(T)

+ fJ1y'(T) =A+

where, obviously, cp(O)

= 0.

= cp

fJo cp(O), ao

(7.3.10)

0


Theorem 7.3.1 reduces the BVP (7.3.1) and (7.3.2) to the problem of finding max[-r,T] lx(t)l and max(o,T) lx'(t)l, where xis the solution of (7.3.1)~ and (7.3.2). In the following we will show some results for these bounds. Theorem 7 .3.2. Assume that (HI) There exists a constant M > 0 such that for every ( t, u, v) E [0, T] x C(J) X Rn with lu(O)I > M and (u(O),v} = 0 implies

(u(O),J(t,u,v)} > 0. Then every solution x of the BVP

(7.3.1)~

max lx(t)l S Mo tE(-r,T]

(7.3.11)

and (7.3.2) satisfies

=2

-\lcplh-r,o] O'O

+ Mo

(7.3.12)

where _ { max{M, ~0 lcp(O)I, Mo= max{M, ,;0 lcp(O)I},

Jo IAI},

if ao, f3o -::/: 0 (7.3.13)

if {30 = 0.

Proof: Assume that A E (0, 1]. Let x be a solution of

r(t)

1 = 21 (x(t),x(t)} = 21x(tW,

(7.3.1)~

t E [O,T].

Suppose that the function 1 takes its maximum value at a point we have

and

and (7.3.2) and

eE (0, T).

Then

Boundary Value Problems for Second Order Functional Equation&

435

or (x~(O),x'(e)) =

o

(7.3.14)

and (7.3.15) Since A > 0, conditions (7.3.14), (7.3.15) and (H1) imply lx(e)l :::; M. That is, if the function ix(t)i takes its maximum value at a point E (0, T), then lx(OI :::;M. Now we suppose that lx(t)i on [0, T] takes its maximum value at the point 0. Then from the first boundary condition we have

e

0 2:: a1r'(O) = (x(O),a1x'(O)} = (x(O), aox(O) - Cf'(O)} = aoix(OW - (x(O), Cf'(O)}

2:: aoix(oW -lx(O)Iilf'(O)I

= lx(O)i{aolx(O)I-IIf'(O)I}. Hence

o.

aoix(O)I - ilf'(O)I :::; Since ao

> 0, 1 lx(O)I :::; -llf'(O)I. ao

Assume that lx(t)i on [0, T] takes its maximum value at the point T. Then, by using the same arguments as above we have ix(T)I :::; M,

if f3o =

lx(T)I :::; ; 0 IAI,

o

if f3o > 0.

Chapter 7

436

Therefore,

]x(T)I ::; max { M, ;o IAI} and thus the theorem is proved, if 0 . is simple and gives the same bounds as in the case..\:/:- 0. Summarizing the above, if ]x(t)] takes its maximum value at a pointe E (O,T], then _ max ]x(t)] ~ Mo tE[O,T)

=

{ max{M, ~o ]cp(O)],

Jo ]AI},

max{M, ~o jcp(O)j},

if ao, f3o :/:- 0 if f3o = 0.

Next we shall prove that x 0 is bounded on C(J). Let x be a solution of the BVP (7.3.1)>. and (7.3.2). From the first boundary condition we have, if a 1 > 0 1 + aoMo) ]x'(O)I :5 -(]cp(O)I 0!1

and consequently, from the boundary condition

llzl][-r,o]

[llct'll[-r,o] + a1]z'(O)I] ~ ...!._ ao ~

2 +Mo. -llcpl![-r,O] ao

Set

2 Mo = -llcpl![-r,OJ ao

+ Mo,

then ]]zll[-r,T) ~Mo. The proof is complete.

D

Next we will establish a priori bounds for max1e[o,T) ]z'(t)l if an a priori bound for max[-r,T) ]z(t)J is given.

BoundaT?J Value Problems for Second Order Functional Equations

437

Theorem 7 .3.3. Assume that there exists a constant M 0 such that

max lx(t)l

tE[-r,TJ

~ Mo

(7.3.16)

for every solution x of the BVP (7.3.1);~.. and (7.3.2), .\ e [0, 1]. Also, we suppose that the continuous function f: [0, T] X C(J) X Rn -+ Rn satisfies the following conditions

(u(O),f(t,u,v)) ~ K1lvl 2 + K2 l(v,j(t,u,v))l ~ (K~Ivl 2 + K~) lvl for any (t,u,v) E [0, T] X C(J) X Rn with are positive constants such that

llull[-r,o)

~

Mo, where K1, K2, K~, K2

Then, there exists a constant M1 independent of.\, such that

max lx'(t)l ~ M1

tE(O,TJ

for every solution x of BVP

(7.3.1);~..

and (7.3.2).

Proof: For a solution x E C 2 [0,T] satisfying the hypotheses, let M = llx'll and let to E [0, T] such that !x'(to)l = M. We will show that M can be bounded independently of x. If u E 0 2 [0, T], we obtain, by a Taylor expansion,

u(to + p.)- u(to) = p.u'(to) +

l to+l' u"(s )(to + p.- s) ds

(7.3.17)

to

provided that to+ p.

e [0, T].

Set u(t)

=I: lx'(s)l 2 ds, then

l to+l' lx'(sW ds = p.lx'(toW + 2 lto+l' (x'(s), x"(s))(t ~

0

~

+ p.- s) ds.

(7.3.18)

On the other hand, integrating by parts we have

to+l'

lt.

to

lx'(sWds = (x(to+p.),x'(to+p.) )-(x(to),x'(to))-

lto+l' (x(s),x"(s))ds. to

Chapter 7

438 By (7.3.16), (H2) and (H3), we have

{7.3.19)

Combining (7.3.18) and (7.3.19) and using (H3) we obtain {7.3.20)

Let us assume that 8M0 :5 M{1- K 1 )T; if this is not the case, the desired bound for llx'll is already obtained. Let us choose 11-'1 such that 11-'IM = 4Mo/(1- Kt). and the sign of 1-' can With the above restriction on M, we then have 11-'1 :5 be chosen so that to + 1-' E (0, T], which guarantees the validity of the Taylor expansion used above. With that choice of 11-'1 the relation (7.3.20) becomes

t

since 11-'1 :5

Since K~
., such that

M1

for every solution x of BVP (7.3.1)>. and (7.3.2). Proof: Let x be a solution of the BVP (7.3.1)>. and (7.3.2), .>. E [0, 1]. Then, since 01 > 0, the first boundary condition gives 1

lx'(O)I :5 -(1~(0)1 + oo Mo) =c. 01

From the Cauchy-Schwarz inequality, if lx'(t)l =/= 0, we have

lx'(t)l'

= l(x'(t),x"(t))l < lx"(t)l -

lx'(t)l

·

Clearly this inequality is also true if lx'(t)l = 0. From (H 4 )

lx'(t)l' :5 lx"(t)l :5 q(t) n(lx'(t)l)

for every t E [0, T].

Thus

lx'(t)l :5 k+

lot q(s)n(lx'(s)l)ds,

t E [O,T]

where k = lx'(O)I. By Bihari's inequality we have

t E [O,T] where

1.

dt

G(s) = ~ n(t),

E:

> 0, s

~

0,

Chapter 7

440 and G- 1 is the inverse mapping of G, which is supposed such that

Hence

max lx'(t)l :5 M1.

tE[O,T)

0


Remark 7.3.1. (H 4 ) can be replaced by a Bernstein-Nagumo type condition: (H 4 )' There is an increasing function '1/J = [0, oo) -+ (0, oo) such that ~ is integrable on [0, oo)

if(t, u, v )I ::::; '1/J(IvD

(7.3.23)

for t E [0, T], llull[-r,o) :5 Mo and

1

00

c

ds

(7.3.24)

.,P(s) > T

where 1

c = - (l 0 has at least one solution. Example 7.3.1. Consider the equation

x"(t) = g(t)x(t)F(xt) + h(t)x'(t)lx'(t)l,

t E (0, 1]

(7.3.25)


441

where g : [0, 1] --+ R is continuous and positive, F : C(J) --+ R is continuous and positive and maps bounded sets into bounded sets and h : [0, 1] --+ R is continuous with maxte[o,I) lh(t)l < 1. Consider the following boundary conditions

:z:o- :z:'(O) = r.p :z:(1) + :z:'(1) =A.

(7.3.26)

In the notations of (7.3.1) and (7.3.2) f(t,u,v) = g(t)u(O)F(u) + h(t)vlvl T= 1,

It is not difficult to prove that (H1)-(H3) are satisfied. Therefore, BVP (7.3.25) and (7.3.26) has a solution by Theorem 7.3.5.

7 .4. Topological Transversality Method In this section we consider the system

:z:"(t) = f(t,x(ui(t)), . .. ,x(uk(t)),x'(uk+I(t)), ... ,x'(uk+m(t))),

t EI

(7.4.1) where I= [a,b] (a< b), f E C[I X (Rn)m+k--+ Rn], u;, i = l, ... ,k + m, are real valued functions defined on I, such that the set {t E I: u;(t) =a or u;(t) = b, i = k + 1, ... , k + m} is finite. We suppose that -oo < ra =

min

l~i:s;k+m

minu;(t) 0 such that, if for every vERn with lvl and (t, u1, ... , Uk+m) E I X (Rn)k+m the following conditions are fulfilled:

>m

(i) if O'j(t) = t for a j E {k + 1, ... , k + m}, then (v, Uj} = 0, (ii) ifo;(t) = Oj(t) E I forai E {1, ... ,k} and aj E {k+1, ... ,k+m}, then (u;, Uj} = 0, and


443

where if u;(t) = t

i = 1, ... ,k.

if u;(t) =F t,

(H 2 ) There exists an index j E {k + 1, ... , k + m} such that Uj(t) :5 t, t E I and

for every (t,u1, ... 1 Ut.+m) E I X (Rn)k+m, with (lutl, ... , luA:I) E [O,Mo]k and where the functions g and n are nonnegative continuous on [0, 00 ), n is nondecreasing. (H3)

ao

=0

==> q1(a)

= 0,

f3o

=0

==> q2(b)

=0

and Ot

> 0.

Then the boundary value problem (7.4.1) and (7.4.2) has at least one solution. To prove this result we will use a fixed point theorem in Chapter 1. We divide the proof into the following lemmas. The following basic results can be found in any elementary differential equations text book. Lemma 7.4.1. problem

The Green's function for the homogeneous boundary value

x"(t)=O, - aox(a)

tEl

+ a1x'(a) =

0

exists and is given by the formula

1 { (f3ot- f3ob- f3t)(aos- aoa +at), G(t,s) =l (f3os- f3ob- f3t)(aot- aoa +at),

s :5 t t :5 s,

Chapter 7

444 where

eis defined by (7.4.3).

Also the following inequalities hold:

where K 1, K 2 are constants depending on ao,

b, f3t

:f 0;

t > b, f3I

:f 0;

= 0;

I: q2(s)exp(~(s- b))ds)exp(-~(t- b)),

Jo q2(t),

1

t E I;

= 0.

We consider, for .A E [0, 1],

x" (t)

= .A f( t, x(

O"J (

t)), ... , x( uk( t) ), x' ( O"k+J ( t) ), ... , x' (O"k+m ( t))),

t E I.

(7.4.1)A In the following lemma we establish an a priori bound for maxtEI lx(t)l, where x is a solution of the boundary value problem (7.4.1)>. and (7.4.2), >. E [0, 1].

Lemma 7.4.3. Assume that (Ht) and (H 3 ) hold. Then every solution x of the


445

boundary value problem (7.4.1).\ and (7.4.2), ,\ E (0, 1], satisfies

max{M, ~.lq1(a)l, J.lq2(b)l}, ifo.o, Po =F 0 maxlx(t)l $ Mo := tEl

{

max{M, J.lq2(b)l},

ifo.o =0

max{M,~0 Iq1(a)l},

if#o=O.

(7.4.4)

Proof: For any,\ E (0, 1], let x be a solution of (7.4.1).\ and (7.4.2) and

1 1 rp(t) = 2 {x(t), x(t)} = 2lx(t)l 2,

tEI

where (-, ·) denotes the Euclidean inner product. If rp takes its maximum value at a point { E (a, b), then we have

and rp"({) = lx'({)l 2 + ,\ {x({), /({, x(u1(e)), ... ,

x(uk({)), x'(ukH({)), ... , x'(uA:+m({)))) $ 0. (7.4.5)

Consider the case that,\> 0 first. Then the above relations and (Hi) imply that lx(e)l sM. If x takes its maximum value at the point a, then from the boundary condition,

and, if o.o = 0, we obtain o.1x'(a) = q1(a). By (Ha), q1(a) = 0 and, because 0.1 > 0, we have x'(a) = 0. Thus rp'(a) = 0. If lx(a)l > M, by (7.4.6) and (H1), we obtain rp"(a) > 0. Hence, there exists a positive c such that rp'(t) > 0 for t E (a, a + c), which means that the function rp is increasing near a which contradicts the fact that rp(a) is maximum. Therefore lrp(a)l $ M. If o.o > 0, 0.1 > 0, since rp(a) is the maximum value of rp, we have

446

Chapter 7

= (x(a),qt(a) + aox(a)) = (x(a),qt(a)) +ao lx(a)l 2 ~ -!x(a)llqt(a)! + ao lx(aW

= -lx(a)l [lqt(a)!- ao !x(a)IJ. Hence, lqt(a)!- aolx(a)l 2':: 0 and consequently,

This inequality is also true for a 1 = 0.

If lx(a)l is the maximwn value of the function lx(t)l, then

Similarly, if lx(b)l is the maximwn value of !x(t)l on I, then

Thus, for A E (0, 1], the lemma is proved.

=

c1t + c2 for some c1, c2 Now we consider (7.4.1) for A= 0. Then x(t) in Rn and t E I. If jx(t)j takes its maximum at E (a, b), then (7.4.6) yields !x'(e)l = 0 and hence x(t) = c2 • The boundary conditions give -aoc2 = qt(a) and f3oc2 = q2(b). Hence either

e

and lx(t)! ~ Mo, t E (a, b) in this case. Next assume that lx(a)l is a maximum value of !x(t)l on [a, b]. If ao = 0, then a1x'(a) = q1 (a) and by (Ha), x'(a) = 0. Thus x(t) = c2 , t E I. The boundary condition at t = b gives {30 c2 = q2 (b) and thus


447

Finally, we consider the case a 0 > 0. But then the previous arguments apply again and yield

Consequently, we find that when the maximum occurs at t = a, then

The same bounds hold also if lx(t)l takes its maximum at t =bon I. The lemma 0 is proved. The following lemma concerns a priori bounds for max 1eiix'(t)l. Lemma 7.4.4. We assume that there exists a constant M 0 such that max lx(t)l ~ Mo IE/

for every solution of the boundazy value problem (7.4.1).x and (7.4.2), A E [0, 1] and assume (H2) holds. Then, if a1 > 0, there exists a constant Mt independent of A, such that

for every solution x of the bounda.zy value problem (7.4.1).x and (7.4.2). Proof: Let x be a solution of the boundary value problem (7.4.1).x and (7.4.2), A E [0, 1]. Since a1 > 0, the first of the boundary conditions implies that

or

Chapter 7

448 Let

C= {

max(~ (lqi(a)l+aoMo), p1 (lq2(b)l+f3oMo), sup lw'(t)l), 1

1

> 0,

PI =0

max(~ (jq1(a)l + aoMo}, sup lw'(t)l), 1

(31

tEE(a)

tEE(a)

(7.4.6) Without loss of generality we assume that M 0 is large enough so that sup lw(t)l ~ Mo. From (7.4.2) and (H2), for every t E I we have tEE(a)

lx"(t)l ~ g(t)O(lx'(cr;(t))l). From the Cauchy-Schwarz inequality, if x' is a nonzero function defined on I, we have

and thus lx'(t)l' ~ g(t)O(x'(cr;(t))l),

for every t E I, with x'(t) =F 0.

It is obvious that the last inequality remains valid even when x'(t)

Therefore, we have lx'(t)l' ~ g(t) O(lx'(cr;(t))l},

for every t E I,

and so lx'(t)l

~ lx'(a)l +

~C+

1'

1'

g(s)O(lx'(cr;(s))l}ds

g(s)O(Ix'(cr;(s))l ds.

Furthermore, we define

,P(r) =

sup 8EE(a)U[a,r]

l:z:'(B)I

for r E [a,b]

= 0 fortE I.

Boundary Value Problem& for Second Order Functional Equations

which is continuous, since t E [a, b) and a;(t)

a1

> 0 implies w~(a)

= wi.(a).

449

So, for .,. E [a, t),

5 t, t E I, we get lx'( -r)l 5

c+

1'

g(s) n( (s )) ds

and hence sup lx'(-r)l5 C+ .. e[co,t)

Since x(t) = w(t), t

J.' g(s)!l((s))ds • "

e E(a) and from the choice of C, we obtain (t)

~C+

1'

g(s)!l((s))ds,

t E I.

By Bihari's inequality we have

(t)

~ G-

1(

G(C) +

1 6

g(s)ds) := Mt

where G is supposed such that

Hence sup,e 1 1x'(t)l

~

M 1 • The proof is complete.

0

Proof of Theorem 7.4.1: By Lemma 7.4.3 and Lemma 7.4.4, there exists a constant d such that 1\xl\ 5 d for every solution x of the boundary value problem (7.4.1).\ and (7.4.2), .X e [0, 1). Consider the subspace X of B given by X = { u E B : u is piecewise twice differentiable on I}

and define

T.\x = .XLx +w

(7.4.7)

Chapter 1

450

where

J: G(t, s )f(s, x(

a1

(s )), ... , x(ak( s )), x'(ak+I (s )),

... ,x'(ak+m(s))), Lx( a),

e(aofat)(t-a)

Lx(t) = 0, e-(f3o/f3l)(t-b)

Lx(b),

tEl,

t b, /31

0,

0

= 0.

It is clear that (7.4.8) defines a mapping T>..: X-+ X for every A E [0, 1]. Define

U

= {u

EX: !lull< d + llwll

+ 1}

and H:(0,1]xU-+X

where

H(.\,u) = T>.. u = .\Lu + w. We shall show that the mapping His a compact homotopy. To this end it is enough to prove that the mapping L: X-+ X, defined by (7.4.9), is compact. Indeed, let {h 11 } be a bounded sequence in X, i.e. Then we can prove that

I!Lh11ll ~

II h 11 II :::;

C, for all v.

C := max{FK,FKCI}

where

F = max lf(t, Ut, ... , Uk+m)l, tEl

i = l, ... ,k

+m

lu•I~C

K

= max{Kt,K2 },

K 1 , K2 are the constants of Lemma 7.4.1,


451

and C1 =max { max tEE(a)

le*(t-a}l, tEE(a) max I!!.Q.. e*(t-a),, O't

(7.4.9) In (7.4.9) cl can be appropriately adjusted when either For all t 1 , t2 E J and arbitrary v we have

\Lh.(t.)- Lh.(t2)\ = \(Lh.)'(t,)- (Lh. )'(t2)\ =

ll' ll'

fr}

or

f3I

is zero.

(Lh.)'(s) dsl :o; 8\t, - t2\, (Lh.)"(s) dsl :o; h\t, - t2\,

where

Hence the sequences {Lh 11 } and {(Lh 11 ) ' } are equicontinuous. Moreover, it is obvious that H(O, au) = {w }. Thus, since w E X, it is clear that H(O, au) c X. On the other hand, for every X E au, i.e., for every x EX with llxll = d + llwll + 1, we cannot have x = H(>..,x) for some>.. E [0, 1]. Indeed, in that case x must be a solution of the boundary value problem (7.4.1).x. and (7.4.2) and hence llxll ~ d. Therefore,

x

=/:-

H(>..,x)

for all x E

afJ, )..

E (0, 1).

The assumptions of the fixed point theorem are then satisfied and the function H(l, ·)has a fixed point in X. This means that there exists at least one x EX such that x = H(l,x), or x = T1x = Lx + w, which implies that xis a solution of the boundary value problem of (7.4.1) and (7.4.2). D Example 7.4.1. Consider the boundary value problem

x"(t) = x 3 (t)

+

x'(t)(x'(t 1)) 2 sin(x'(t + 1))- 1 1 + t + lx'(t)l '

t

E [0, 1]

Chapter 7

452

- aox(t) f3ox(t)

+ a1x'(t) =

+ {31x'(t) =

q1(t),

q2(t),

t E [-1,0),

t E [1,2].

It is easy to check that (H 1 ) holds with M = 1. Also, if we set

clearly, for every (t,ut,u2,u3,u,) E [0,1] X R4 with !uti E [0, 1], we have

and consequently, (H2 ) holds with n(s) = 2 + s 2 • In order to have a solution for the above boundary value problem it is enough to check that

G(C) + 1 E DomG- 1 , where C is defined by (7.4.6). For example, if ao = 1, a 1 = 1, qt(t) = 0 and f3t = 0, the above problem has at least one solution since 0 < C = 1 < v'2 tan( v'2 (~- 1)).

7.5. Boundary Value Problems for Singular Equations In this section we consider the second order functional differential equation

x" + f(t, x( r(t)))

= 0,

0$t$l

(7.5.1)

under the assumption (H1) f(t,x): (0, 1) x (O,oo)--+ (O,oo) is continuous and decreasing in x for each fixed t and integrable on [0, 1] in t for each fixed x. And lim f(t, x) = oo .. -o+

uniformly on compact subsets of (0, 1)

and lim f( t, x) = 0 uniformly on compact subsets of (0, 1).

z-oo

Boundary Value

Problem~

for Second Order Fu.nctional

Equation~

453

(H2) T(t) is continuous on (0, 1] satisfying inf T(t)

tE[D,l]

0. tE[O,l]

From (H2) we see that the set E = {t E (0, 1] : T(t) E (0, 1)} satisfies mesE > 0. According to the continuity of T(t) there exist closed intervals I C E and J C (0, 1) with mesi > 0 such that t E I means T(t) E J. Denote a = min{O, inf T(t)}, b = max{1, sup T(t)}. tE[O,l]

tE[O,l]

The boundary conditions considered here are

= ~t(t),

t E (a,O]

-yx( t) + c5x' (t) = v( t),

t E (1, b]

ax(t)- (3x'(t)

(7.5.2)

which satisfy (H3) a, (3, -y, c5 are nonnegative constants, with

~t(t)

and v(t) are continuous functions defined on [a, 0] and [1, b], respectively with ~t(O) = v(1) = 0, and satisfying ~t(t) ~ 0 for (3 = 0 and ft0 e-J•Jl(s)ds ~ 0 for (3 > 0 and v(t) ~ 0 for c5 = 0 and J1t ef•v(s)ds ~ 0 for c5 > 0. Note that boundary condition (7.5.2) gives that

{

ax(O) - (3x'(O) = 0 (7.5.3)

-yx(1) + c5x'(1) = 0.

=

If r(t) t, this coincides with the usual linear BVP for an ODE. By solving the linear equations in (7.5.2) we see that (7.5.2) is equivalent to

(3>0

t E [a, 0] (3=0

(7.5.4)

Chapter 7

454

and

t E [1, b].

(7.5.5)

6=0

(7.5.4) and (7.5.5) imply that any function x(t) satisfying (7.5.2) with x(O) ;:=: 0 and x(1) ;:=: 0 will be nonnegative on [a, 0] U [1, b] (if x(O) > 0, x(1) > 0, then x(t) > 0 on [a, OJ U [1, b)). By a positive solution of the problem (7.5.1) and (7.5.2) we mean a function in C[a, b] n C 2 (0, 1) which is nonnegative in (a, b) and positive in (0, 1) and which satisfies the equation (7.5.1) and the boundary condition (7.5.2). Remark 7.5.1. As we will see later for the case that /3 > 0 (or 6 > 0) the solutions are actually in C[a, b]nC 1 [a, 1]nC 2 (0, 1) (or C[a, b)nC 1 [0, b]nC 2 (0, 1)). Although the expressions for boundary functions are different for different values of /3, 6, the conclusion concerning existence and uniqueness of a positive solution are exactly the same. In the sequel, we will state the results for the general case and give the proofs only for the case that j3 > 0 and 6 > 0, since the other cases can be done in a similar way. A fixed point theorem on cones will be used which we outline below. Lemma 7.5.1. Let X be a Banach space, K a normal cone in X, D a subset of K such that if x, y are elements of D, x $ y, then (x, y} is contained in D, and let T : D ~ K be a continuous decreasing mapping which is compact on any closed order interval contained in D. Suppose that there exists x 0 ED such that T 2 xo is defined (where T 2 x 0 = T(Tx 0 )) and furthermore Tx 0 , T 2 x 0 are (order) comparable to xo. Then T has a fixed point in D provided either (I) Txo $ Xo and T 2xo $ xo or Txo 2:: x 0 and T 2xo 2:: xo, or (II) The complete sequence of iterates {Tnxo}~=O is defined and there exists Yo E D such that Tyo E D and Yo $ Tnx 0 for all n.

We shall now give an existence theorem for boundary value problems (7.5.1) and (7.5.2) under the hypotheses (H1)-(H3) and j3 > 0, 6 > 0.


455

Let Yt : [a, b] --+ [0, oo) be defined by

Yt(t) = {

t-a

if a :5 t :5

b-t

if

t

t :5 t :5 b,

= 8g1 • We will assume further that

and for (}

> 0, gg( t) is defined by gg

(H4) 0
0.

With the assumption (H3) the boundary value problem x" = 0

ax(O)- {3x 1(0) = 0

(7.5.6)

-yx(1) + 6x'(1) = 0

has a Green's function G: [0, 1]

X

[0, 1]--+ [0, oo) given by

~('Y + 6- -yt)(/3 +as), G(t,s) = {

1

-({3 + at)( 'Y + 6 - -ys ),

t:5s:51.

p

It is clear that G(t,s) > 0 for (t,s) E (0,1) X (0,1), and G(t,s) satisfies condition (7.5.3). We seek to transform (7.5.1) and (7.5.2) into an integral equation via the use of the Green's function and then find a positive solution by using Lemma 7.5.1. Denote by X the Banach space of real-valued continuous functions defined on [a, b] with supremum norm 11·11, K be the cone in X of nonnegative functions x( s) which are differentiable on [a, 0] U [1, b] and satisfy the differential equations

ax(s)- f3x'(s) = kJ.L(s),

s E [a,O]

-yx(s) + 6x'(s) = hv(s),

s E [1,b]

and

for some k, hER. Obviously, [(is a normed cone. Define a subset of [(by D

= {tp E [(

: tp(t) ~ gg(t) for some 8

> 0, t E [a,b]}.

Chapter 7

456

Then we can define an operator T : D

e!Jt(~

1 1

Tcp(t) =

1°

K by

-+

t E [a, OJ

e-t•Jt(s)ds +cp(o)).

tE(O,l)

G(t,s)f(s,cp(r(s)))ds,

e-tt (~

1'

e-l•v(s) ds

+ e-lcp(l)).

(7.5.7)

t E [l,b).

Noting that G(t, s) is the Green's function for the boundary value problem (7.5.6) and that the functions in (7.5.4) and (7.5.5) satisfy the condition (7.5.3) we see that any fixed point cp of the operator Tis in C 1 [a, b]nC 2 (0, 1), and hence satisfies

cp(O) =

1

cp(1) =

1

1

G(O,s)f(s,cp(r(s))) ds

and 1

G(1, 8)f( 8, cp( r( 8))) d8.

Thus for the fixed point 0 for t E (a, b), ll 0 and C,On(O) - 0 as n - oo. (7.5.2) gives that cp~(O) - 0 as n - oo, i.e., there exists a L > 0 such that cp~(O) $ L for all n. Since cp~(t) < 0 fort E (0, 1), we have cp~(t) $ L for all n and t E (0, 1). This together with C,On(O) - 0 as n - oo gives that C,On(t) are uniformly bounded on [0, 1], which contradicts that llc,onll - oo as n - oo. Similarly, we conclude that 'Pn(1) ~ i 2 > 0 for all n. In view of the fact that C,On(t) is concave down on (0, 1) we see that there exists a 8 > 0, which is independent of n, such that

C,On(t) Therefore fortE [0, 1] and n

~

gs(t),

~no,

1 1

C,On(t) = Tc,on(t) =

=(

I+ I

J1

J[o,1]\1

$1M· $1

te(0,1)\I.

+M

G(t,s)f(s,c,on(T(s)))ds )G(t,s)f(s,c, on(r(s)))ds

M~IIdt+

1 1

1 1

G(t,s)f(s,gs (s))ds

f(s,gs(s))ds < oo,

contradicting C,On(tn) - oo. This completes the proof.

0

Theorem 7.5.2. Assume (Hl)-(H4) hold. Then the BVP (7.5.1), (7.5.2) has at least one positive solution. Proof: For each n, let

a-t a

1/Jn(t) =

1 1

11 G(O,s)f(s,n )ds,

Jo

G(t,s)f(s,n) ds,

t

b-t b _ 1 Jo G(1,s)f(s,n )ds,

t E (a, OJ

t E (0, 1)

t E [1,b].

(7.5.10)


459

Then '1/Jn(t) is continuous on [a, b], and from (H1), it follows that 'I/Jn+1 :::; '1/ln, '1/Jn(t) > 0 for t E (a, b), and lim '1/ln(t) = 0 uniformly on [a, b]. Define fn : n--+oo (0, 1) X (0, 00) --+ (0, 00) by

fn(t,s)

= f(t,max(s,'l/ln(r(t)))).

and observe that fn is continuous and by (H1), for (t,s) E (0, 1)

fn(t, s):::; f(t, s)

and

X

(0, oo),

fn(t,s):::; f(t,'f/Jn(r(t))).

Now define operators Tn : K --+ K by

+ Tn 0

where I is defined in (H2). According to (H1) there exists a h > 0 such that t E I implies that f( t, h) ~ mill" By assumption there exists a no such that n ~ no implies that 0 < lf'n(r(t)) < h fort E I since r(t) E J C (0, 1). The definition

Chapter 7

460

of

tPn

shows that there exists a n 1 ~ no such that t/Jn(r(t)) < Then for t E I and n ~ nl!

c fort

E I and

n ~ n1.

ct'n(t) =

1

~

1

1

G(t,s)fn(s,cpn(r(s)))ds

G(t,s)fn(s,cpn(r(s)))ds

~m

jJ(s,max(cpn(r(s)),.,Pn(r(s)))) ds

~m

jf(s,c)ds

~1

contradicting that lim ct'n(t) = 0 on [0, 1]. n-+oo From the property of tpn(t), it is easy to see that there is a 8 > 0, independent of n such that tpn(t) ~ gs(t), t E [a, b]. Let

e!J 1

p(t) =

(~

1°

e-fs JL(s) ds

+ gs(O)),

t E (a,O]

gs(t),

t E (0, 1)

e-ft(~ lt eT"v(s) ds + eT gs(1)).

t E [1,b],

and

eft(~ g(t) =

1°

e-f JL(s) ds +

R).

R, e-ft (

t E [a,O] tE(0,1)

~ lt eT"v(s) ds + ef

R),

t E (l,b].

Then p, q E K, and ct'n E (p, q). It is easy to see that T is compact on this interval. Then by going to a subsequence if necessary, we may assume that lim Tcpn exists, and we denote it by cp•. n-+oo The last step is to show that lim (Tcpn - ct'n) = 0, since, if this is true, n-+oo

then we have that cp• E (p, q) and that


461

To see that lim (T'f'n - 'f'n) = 0, choose (J > 0 such that for all n, 'Pn(t) n-+oo 99(t), t E [a, b]. Let c > 0 be given and choose c such that 0 < c < 1 and

2M[1~ j(s,g9(r(s))) ds+ l~J(s,g9(r(s))) ds] Then there exists a n 0 such that if n

~ n0 ,

Oscillation Theory for Functional Differential Equations

Oscillation Theory for Functional Differential Equations

Nonoscillation and Oscillation: Theory for Functional Differential Equations

Oscillation Theory for Difference and Functional Differential Equations

Oscillation theory for difference and functional differential equations

Oscillation Theory of Partial Differential Equations

Oscillation theory of operator-differential equations

Oscillation Theory for Neutral Differential Equations with Delay

Oscillation theory for neutral differential equations with delay

Stochastic functional differential equations

Nonoscillation and Oscillation Theory for Functional Differential Equations (Pure and Applied Mathematics)

Comparison and Oscillation Theory of Linear Differential Equations

Boundary Value Problems for Functional Differential Equations

Functional Spaces for the Theory of Elliptic Partial Differential Equations

Functional Spaces for the Theory of Elliptic Partial Differential Equations

Oscillation Theory of Delay Differential Equations: With Applications

Functional methods in differential equations

Functional Differential Equations and Bifurcation

Stability of functional differential equations

Stability of functional differential equations

Functional Methods in Differential Equations