Numerical-analytic methods in the theory of boundary-value problems

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Numerical-Analytic Methods in the Theory of Boundary-Value Problems

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Numerical-Analytic Methods in the Theory of Boundary-Value Problems M. Ronto University of Miskolc, Hungary

A. M. Samoilenko National Academy of Sciences , Ukraine

World Scientific Singapore & NewJersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

NUMERICAL-ANALYTIC METHODS IN THE THEORY OF BOUNDARY-VALUE PROBLEMS Copyright m 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981 -02-3676-X

This book is printed on acid -free paper.

Printed in Singapore by Uto-Print

PREFACE

Formulation and development of constructive methods is one of new directions in the contemporary mathematical analysis and simulation. In spite of the fact that investigations in this field were carried out for only several decades, the class of constructive mathematical methods draws more and more attention. It is likely that there is no accepted definition which would strictly restrict the class of constructive methods. Nevertheless, this term becomes more and more common. Apparently, the most natural way is to regard constructive methods as certain methods for the construction of solutions of different classes of equations and investigation of the existence and properties of exact and approximate solutions. Furthermore, the main characteristic of constructive methods is the fact that they enable one to completely solve the problem (up to the numerical values) and practically verify the theoretical background and conditions that guarantee the applicability of these methods to specific classes of problems. Apparently, for the first time, the constructive side of methods attracted attention in the fields of mathematics that are now known as the theory of nonlinear oscillations and nonlinear mechanics. It is widely recognized that asymptotic methods using different averaging schemes became, in fact, one of the main tools of constructive investigation and construction of solutions of various problems of nonlinear mechanics. The monograph [GrR3], for example, convincingly points to this fact. In the monograph mentioned, the constructivity of methods is investigated with the use of asymptotic decompositions, the averaging principle, various iterative versions of the method of LyapunovPoincare" series in powers of a small parameter, and the iteration method with accelerated convergence. These investigations are mainly aimed at studying periodic and quasiperiodic solutions. It is clear that there is an urgent need in developing constructive methods for other branches of the theory of differential equations, in particular, for the theory of boundaryvalue problems for ordinary differential equations. The analysis of the contemporary state of methods for the investigation of boundary-value problems convincingly shows that the classes of analytic, functional-analytic, numerical, and numerical-analytic methods are most often used in this field. Obviously, each group of these methods has advantages and disadvantages. However, it should be noted that, in the theory of boundaryvalue problems, the numerical-analytic methods compare favorably with other methods by their constructivity both at the stage of construction of solutions and in the course of v

vi

Preface

investigation of the principal qualitative problems such as establishing the existence of solutions, the verification of convergence of approximate solutions to exact solutions, and obtaining error estimates for approximate solutions that can be practically verified. All problems indicated above demonstrate that the numerical-analytic methods open good prospects of further development of constructive methods for the investigation of solutions of boundary-value problems for ordinary differential equations. I am pleased to introduce this book to readers. It is written by the well-known experts in the field of numerical-analytic methods and is a natural continuation of their works [SaRI], [SaR2], and [SaR4]. The present monograph is characterized by the perfect combination of the most established elements of the theory of the numerical-analytic method of successive approximations, which has proved to be very efficient; new directions of research are also described. In view of the small volume of this monograph, the authors consider a narrow range of problems concerning only the generalization of nonlinear, boundary-value problems for ordinary differential equations to new classes. One of the advantages of this work is the fact that the authors demonstrate the possibility of application of the numerical-analytic schemes under consideration not only to classic boundary-value problems, but also to nonstandard boundary-value problems, e.g., boundary-value problems with parameters in boundary conditions or pulse influence. Another distinctive feature of this book is the development of the idea of possibility and expediency of combining various numerical-analytic methods for the investigation of periodic solutions and solutions of nonlinear boundary-value problems of the general form. Clearly, this monograph contains an interesting and necessary material, which will be interesting for experts in the theory of boundary-value problems and nonlinear oscillations and will contribute to the further development of constructive numerical-analytic methods for the investigation of periodic boundary-value problems and boundary-value problems of general form. Academician Yu. A. Mitropolsky

comm Preface

v

Introduction

1

Chapter 1. NUMERICAL-ANALYTIC METHOD OF SUCCESSIVE APPROXIMATIONS FOR TWO-POINT BOUNDARY-VALUE PROBLEMS

19

1. Abstract Scheme of the Method

19

2. Choice of the Form of Successive Approximations and Their Uniform Convergence

26

3. Sufficient Conditions for the Existence of Solutions

41

4. Necessary Conditions for the Solvability of the Boundary-Value Problem

45

5. Error of Calculation of the Initial Value of a Solution

52

6. Special Types of Successive Approximations and Estimates

64

7. Numerical-Analytic Method in the Case of Nonlinear Two-Point Boundary Conditions

72

8. Boundary-Value Problems with Small Parameter

81

Chapter 2. MODIFICATION OF THE NUMERICAL-ANALYTIC METHOD FOR TWO-POINT BOUNDARY-VALUE PROBLEMS

89 89

9. Periodic Boundary-Value Problem 10. Theorems on the Properties of Determining Functions of a Periodic Boundary-Value Problem

98

11. Solvability of the Approximate Determining Equation and the Error of the Initial Value of a Periodic Solution

105

vu

viii

Contents

12. Modification of the Method for Two-Point Problems 115 13. Relationship between Exact and Approximate Determining Equations

123

14. Determination of Initial Values of Solutions of Two-Point Boundary-Value Problems 131 15. Realization of the Method for Systems of Two Equations 142 Chapter 3. NUMERICAL -ANALYTIC METHOD FOR BOUNDARY -VALUE PROBLEMS WITH PARAMETERS IN BOUNDARY CONDITIONS

149

16. Successive Approximations for Problems with One Parameter in Linear Boundary Conditions 150 17. Sufficient Solvability Conditions and Determination of the Initial Value of a Solution of the Boundary-Value Problem with Parameter 158 18. Boundary-Value Problems with Nonfixed Right Boundary 168 19. Solvability Theorems for Problems with Nonfixed Right Boundary 177 20. The Case of Several Parameters in Boundary Conditions 193 21. Linear Dependence of Boundary Conditions on Two Parameters 209 Chapter 4. COLLOCATION METHOD FOR BOUNDARY-VALUE PROBLEMS WITH IMPULSES 221 22. Green Function of a Homogeneous Two-Point Boundary-Value Problem

222

23. Inhomogeneous Linear Impulsive Boundary-Value Problems 229 24. Convergence of the Algebraic Collocation Method for Nonlinear Systems 234 25. Method of Trigonometric Collocation for Periodic Systems 245 26. Practical Solution of Impulsive Problems 250 27. Green Function for a Three-Point Boundary-Value Problem with Single Impulse Influence 257 28. Semihomogeneous Linear Two-Point Boundary-Value Problem with m-Impulse Influence

271

Contents

ix

29. Inhomogeneous Linear m-Impulse Two-Point Boundary-Value Problem

279

30. Multipoint m-Impulse Boundary-Value Problem

280

31. Equations with Piecewise-Continuous Right-Hand Side

286

32. Construction of Solutions of Two-Impulse Systems

308

Appendix. THE THEORY OF THE NUMERICAL -ANALYTIC METHOD : ACHIEVEMENTS AND NEW TRENDS OF DEVELOPMENT

317

Al. History of the Method of Periodic Successive Approximations

317

A2. Relation to Other Investigations

329

A3. Application of Modifications of the Numerical-Analytic Method to the Investigation of Various Boundary-Value Problems

339

REFERENCES

415

IIITRODUCTIOII

The development of the theory of boundary-value problems for ordinary differential equations was initiated by the works of G. Floquet [Flo], A. M. Lyapunov [Lyal], [Lya2], S. N. Bernstein [Ber], G. D. Birkhoff [Bir], E. L. Bunitsky [Bunl], [Bun2], D.Jackson [Jacl], [Jac2], G. A. Bliss [Blil], [Bli2], G. D. Birkhoff and R. E. Langer [BiL], and by E. Kamke [Kam], which appeared at the end of the 19th century and in the first decades of the 20th century. The analysis of bibliography shows that boundary-value problems have always attracted particular attention. It is natural that, at the first stage, the theory of linear boundary-value problems was developed. Solvability conditions for homogeneous and inhomogeneous boundary-value problems were studied for scalar second- and higher-order differential equations and for systems of first-order differential equations. Periodic solutions of differential equations with periodic coefficients were investigated, i.e., in fact, boundary-value problems with periodic boundary conditions were considered. On the basis of the properties of differential operators generated by given differential equations and boundary conditions, spectral properties of boundary-value problems and their relation to the corresponding conjugate problems were investigated. Much attention was given to the construction of Green functions and generalized Green functions for various differential operators and to the investigation of their properties. The theory of nonlinear boundary-value problems was also gradually developed in depth. Methods that enable one to study the problem of unique existence of solutions, analyze the oscillation properties of solutions, construct approximate solutions, and estimate errors were developed. At present, the theory of boundary-value problems possesses a fairly powerful stock of diverse methods. These methods can conventionally be classified into several main groups, namely, analytic methods, functional-analytic methods, numerical methods, and numerical-analytic methods. It should be noted that analytic and functional-analytic methods are mainly aimed at the investigation of qualitative problems of unique existence of solutions and their continuous dependence on parameters. The group of numerical methods is aimed at the direct calculation of numerical values of approximate solutions. Finally, numerical-analytic methods are fairly universal and can be used for both the investigation of the problem of existence and practical construction of solutions. Numerous profound monographs, theses, surveys, and articles that reflect the state of the theory and methods for the investigation of various boundary-value problems in the 1

2 Introduction directions indicated above were written by many mathematicians from different countries. We do not even pretend to give a complete list of the most important works; we only mention works in which a fairly complete bibliography can be found and which reflect the most typical approaches , as well as investigations of individual scientific schools. Apparently , the problems that were first comprehensively investigated were periodic boundary-value problems, which play an extremely important role in the theory of oscillations and nonlinear mechanics . In fact, all methods for studying periodic solutions are, at the same time , methods for the investigation of the corresponding periodic boundaryvalue problems . This is related to the known fact [Har] that a T-periodic solution x(t) of the differential equation x = f(t,x), f(t+T,x) = f(t,x),

(0.1)

satisfies the periodic boundary condition x(0) = x(T)

(0.2)

and, conversely , if x(t ) is defined for t E [0, TI and is a solution of the periodic boundary-value problem (0.1), (0.2), then it is the restriction of a T-periodic solution of equation (0.1) defined for t E (- -, oo) to the segment [0, T]. Among the most powerful and universal analytic methods in the theory of oscillations, one should mention asymptotic methods of nonlinear mechanics, the method of a small parameter, and averaging methods . A notable contribution to the development of these methods was made by mathematicians of the Kiev scientific school. Asymptotic methods created for various types of equations in the fundamental works of N. M. Krylov and N . N. Bogolyubov [KrB], N. N. Bogolyubov and Yu. A . Mitropolsky [BoM], and by Yu. A. Mitropolsky [Mit] were developed and generalized in the monographs of the following authors : N. N. Bogolyubov , Yu. A. Mitropolsky , and A. M. Samoilenko [BMS ], V. I. Arnold [Am], A. N. Filatov [Fil], Yu. A. Mitropolsky and 0. B. Lykova [MiLy ], E. F. Mishchenko and N. Kh . Rozov [MiR], Yu. A. Mitropolsky and B. I. Moseenkov [MiMo], Yu. A. Mitropolsky and D. I. Martynyuk [MiM], N. N. Moiseev [Moi], S . A. Lomov [Lom ], N.J. Shkil, A. N. Voronoi , and V. N. Leifura [SVL], A. V. Skorokhod [Sko], V. S. Korolyuk , N. S. Bratiichuk, and V . Pirdzhanov [KBP], A. M. Samoilenko [Sams] , Yu. A. Mitropolsky and A. K. Lopatin [MiLl, Yu. A. Mitropolsky, A. M. Samoilenko , and D. I. Martynyuk [MSM]. In the theory of oscillations and related fields , analytic methods were developed and studied in the monographs of L. Cesari [Ces2], Ph. Hartman [Har], V. A. Pliss [Pli2], V. Vazov [Vaz ], A. D. Myshkis [Mysi], A . B. Vasil ' eva and V.F. Butuzov [VaB], K. G. Valeev and 0. A. Zhautykov [VaZ], A. I. Pavlyuk, V. M. Burym, and Yu. A. Pasenchenko [PBP], I. G. Malkin [Mal], R. Z. Khasminsky [Kha], A. P. Proskuryakov [Pro], Yu. A. Mitropolsky , A. M. Samoilenko, and V. L. Kulik [MSK], A. A. Martynyuk [Mara], E. F. Tsarkov [Tsa], and D. G. Korenevsky [Kor].

Introduction 3 The application of a combination of the Lyapunov method, the Poincare method of small parameter, and the averaging method to nonautonomous quasilinear systems and autonomous equations was considered in the monograph of V. M. Starzhinsky [Sta]. Sufficient existence conditions and asymptotics of periodic solutions of differential and integro-differential equations with small parameter are presented in the book of Ya. V. Bykov and D. Ruzikulov [ByR]. Some practical aspects of the use of analytic methods based on the method of perturbations, averaging, and iterations in the theory of nonlinear oscillations are described in the works by T. Hayashi [Hay], M. Farkas [Farl], M. Farkas and H. I. Freedman [FaF], Yu. V. Trubnikov and A. I. Perov [TrP], and V. Volterra [Vol]. The analytic theory of linear periodic boundary-value problems is developed in the monographs of N. P. Erugin [Eru], V. A. Yakubovich and V. M. Starzhinsky [YaS], M. V. Fedoryuk [Fed], S. F. Feshchenko, N. I. Shkil, and L. D. Nikolenko [FSN], and G. S. Zhukova [Zhu]. Analytic methods were successfully used in the investigation of the interrelation between the problems of existence and sign-constancy of Green functions and the problems of existence and uniqueness of solutions of periodic and nonperiodic boundary-value problems in the scalar and vector cases (A. Ya. Khokhryakov [Kho], A. I. Perov [Perol], A. V. Kibenko [Kib2]). Some applied problems in the theory of nnlinear oscillations were investigated in the monograph of E. N. Rozenvasser [Roz] with the use of the methods of integral equations. Numerous works are devoted to the investigation of nonperiodic boundary-value problems, in particular, with nonseparable boundary conditions. In these works, the problem of unique existence of solutions and the properties of the Green function are mainly analyzed (see, e.g., the works by G. N. Zhevlakov, Yu. V. Komlenko, and E. L. Tonkov [ZKT], V. M.Zubov [Zub], and Yu. V. Pokomyi [Pokl]). A considerable contribution to the development of one of the basic directions in analytic methods for the investigation of boundary-value problems, namely, methods based on a priori estimates and differential inequalities, was made by the mathematicians of the Riga school. The results of these investigations with detailed bibliographical notes are presented in the monographs of V. V. Gudkov, Yu. A. Klokov, A. Ya. Lepin, and V. D. Ponomarev [GKLP], N. I. Vasil'ev and Yu. A. Klokov [VaK], and A. Ya. Lepin and L. A. Lepin [LeL]. Two-point boundary-value problems for scalar second-order equations and for systems of first-oder and second-order equations whose right-hand sides satisfy either conditions of continuity or the Caratheodory conditions were studied in [GKLP]. With the extensive use of the method of a priori estimates, the notion of lower and upper solutions introduced by M. Nagumo [Nag], and the inclination functions, necessary and sufficient conditions for the unique solvability of the boundary-value problems were obtained. In the same work, quasilinear multipoint boundary-value problems for systems of equations of the following form are also considered: x = A(t)x + g( t,x), t r= [a, b], m

I A.x(t,) = g (x ( tt ), ... , x(tm)),

1=1

4 Introduction n

t r Car (I X Rn), ( E C" (Rnm) Recall that sufficient conditions for the solvability of the scalar two-point problem z = f (t, x, x ), x (a) = A, x (b) = B under the assumption of analyticity of the righthand side f (t, x, i) were established in [Ber]; in [Nag], they were generalized to the case where the right-hand side is a continuous function. Further results in this direction are described, in particular, in the works of Z. Opial [Opi] and I. T. Kiguradze [Kigl], [Kig2]. The results obtained in [GKLP] were generalized and developed in [VaK] and [LeL]. In these works, on the basis of the method of a priori estimates, parallel with the problem of existence, the problem of continuous dependence of solutions on the data of problems and boundary-value problems with nonsummable singularities were considered. Boundary-value problems with a condition at infinity were studied in detail by Yu. A. Klokov [Klo]. Some problems in the theory of linear boundary-value problems with a condition at infinity for equations in a Banach space were studied by G. V. Radzievsky [Rad]. In analytic methods for the investigation of boundary-value problems, one can also use the estimates obtained by Ya. D. Mamedov, S. Ashirov, and S. Atdaev [MAA]. Fundamental results in the theory of nonlinear singular differential equations with functional two-point and multipoint boundary conditions are presented in the known monograph of I. T. Kiguradze [Kig2] and in the works of I. T. Kiguradze [Kig3], I. T. Kiguradze and B. L. Shekhter [KiS], T. A. Chanturiya [Cha], M. A. Kakabadze [Kak], B. L. Shekhter [She], Sh. M. Gelashvili and I. T. Kiguradze [GeK], M. T. Ashordiya [Ash], and D. G. Bitsadze and I. T. Kiguradze [BiK]. An independent direction in the development of the theory of boundary-value problems for differential systems generalized in the Kurzweil sense was developed by J. Kurzweil [Kur], S. Schwabik, M. Tvrdy, and O. Vejvoda [STV], and S. Schwabik [Sch]. Important and interesting results for boundary-value problems in the case of functional differential equations were obtained by N. V. Azbelev [Azbl], [Azb2], N. V. Azbelev and V. P. Maksimov [AzM], N. V. Azbelev and L. F. Rakhmatullina [AzR], L. F. Rakhmatullina [Rak], V. P. Maksimov [Mak], and M. E. Drakhlin [Dra] and, for the problem of existence of periodic solutions of functional differential equations with unbounded delay, by J. Mawhin [Maw2], T. A. Burton and L. Hatvani [BuH], and L. Hatvani and T. Krisztin [HaK]. Some aspects of the theory of boundary-value problems for differential-difference and functional differential equations were also developed by A. N. Sharkovsky, Yu. A. Maistrenko, and E. Yu. Romanenko [SMR], G. P. Pelyukh and A. N. Sharkovsky [PeS], G. A. Kamensky and A. D. Myshkis [KaM], M. I. Kamensky [Kam], and V. G. Kurbatov [Kur]. The idea of a priori estimates and differential inequalities, realized for some problems in the monograph of S. A. Chaplygin [Chap], was further developed by K. W. Chang and F. A. Howes [ChH] for the investigation of the problem of existence and asymptotic behavior of certain classes of singularly perturbed boundary-value problems. A combi-

Introduction 5 nation of asymptotic and qualitative methods was extensively used by V. V. Strygin and V. A. Sobolev [ StS] for the investigation of singularly perturbed systems. Analytic methods for the investigation of differential equations with discontinuous right-hand side and differential inclusions were developed by A. F. Filippov [Fili] and A. A. Tolstonogov [Toll. The investigation and solution of problems related to boundary-value problems were also performed in the monographs of F. V . Atkinson [Atk] and A. Kufner and S. Fucik [KuF] and in the works of Yu. V. Pokornyi [Pok2], A. I. Perov and A. V. Kibenko [PeK], Yu. V. Komlenko [ Kom], R. E. Gaines and J. Mawhin [GaM1] , and A. D. Wood and F. D. Zaman [WoZ]. The analytic theory of periodic solutions and boundary-value problems, the problems of dichotomy , reducibility , and branching were considered in the works of E. I. Grudo [Gru], N. A. Izobov [Izo ], V. M. Millionshchikov [Mil], Z. P. Ordynskaya [Ord], A. M. Samoilenko and R. I. Petrishin [Safe] , A. M. Samoilenko and Yu. V. Teplinsky [SaT1], and V. I.Tkachenko [Tka]. Some questions related to finding the initial conditions of periodic solutions were considered in the book by V. I. Mironenko [Mir]. The class of bilateral processes of successive approximations monotonically convergent to the required solutions can also be attributed to analytic methods . The theory of such methods for periodic solutions is presented in the monograph of N. S. Kurpel and B. A. Shuvar [KuS]. The monograph of V. I. Fushchich, V. N. Shtelen, and N. I. Serov [FSS] also enriched the arsenal of analytic methods. In this book, on the basis of symmetry analysis , the authors reduced certain multidimensional problems of mathematical physics to ordinary differential equations and constructed their exact solutions. The group of functional- analytic methods widely uses the apparatus of functional analysis , topological notions , and the theory of approximate methods for solving operator equations. The foundation for these methods, in their contemporary conception , for periodic boundary-value problems was laid by M. A. Krasnoselsky [KVZRS], L. Cesari [CesI], and J. Mawhin [Maw4] . For periodic and multipoint boundary-value problems in the resonance and nonresonance cases , these methods were developed by Fam Ki Anh [Fam], A. I. Kolosov [Kol ], and V. Ya. Derr [Der] . The method of point mappings, which is widely used in the theory of oscillations , can also be attributed to functional -analytic methods; this method is presented in the monographs of A. A. Andronov, A. A. Vitt, and S. E. Khaikin [AVK], Yu . I. Neimark [Neil, and N. V. Butenin, Yu. I. Neimark, and N. A. Fufaev [ BNF]. For differential equations, these methods were also developed by A. D. Bryuno [Bry], K. S. Sibirsky [Sib], and N. I. Vulpe [Vul]. A very important class of functional-analytic methods is formed of the methods that use the properties of topological degree , the coincidence degree, and the continuation theorem [Maw4]. With the use of this apparatus, interesting results for periodic and general boundary -value problems were obtained by A. Capietto, J. Mawhin, and F.Zanolin [CMZI]-[CMZ4] and A. Capietto and F. Zanolin [CaZ]. Under the assumption of the existence of solutions, the group of numerical methods gives practical algorithms for approximate construction of solutions of boundary-value problems . One part of numerical methods is aimed at the determination of the initial

6 Introduction values of the required solutions , i.e., at the reduction of boundary -value problems to Cauchy initial-value problems. The other part of numerical methods is aimed at finding solutions of boundary-value problems on the entire domain of variation of the independent variable. The development and justification of various numerical schemes for solving boundary-value problems such as the sweep method, the shooting method , the immersion method, the method of reduction to Cauchy problems, the finite-difference and quadrature-difference methods, the method of subdomains, the method of spline approximation, the Newton method, the quasilinearization method, and eigenvalue problems are considered in numerous works, e.g., the works of N. S. Bakhvalov [Bak], J. K. Batcher, J. L. Lambert, A. Prottero, et al. [BLP], B. N. Pshenichnyi and Yu. M. Danilin [PsD], I. Taufer [Tau], J. Casti and R. Calaba [CaK], E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders [DMS], F. P. Vasil'ev [Vas], J. M. Ortega and W. G. Poole [OrP], and G. E. Pukhov [Puk]. Various aspects of numerical methods and their applications were studied by A. A. Abramov and V. B. Andreev [AbA], Z.Aktas and H. J. Stetter [AkS], G. M. Vainikko [Vai2], A. M. Samoilenko and V. A. Ronto [SaRol], M. Ronto, E. P. Semagina, and E. N. Dzhigun [RSD]. Among contemporary methods for the investigation of nonlinear boundary-value problems, an important place belongs to numerical-analytic methods for the investigation of the problem of existence and finding approximate solutions. We understand numerical-analytic methods as methods that enable one to represent the required solution in an analytic form, although some of its parameters or coefficients are determined numerically. According to this definition, first of all, all direct methods of mathematical physics, i .e., all variational and projective methods , including the collocation method, can be attributed to the group of numerical-analytic methods. Recall that, according to S. L. Sobolev [Sobo], direct methods of mathematical physics are defined as methods for solving problems of the theory of integral and differential equations based on the replacement of these equations by algebraic ones . It is clear that, in the sense of this definition, finite-difference methods for solving boundary -value problems can also be attributed to direct methods. We note that the history of development and application of the theory of direct methods and relevant questions are presented in detail in the monograph of A. Yu . Luchka and T. F. Luchka [LuL]. On the basis of projection and projection-iterative methods, the numerical-analytic methods for periodic and more general boundary-value problems were developed by M. Urabe [Ural], [Ura2], A. Yu. Luchka [Luc I], A. I.Perov [Pero2], A. M. Samoilenko and O. D. Nurzhanov [SaN], A. M. Samoilenko and 1. 0. Parasyuk [SaPa], and I. A. Lukovsky, M. Ya. Barnyak, and A. N. Komarenko [LBK]. In the theory of periodic solutions, numerical-analytic schemes on the basis of iterative methods were, apparently, first developed by L. Cesari [Ces2], N. N. Bogolyubov, Jr., and B. I. Sadovnikov [BoS], J. K. Hale [Hall], [Ha12], and A. M. Samoilenko [Saml][Sam3] . Later, numerical-analytic methods for periodic and general boundary-value problems were developed on the basis of iterative schemes by A. M. Samoilenko and V. A. Ronto [SaRo2], A. M. Samoilenko and V. N. Laptinsky [SaLa], Le lyong Tai [Le13],

Introduction 7 A. A. Boichuk [Boi], E. P. Trofimchuk [TroI], and A. M. Samoilenko and B. P. Tkach [SaTk]. A significant contribution to the development of numerical-analytic and functionalanalytic methods and their application to the numerical solution of nonlinear boundaryvalue problems for ordinary differential equations was made by the mathematicians of the Tartu school. Theoretical problems related to the theory of approximate methods for solving operator equations and discretization and approximation methods were investigated in the works of G. M. Vainikko [Vail], [Vai4], G. M. Vainikko, A. Pedas, and P. Uba [VPU], G. M. Vainikko and A. Yu. Veretennikov [VaV], and M. A. Krasnoselsky, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii, and V. Ya. Stetsenko [KVZRS]. In these works, as well as in the works G. M. Vainikko [Vai2], [Vai3], [Vai5] and G. M. Vainikko and P. Kh. Miilda [VaM], the general results obtained for operator equations were used and various methods for solving broad classes of initial-value and boundary-value problems in the case of differential, integral , and integro-differential equations were justified. The main attention was given to the Galerkin method, method of mechanical quadratures, method of subdomains, and difference methods . The collocation method for scalar linear and nonlinear differential equations of the in th order in the case of twopoint linear boundary conditions was first justified in [Vail] and [Vai2]; in particular, in the work of G. M. Vainikko [Vai6], these results were generalized to the case of nonlinear boundary conditions for a problem of the form Au := u(m)- f (t,u,u(l),...,u (m_l)) = 0, a 5 t 0, and satisfies the Lipschitz condition with a matrix K = { K ii >- 0, i, j = 1, 2, ... , n }, i.e.,

Jf(t,x)- f(t,x")I

sup x 0 eD,

1

IT

( R2 +KIE+

K(E-Q)-1)(R1+R2)]Ixo-x+()2 Qw(x)

is true because, by virtue of Theorem 4.2, such subsets cannot contain the point xo that determines the initial value of a solution of the boundary-value problem (2.1), (2.2) according to (4.1). The other subsets Di form a certain set Mm, which tends as i, m - oo to the set M(xo) determining the initial values of a solution of the boundary-value problem (2.1), (2.2) according to (4.1). Any point x0 = zoe Mm can be regarded as an approximate value of the point xo = xo . In this case, the estimate

I zo -xoI _ 1. The statement formulated above is analogous to the sufficient condition given in Theorem 2.3. Note that we can also establish analogs of Theorems 2.2 and 2.3. Moreover, in the general case, one should consider the sequence of functions constructed for the boundary -value problem in the form (2.8) instead of sequence (7.18) obtained on the basis of (6.1). Thus, as in the case of boundary -value problems with the linear boundary conditions (2.2), (0.2), (0.3), the problem of existence and approximate solution of the nonlinear differential equation (7.1) satisfying the nonlinear boundary conditions (7.2) reduces to the problem of finding the roots of the determining equations (7.27), (7 .28) and construction of successive approximations with the use of the recurrence formula (7.18). Along with equations (7.27), (7. 28), we consider the system of approximate determining equations F(xo,Yo) = 0, A. (xo , Yo) = 0, (7.29) where the function A. (xo, Yo)

(C~lA+E)x0

1T f(t,xm(t,x0,y0) T

J

+ h(t,y0))-

dh(t,y0j dt dt

(7.30)

0

differs from (7.22) by the fact that xm(t, x0 , yo) is used instead of the limit function. The following statement on sufficient conditions for the solvability of the nonlinear boundary-value problem (7.1), (7.2) is true: Theorem 7.2. Suppose that that the right-hand side of the system of differential equations (7.1) and the boundary conditions (7.2) are such that the functions f ( t, z) and g (u, v) satisfy the conditions of Theorem 7.1. Moreover, assume that, in the domain Di = Dp xQ there exists a convex closed domain

(7.31)

78 Numerical-Analytic Method of Successive Approximations

Chapter 1

D3 = L x O 'c Dp x 521, (7.32) such that, for a certain m ? 1, the approximate determining equations (7.29) have a unique solution wo = wom, w0 = (x0, Yo) ' wom = (xom , yon ) of nonzero index in D2 . If the inequality ()2[Qrn+1(E_Qy. »inf l&m( xo ,Yo) I > 1M^ KI(C'A+E)xoll ES2

(7.33)

holds for the vector function (7.30) on the boundary S2 of the domain D2, then the nonlinear boundary-value problem (7.1), (7.2 ) has a solution z = z *(t) for which the initial value z *(0) = z0 determined by a formula of the form (7.27) is given by a parameter x0 = xa belonging to the set and a parameter Y0 = yp contained in the domain Q'I . This statement can be proved by analogy with the proof of Theorem 3.1 by showing the homotopy of the vector fields generated by mappings (7.27)-(7.29) on the boundary of domain (7.32) in the case where condition (7.33) is satisfied. For the nonlinear boundary-value problem (7.1), (7.2), we can obtain an analog of Theorems 4.2 and 5. 2, on the basis of which we can give an algorithm for finding approximations of the values of the parameters x0 and yo determining the required initial value of the solution according to (7.26). Let us now clarify conditions that guarantee the solvability of the approximate system of determining equations (7.29). Assume that the functions f (t, z) and g (u, v) satisfy certain additional conditions , namely, let the vector function 'bm (x0, yo ) = (F(xo, y0), Am (x0 , y0)) , (7.34) which generates the mapping (D: R2n -* R2", be differentiable with respect to w0 = (xo, Yo) in domain (7.31). Assume that a point w o. = x'om = (xOm' Yom) is an approximate solution of the system of approximate determining equations (7.29) which gives the residual II m(xOm . YO m )

I

- 1. For the periodic boundary-value problem (2.1), (0.2), one can improve estimate (6.9) by proper modification of the iterative process (6.8). This modification is based on the combination of the numerical-analytic method of successive approximations with one of the most efficient (in the sense of calculations) projective methods, namely, the method of trigonometric collocation, which is also a numerical -analytic method according to our classification. It is known [SaRl ] that if there exists a T-periodic solution x*(t) of problem (2.1), (0.2), then, according to the method of trigonometric collocation , it can be found in the form of a vector trigonometric polynomial of degree k k

lajcos jou + bj sinjwt),

xk(t) = ao +

T=2.

(9.4)

j=1

In this case, the unknown coefficients (parameters) a j = (aj1, ... , a j") and bj = (bj 1, ... , bj") are determined from the system of Nn, N = 2k + 1, determining equations

xk(ti) = f(t1,Xk(t, )),

t; = iN, i = 0, 1, ... N- 1, (9.5)

and, as k-co, the approximate solutions Xk(t) uniformly converge to the exact solution x*(t), and ik(t) converge to x*(t) in mean square.

91

Periodic Boundary-Value Problem

Section 9

For the rate of convergence, the following estimates are true: IIxk(t)- x *(t)IIC
0, together with its J3neighbourhood, where

Section 16

Successive Approximations for Problems with One Parameter

151

R = 2M + R 1(xo, A), 131(x0, ? ) = I -1 [C-d-(C- 'A+XE)x,)jj.

(16.5)

Denote by G the set of (n -1) -dimensional vectors YO = (x02, ... , xOn) such that x0 = col (x01, yo) belongs to the domain Do. We also assume that, as in Section 2, D0#0

(16.6)

A.(Q) < 1,

(16.7)

and

where ? (Q) is the maximum eigenvalue of the matrix Q = T K . n Consider the sequence of functions

x,n (t, Y0 , )L) = x0 + J I .f(t, x.-1(t, Yo' ^)) 0 T - T,I .f(s, xm-1(s, Y0,)L))ds dt + 0 m=1,2,...,

7, [C-ld - (C-'A +XE)x0j,

(16.8)

x0(t, Y0, X) = x0 E DP,

where the first component of the vector x0 is defined by (16.3). It can be directly verified that all functions xm (t, y0, X) depending on the parameter that appears in the boundary conditions (16.2) and on an (n - 1)-dimensional vector y0 (or an n-dimensional vector x0 with fixed first coordinate) satisfy the boundary conditions (16.2) and (16.3) for arbitrary values of y0E G and XE I X. The following statement concerning the convergence of successive approximations (16.8) is true: Theorem 16.1. If the right-hand side f (t, x) of the system of differential equations (16.1) is defined and continuous in domain (16.4) and conditions (16.5)-(16.7) are satisfied for the boundary-value problem (16.1)-(16. 3), then a sequence of functions xm (t, y0, ?) of the form (16.8) satisfying the boundary conditions (16.2), (16.3) for arbitrary y 0 E G and A. E I X converges uniformly as m - co with respect to the domain

152

Numerical-Analytic Method for Problems with Parameters (t, y0, A.) E [0, T] x G x I.

Chapter 3 (16.9)

to the limit function x*(t, yo, A.). Furthermore, x* (t, y0, X ) is a solution of the integral equation

x(t) =

T

x0

+ J { f(tx(r)) - 1 J f(s, x(s))ds TO

+ A1T [C-'d - (C-'A +AE)x0] }dt,

( 16.10)

which passes through the point x*(t, y0, A.) = x0 = (x01, y0) for t = 0. Moreover, x*(t, y0, A.) satisfies the boundary conditions (16.2), (16. 3), i.e., is a solution of the boundary-value problem x = f(t,x)+A(y0,A.),

Ax(0)+A,Cx(T) = d,

x1( 0) = x01, (16.11)

where

A(Y0,.) = AL [C-Id - (C-1A+AE)x0] - T Jf(t, x(t))dt. T 0 In this case, the deviation of the limit function x* (t, y0, A.) from xm (t, y0, A,) for all m = 1, 2, ... is estimated coordinatewise by the inequality

x* (t, Y0, %) - X. (t, YO, X ) I 0, together with their (3-neighbourhoods, where

(3 = 2M+1j ( xo), 31(xo ) = I C-1 d-(C-'A+E) x0I ,

(18.4)

and the first component of the vector x0 is defined in (18.3). Denote by G the set of (n- I)-dimensional vectors y0=(x02..... xOn) such that the vectors x0 = (x01, yo) belong to Da for x01 defined by (18.3). Furthermore, we make certain assumptions typical of the numerical-analytic method under consideration , namely, we assume that the set Dp is nonempty and the maximum eigenvalue of the matrix Q = T K is less than 1, i.e., n Dp # 0,

?.(Q) < 1. (18.5)

Consider the sequence of functions xm (t, y0, ? ,) depending on t, y0, and ? as follows:

170

Numerical-Analytic Method for Problems with Parameters

xm

(t, YO, ?U)

= x0 +

f

Chapter 3

f(t'x'._I(t' yO'x)) - x f .f (s, xm-1(S, y0, ? ))ds I dt 1

+[C-ld-(C- A+E)xo], m=1,2,...,

(18.6)

x0(t,Yo, ?) = x0 = (x01,Y0)•

Substituting ( 18.6) in (18.2) and ( 18.3), one can easily see that , for all m = 1, 2, ... , the functions xm (t, yo , ?.) satisfy the given boundary conditions for any X E [X 1, T] and YoE G. Let us prove that a sequence of functions x m (t, yo, ?,) of the form ( 18.6) is uniformly convergent. Theorem 18.1. Suppose that conditions (2.4), (18.4), and (18.5) are satisfied in domain (16.4) for the boundary-value problem (18.1) -(18.3). Then a sequence of functions xm (t, y0, ?.) of the form (18.6) satisfies the boundary conditions (18.2) and (18.3) for all m = 1, 2, ... and arbitrary y o E G and X e [%j, T] and converges uniformly to the limit function x * (t, y o, A.) as m -+ oo with respect to the domain

(t, y0,X)E [ 0,X]x Gx[ X1, T].

(18.7)

Furthermore, f (t, yo , ?) is a solution of the integral equation

x(t) = xo + j {f(tx(t)) - f f(s,x(s))ds + [C-1d-(C-IA+E) xo]}dt

(18.8)

that passes through the point x* (0, yo, ? ,) = xo = (x01, yo) for t = 0, and x*(t, y0, X) satisfies the boundary conditions (18.2) and (18.3), i.e., x * (t, yo, X) is a solution of the boundary-value problem

x = f(t,x)+A(yo, &), Ax(0)+Cx(A) = d,

x1(0)= x01, (18.9)

where X A(Yo,k) = [C-'d-(C-IA+E)xo] - -f f(t,x(t))dt.

0

For the deviation of xm (t, yo, 7.) from x*(t, yo, X) in domain (18.7), the following coordinatewise estimate is true:

Section 18

171

Boundary- Value Problems with Nonfrxed Right Boundary

x*(t,Yo, X. (t, Yo' ') I a(t,A)W1(Yo,?) = E(x*(t,Yo,?), xm(t,Yo,) ,)),

( 18.10)

where W1(Y0,?) = Qm(E-Q)-1M+KQm- 1(E-Q)-1N1(xo),

Q = ^ K, ot(t, ^) = 3 tttl 1 - I, and î1(x0) is a vector of the form (18.4). Proof. We have already verified that all functions xm (t, yo, a,) satisfy the boundary conditions ( 18.2) and (18.3) for arbitrary A,E [? 1, T] and yor G. The inequality

Ix1(t,Yo,),)- xoI A.) _

[C-'d-(C- 'A+E)xo ] -

f f(t,xm(t,Yo'.))dt, 0

(19.1)

where xm (t, yo, A,) is calculated by using (18.6). It is natural to define the approximate determining equation for the boundary-value problem ( 18.1)-(18 .3) with the use of function ( 19.1) in the form Am(YO,A) = 0.

(19.2)

Theorem 19.1. Suppose that all conditions of Theorem 18.1 are satisfied for the boundary-value problem with nonfixed right boundary (18.1)-(18.3). Furthermore, assume that one can indicate a convex closed domain A' = G'x [.i,?4] c Gx[ T] ,

(19.3)

such that, for a certain m >- 1, the approximate determining equation (19.2) has a unique solution (yo, %) = (yom, Am) in (19.3). Also assume that the index of this solution is not equal to zero and the following inequality holds on the boundary S' of the domain A':

(YOinfE S,

2 3) IAm(Y0,a')I > (,, QW(YO,A.),

where Wt (yo, % ) is defined according to (18 . 10) and (18.4), and a = K.

(19.4)

178


Chapter 3

Then the boundary-value problem (18.1)-(18. 3) has a solution, i.e., the pair (x*(t), X*), and the initial value of this solution x*(0) = (x01, Yo) (19.5) is such that the ( n - 1)-dimensional vector yo belongs to G' and the corresponding parameter A* lies in the segment [a,,, ?,2]. Proof. We use the scheme of the proof of Theorem 17.1. Estimating the difference between the exact and approximate determining functions, we can obtain from (18.28) and (19.1) that

A(yo,^)- A .(Yo,?)

j[f(t,x*(t,y0,?))-f(t,xm(t,Yo,??))}dt.

(19.6)

0

In view of the Lipschitz condition (2.4), relation (18.10), and the equality

f a(t,X)dt = 1 l- !)dt = 9 231t ( 0 obtained from (10 .6) for T = ?,, equality ( 19.6) yields a componentwise estimate of the form

IA(YN,^)- A m(YO,^ ))
Y4)j +

2 V- X" M

,

( 19.16)

Section 19

Solvability Theorems for Problems with Nonfixed Right Boundary

181

where

a3(?', A", A, Yo') =

_ ^"[(^,,- ?')C ld+(C-'A +E)(?'xo'-

X"x )]

,

and a,(t, ?.') is a function of the form (18.15). If ?. > X", then (18.6) with m = 1 yields xl(t, Yo, .')-xt(t, Yo 'r )

_ (xo-x4') = t [(At' xo)-f(t,xo^) - f (f(s,xo)-. f(s,x4'))ds dt (A.' - A.") '° X. + VV, $ f(s,xo)ds - , $ f(s,xo)ds 0 A" + 7^"[(%"-.')C-d+(C-'A+E)( ^,xg_ A."x )]. (19.17) By analogy with ( 19.16), equality (19.17) yields

I xI(t, Yo, A.')-x1(t, Yo , A.") I

X', X2(t, YO , )) -X2(t, Yo, ?")

_

(4 - xo)

+ J (f(t,X1(t, Yo, ?'))-f(t, X1(t, Yo,k"))) 0

I X, - f (f (s, x1( s, yo, ?')) - f (s, X1(s, yo, X "))) ds dt 0

+

) ,

J f(s, X1(s,

"

0

yo, )."))ds

+ f

X,

f(s, X1(s, yo, A"))ds

+ X„[(A."-X')C 1d+(C'A+E)(?'4-%"X;)]

X2(t,

A, ?') -X2(t, Yo , ?") I t I4-4I + K 1- )JIX \\ 0

1 (t, yo,))-Xl(t,yg X")Idt

,l'

+ -' f I X1(t, Yo, )) - X1(t Yoe 7") I dt

+ a3() , ?" , V., V.") +

W) - xm( t, Y011, X") t, ' ^')-x t 0+ y"")Idt J Ix Ixo-xo'I + K 11- -L ^,) m-1( Y0+ m-1(+ 0

184


Chapter 3

2.' + Xt .11 xm-1(t, y0, A.') - xm-1(t, Y0, a.") I d t t

+ ()L', V Yo, Yoh +

[

X', relation ( 18.28) immediately yields

0(Y0I X") - A(Yo, k")

J [C_id+(C-'A+E)xo] - ^, f f(s,x* (s,y6,?'))dt 0

- .""[C-'d-(C-'A+E)xo' ] + J", jf(s,x*(s,Yo,X""))dt

e

1 C'dtf." - V) + (CT]A + E){\%- \'x%) XT' -, j [f (t, x*(t, Yo, ?") - f (t, x*(t, yo X"))] dt 0 %, +

x°

r

f(t, x*(t,Yo,? "))dt + f f( t, x*(t, yo, ).")) dt. 0 x

(19.27)

By virtue of estimate (19.13), it follows from (19.27) that

IA(YoI V)-o(Yo,A."')I = aa( A,' ^", YO' , yo) + IX" K

(yo,? ,?2) E Gx [A',Ai]x [A' ,T],

the limit functions x*(t, yo, &') and x*(t, yo, ") of sequences xm(t, xm(t, yo, a.") of the form (20.6) satisfy the inequality

(20.11)

yo, ?')

and

x* (t, Yo, - x* (t, Yo, X")

< [ E + a1(t, 74)K(E xo 41 + a6(?", A. ", Y6, y6')] where y4 = max (?4,.2),

&1(t, Y4), is a function of the form ( 18.16), and

(20.12)

198


Chapter 3

a6(A',A.", Yo, Yo) = a7(A.',A.", A, YD + 21 X2-"zIM, (20.13) 73 a7( ,y ,yo)

2 -A2)C-d+C- A(A 1 ?62x0-A. A. x0 )+

i2, Xz

12x0- AZxO]

,

73 = max (?4, A2) •

Proof. To obtain inequality (20.12), we estimate the deviation of the functions xm(t, yo', a,') and xm(t, Y0, A.").

If we set m = 1 in ( 20.6) and assume that A2 > '2, then we get x1(t, Yo, A.') - x1(t, Yo, A.")

1 + f (f(t,x)_f(t,4)) f (f(s, 4 )- f(s,xo)) dsldt 2 r

_

(xo-xo')

1

X,

X.

(.f( s, 4)) ds +2 (f(s, 4)) ds ;Lj

+

11 11 1 t , [(A2 -A.2)C- d+C- A(^11 A62x0 -a,1 ,2x01+' )+ CA2x0 -A,2x0]. 2 X0

21 2

(20.14) By analogy with (19.16), relation (20.14) yields

I xt (r, yo A,') -x1(t Yo„, A.")I t

f

7l'

l

" " ) f I xo-x4 Idt +i I x0-xO dt

i

i -x0l +t K 1--

2

0

2

t

2 A2 - A.2

+ ^„ M+a7 (^ ,)L ", YO,Yo) X2

_ [E+a1(t,A.2)K] x0-xoI+ a7(A.,^,",y0,YO) + 2 X ^..^2IM. (20.15) 2

Section 20

The Case of Several Parameters in Boundary Conditions

199

If ?2 > AZ , then relation (20.6) with m = 1 yields x1(t,Yy,? )-x1(t, Yo,^ )I 21 X'2 - Xt2'1 A2 , relation (20.24) yields

(20.29)


204

Chapter 3

0(Yo> X) _'&( Yo, )

22 l(X2 - a,2) C- d + C- A (a1 ?2xo - 2t 1,2x0) + 1,2x0 - A 2x01

X2 + 2 f [f(t,x*(t,Yo 0

f(t,x*(t,Yo• "))^dr

„ a,2 X2

+ X2^- 7^2

r f(t, x*(t, Yo, &'))dt + 1 J f(t, x*(t, Yo, X'))dt. a 2a 2 0 a,2 ;L2

(20.30)

Hence,

(Yoe ) (Yo, ) a( i, , Yo^Yo) X, 2

)L;Z + .. K [E+al(t,74)K(E-Q)-1^ Îxo -xoI + a6 y0,y0dt f20

, < 11, 2" , „) + 2X2 -X2 M - as I, , Yo,yo 2.2

2

+ K E +9 44 K(E -Q)-' 2

„ „ „ [I x0- xoI+ab (X,2. ,Yo,Yo )J,

(20.31)

It is obvious that, for arbitrary pairs (20.11), independently of the relationship between A,2 and 22 , inequalities (20.29) and (20.31) indeed yield estimate (20.25), which completes the proof of Theorem 20.5. Theorem 20.6. Suppose that all conditions of Theorem 20.1 are satisfied for the boundary-value problem (20.1)-(20.3) with two parameters and a nonfixed right boundary point. Then, in order that a certain subdomain of the domain of definition of the given problem A" = G"x [Ai, Ai ] x [A2, A2 ] c G x [Ai, Ai ] x [A2, T^ (20.32)

The Case of Several Parameters in Boundary Conditions

Section 20

205

contain the pair (yo,X*) that determines a solution x = x*(f) of the boundary-value problem (20. l)-(20.3) with X=X* = (X\,X*2) that passes through the point x'(0) = (x0l,x02,yt),

X

= X* =

(20.43)

where E(x* (t, YO, A,), xm(t, yo, X)) is calculated according to (20. 8) and (20.5), a6(A,, yo, Y0) is calculated according to (20.13), y4 = max(,,, ^), Q = Y4 K, and x * (t, XE fi 7t

yo, A.) is the limit function of sequence (20.6). As in the case of Theorem 19.4, the proof of Theorem 20.7 is based on the use of inequality ( 17.28). Estimating the first term of it according to (20 .8) and the second term according to (20 .23) and taking (20.42) into account, we arrive at estimate (20.43). It is natural to choose the exact solution

(yo., A 'm) of the approximate determining

equation (20.10) as (yo , A,). In this case , we have x* (t, Y0, A,*) - Xm(t, YOm+ )1.)

E(x*(t, Yo, V), xm(t, Yo, X*)) _ 1

l

+ E+ ai(t, 7a)Kia i][Ixo XOm +a6(A,*,A,m,YoYom)] J i=0 sup Yo

E

G', ;L E P

I

E ( x *(t ,Yo, A.), xm(t,Yo,

A.))

1

+ [ E+&,Q, y4)K j Q' i =o

] [I xo - xo

m

l +a6 (A.+A'm, Yo, YOm)J

where 74 = max (A.*, A.m), T4 = max (A,, .m), 71.E P

Section 21

209

Linear Dependence of Boundary Conditions on Two Parameters

a1(t, y4) and &1(t, y4) are functions of the form (18.16), and

Q = 'Y4 K, l4K. it It

21. Linear Dependence of Boundary Conditions on Two Parameters Note that a specific form of the dependence of boundary conditions on controlling parameters for the numerical-analytic method under consideration essentially determines the form of successive approximations xm (t, yo, A.) on the basis of which we construct approximate solutions and investigate the solvability of boundary-value problems with parameters. The parameters may enter given boundary conditions linearly (as, e.g., in problem (16.1), (16.2)) and the successive approximations (16.8) nonlinearly. Consider the following two-point boundary-value problem with two parameters that enter the boundary conditions linearly: x = f(t,x),

(21.1)

x,fE R",

A.1Ax(0 )+7.2Cx( T) = d, detC# 0 ,

d#0, (21.2)

x1(0) = x01, x2(0) = x02•

(21.3)

We pose the problem of finding a solution x = x* (t) of equation (21.1) continuously differentiable on the segment [ 0, T] and the values of the parameters A.1 = Ai and (21.2) and (21.3). A.2 = A:2 for which the required solution satisfies conditions Under conditions typical of the numerical- analytic method under consideration, we assume that the function f (t, x) is continuous in domain (16.4) and satisfies conditions of the form (2.4). Let D0 denote the set of points x0 = (x01, x02, y0), y0 = ( x02, ••• x0r, ) contained in the domain D for any values of A. from a certain domain A, (.1,.2)E [A1', A"] X [A',A"] = A, A2' >0, (21.4) together with its

a-neighbourhood, where R = 2111 + P1(x0, X),

R t (x0, 7^ ) =

I-L[2 C-ld -

(21.5) (X,C-'A + '%2E) XOIII


210

Chapter 3

and let G be the set of ( n - 2)-dimensional vectors yo such that xo = (x01, x02, y0) e D0. We also assume that inequalities of the form (18.5) are true for the set D5 defined by vector (21.5) and the matrix K appearing in (2.4). We choose a sequence of functions x, (t, yo , X) satisfying the boundary conditions (21.2) for arbitrary values of X = (X 1, X 2) from domain (21.4). For this purpose, we consider the sequence of functions

xm

t (t, Yo, ^) = x 0 + f f(t, xm -1(t, yo, X)) dt + at,

0

(21.6) m = 1, 2, ... , xo (t, Y0, ?) = x0, where the vector parameter a = (a I,... , a„) is chosen so that all functions (21.6) satisfy the boundary conditions (21.2), (21.3) for arbitrary x0 and Xe A. Substituting (21.6) in (21.2), we obtain the following system of algebraic equations for the calculation of a: rr T ^.lAxo + ^,2C xo + f f(t, xm -1(t, yo, % )) dt + aT = d,

L

0

whence a = -ZT[C-'d -(O1C-'A+X 2E)xo] - T f At' xm-1(t, Yo, X)) dt.

(21.7)

Thus, relations (21.6) and (21.7) enable one to construct the sequence of functions

X.

(t, Y0, ^) = X() +

f [f(t,

X xm -1(t ,

f

Yo, ))dt - T f (s> xm-1(S, Yo, A.))ds1 dt

+ _,2T1C-ld-(XIC-lA+), 2E)xo1 ,

m = 1, 2, ... , (21.8)

x0(t, Y0, ?.) = xo,

which satisfy the given boundary conditions for arbitrary y0 and Xe A. For the boundary-value problem (21.2)-(21.3), we can also formulate analogs of all statements of the previous section concerning problem (20.1)-(20.3).

Section 21

211

Linear Dependence of Boundary Conditions on Two Parameters

Theorem 21.1. Suppose that, for the boundary -value problem (21.1)-(21.3) with two parameters entering linearly, the values of the vectors M, K, and d, the square matrices A and C, the scalar T, and the domain of definition (16.4), (21.4) are such that this problem satisfies conditions (2.4), (18.5), and (21.5). Then a sequence of functions xm (t, y0, A,) of the form (21.8) satisfies the boundary conditions (21.2) and (21.3) for all m = 0, 1, 2, ..., y0 a G, and A, C= A. As m 00, it converges uniformly with respect to the domain (t,yO,A.)E [0,T]xGxA (21.9) to the limit function x*(t, y0, A,). The function x* (t, y0, X) passes through the point x* (0, y0, %) = x0 = (x01, x02, y0) for t = 0, satisfies the boundary conditions (21.2) and (21.3), and is a solution of the integral equation

- 1-Jf(s, x( s)) ds x(t) = xo + J{f(tx(t)) To 0 + -ZT[C-'d-(A,IC-'A+A, 2E)xo]}dt,

i.e., x *(t, y0, A.) is a solution of a "perturbed" boundary- value problem of the form x = f(t,x)+ A(yO,A ), x1(0) = x01,

.1Ax(0 )+ A.2Cx(T) = d, x2(0) = x02,

where T

A(YO,X) =

--L[CX2T

1d-(.1C-1A+A. 2E)x0] - TJf(t ,x(t))dt.

Moreover, in domain (21.9), the error of the m th approximation satisfies the inequality

Ix*(t,yo,A,) - xm(t,Y0,A)I

- 1, the approximate determining equation T Am(Y0,X ) _ ^ZT[C-'d-(.1C-'A+. 2E)xo] - Tj.f(t,xm (t,yo, A.))dt

(21.14)

has a unique solution (yo, A') = (Yom, X. I A. m = ()Lm 1, A. m 2) in domain (21.13). Also assume that the index of this solution of equation (21.14) is not equal to zero and the following inequality holds on the boundary S 1 of the domain A': 2 inf IAm(yo,A.)I > (!) QW3(y0,A.),

(YO,2,)es' 3

where Am(yo, A,) is calculated according to (21.14), W3(yo, A,) is the vector appearing in inequality (21.10), and Q = T K. tc Then the boundary-value problem (21.1)-(21. 3) has a solution (x*(t), A.*) such that, for t = 0, the initial value of this solution x * (0) = (x01, x02, yo) is determined by an ( n - 2) -dimensional vector yo lying in the domain G' and the parameters Ai and A2 in (21.2) are such that A,; E and A! 2,E [ A2, AZ ^. Lemma 21 .1. If the conditions of Theorem 21.1 are satisfied, then, for arbitrary pairs (yo, A.'), (yo,A,")E Gx [A',A'1] x[A',A2(21.15) the following inequality holds for the difference between the limit functions x*(t, yp, A,') and x*(t, yo, A.") of sequences of the form (21.8):

I x*(t, yo, 7.') - x*(t, yo 7.") I ,

and b21) must be such that

256

Collocation Method for Boundary- Value Problems with Impulses aot) = a0(2) -0.1, bit) = bi2),

a^t^ = a^2^ +0.1,

Chapter 4

bet) = b22^

In this case, relation (26.17) takes the form x12(t) = a02> -0.1+(ai2) +0.1)cos4t+bi2isin4t+ b22)sin8t , tE 01 fl, (26.18) x22(t ) = ao2i+a^2icos4t + 42isin4t+b221sin8t , tE [42 As collocation nodes, we choose the following points: ti = 0.0625x, t2 = 0.1875x, t3 = 0.3125x, t4 = 0.43757n, (26.19) Substituting (26.18) in (26.14) at points (26.19), we obtain the following nonlinear determining system: 0.49307 [ao2) + 1 .4121a(2) + b22) + 0.04142

]2

+ 0.70711(a12) + b(2)) + 4b22) = 0.07318, 0.49307 [a?) + 1 .41421 bf 2) - 42) - 0.1] 2

+ 0.70711(bb2)-a(2))+4622) = 0 .07318,

2.02811 [ 2)-1.41421 ai2)+b22'l2

- 0.70711(a(, 2) +4 2)) +4 2.02811

[a(2) -

b? )

= -0.04515,

22,

= - 0.0591.

22

1.412142) - b )j 2 + 0.70711 lai2) -b12)) -46

Solving the last system by the Newton method and choosing ao2) = 0 . 1, a^2^ = 0, 02) = 0.1, and b22) = 0 as the zero approximation, on the fourth iteration we obtain the following approximate values of the unknown coefficients: apt) = 0. 10602, bt(2) = 0.08954 ,

a(2) = 0.00162, b22) = -0.00056.

Section 27

Green Function for a Three-Point Boundary-Value Problem

Substituting these values of aff\

257

a{2\ b{2\ and b^) in (26.18), we obtain the ap-

proximate — -periodic solution

jc12(t) = 0.00602 + 0.10162cos4r + 0.089541sin4r-0.00056sin8r,

re [ o , - l

= 0.106023 + 0.00162cos4r + 0.089541sin4r-0.00056sin8r,

re [ - , - ] •

JC 2 2 (0

The error of the obtained approximate solution as compared with the exact —periodic solution of problem (26.14), (26.15)

x°(0 =

x,°(r) = 0.1 (sin 4/ + cos4r),

t e [o, - 1

x°2(t) = 0.1e sin4 ',

' e [f'?}

does not exceed 0.03, which means that reasonable accuracy is attained even if we choose m = 2 in the approximate solution (26.16).

27.

Green Function for a Three-Point Boundary-Value Problem with Single Impulse Influence

Consider a three-point boundary-value problem for a linear inhomogeneous system of differential equations with impulse influence at the point t = x, of the form ^ = A(t)x + f{t), dt Cx(ix - 0 ) + Dx(x, +0) = 0 ,

r * x,,

te[a,b],

x, e (a, b),

3,x(a) + £,*(*,) + B^fc) = 0,

delD * 0,

t * x,,

(27.1) (27.2) (27.3)

where A(r) is an ( n x n ) matrix function continuous for r e [a, fc], C, D, fl0, Bx, and B 2 areconstant ( n x n ) matrices, x = (xx,x2 xn)e Rn, and f(t) is a piecewise-continuous vector function with discontinuity of the first kind at r = x,:

258 Collocation Method for Boundary-Value Problems with Impulses

f(t) _

f(t)=(f 1(t),..., fn(t)), t

1

f2(t) = (f21(t),..., f2n( t )),

Chapter 4

e [a, t1 ],

t E [ t1, b ]•

Let us find a solution of the semihomogeneous impulsive boundary-value problem (27.1)-(27.3). First, we construct the Green function for the corresponding homogeneous impulsive boundary-value problem (27.4), (27.2), (27.3): dx dt = A(t)x,

x c Rn,

t * TI,

t e [a , b]. (27.4)

The form of the Green function depends on the time of impulse influence t =,c 1 with respect to the point t = t 1 from the boundary conditions (27.3). There are two possible cases. Consider the first case where ti 1 E (t 1, b ). Denote by x1(t), t e [ a, 't1 ],

*(0 = x2 (t),

tE[t1,b],

the solution of the impulsive differential equation (27.4). We assume that x(t) belongs to the space C1 [ a, b ] of vector functions piecewise-continuous on [a, b ] with discontinuity of the first kind at t = ti 1, namely,

x1(t) = (x11(t),...,x1n(t)) E C[a,t1],

x2(t) = (x21(t),...,XN(t)) E C[til,b],

where C[ a , T1 ] and C[ i1, b ] are the spaces of vector functions continuous on the closed intervals [ a, i1 ] and [t1 , b ] , respectively. The following auxiliary statement is true: Lemma 27.1. If the homogeneous single-impulse three -point boundary-value problem (27.4), (27.2), (27.3) has only the trivial solution x (t)=_0 (x 1(t) _= 0, X2(t)=O) and ti 1 E (t 1, b), then the matrix L = Bo + B1cb(tl) - B2'D(b)L1,

I,1 = (D-1(il)D-1C(D(ri)

is nonsingular, i.e., det L * 0, (27.5)

Section 27

Green Function for a Three-Point Boundary- Value Problem

259

where fi(t) is the fundamental matrix of the homogeneous system (27.4) without impulse influence normalized at t = a ( i.e., O (a) = E). Proof. It is well known that, in terms of the fundamental matrix (D (t), a solution of the homogeneous system of differential equations (27.4) can be represented as follows:

x1(t) = (D(t)(D-'(a)x1(a) = cb(t)xl(a),

t E [a, tit ] , (27.6)

x2(t) = ci)( t)(D-'(ti1)x2 ( t1), t e [tit , b]. (27.7) By using relations (27.6) and (27.7), and the jump condition (27.2), we obtain x2(tit) = - D-' C4)(ti1)x1(a)• Substituting this relation into (27.7), we obviously get x2(t) = - 4D(t)L1x1(a ),

t e [ ti1, b ] . (27.8)

Since t 1 < ti 1 , we have

x(t1) = (D(tl )xt (a), x(b) = - cD(b)L1x1(a), and, therefore , taking the boundary conditions (27.3) into account, we get [ Bo + B14)(t1) - kb(b)L1 ]x1(a) = 0. By assumption, the homogeneous three -point boundary-value problem (27.4), (27.2), (27.3) possesses only the trivial solution . Hence, the last equation implies that condition (27.5) is satisfied. Lemma 27. 1 is proved. We introduce the matrix function of two variables G(t, s) (_ 1)'+i4(t)IL- 1[ B24(b)L1- j - B1cI'(t1)S(t1 - s )14)-,(S), t !5 s' (-1)t+jcb(t)[ L.L ' (B2c(b)L1-

(27.9) j - B1(D(tl)S(tl - s)) + L, - j ](D-' (s), t? s,

defined for all ii t 1, the domain of definition of the Green function a t 1 is displayed in Fig. 3. It is not difficult to verify that the Green function obtained can be written in the form (27.9) with the use of the 8-function. The theorem is proved. Now consider the second case where ti 1 e (a, t I ). The following statement is analogous to Lemma 27.1: Lemma 27.2. If the homogeneous single-impulse three-point boundary-value prob-

264 Collocation Method for Boundary- Value Problems with Impulses

Chapter 4

lem (27.4), (27.2), (27.3) has only the trivial solution x (t) = 0 (x 1(t) = 0, x2(t) == 0) and T 1 E ( a, t 1), then det M = det (Bo - B14)(t1)L1- B2,0(b)L1) * 0,

(27.18)

where L1 =-1(T1)D-1Ct(T1) Proof. As shown above, a solution of the homogeneous impulsive equation (27.4) on the intervals t E [a, T 1 ] and t E [ T 1, b ] is given by relations (27.6) and (27.8), respectively. Taking the inequality T 1 < t 1 into account, we get x(t1) = x2(tl) = -(D(t1)L1x1(a), x(b) = x2(b) = -(D(b)Ltx1(a)• Therefore, by virtue of the boundary conditions (27.3), we have (Bo - B1(b(t1)L1 - B2'D(b)L1)x1(a) = 0, which immediately implies condition (27.18). Lemma 27.2 is proved. Let us show that, in the case considered, the Green function G(t, s) has the following form for Ti:5 t:5 Ti+1 and Tj:s ti 1. The explicit form of this function in all intervals of variation of the variables t and s is presented in Fig. 5. One can readily verify that the Green function obtained can be written in the form (27.19). Theorem 27.2 is proved. By using Green functions of the form (27.9), (27. 19) constructed above , one can find a solution of the semihomogeneous boundary -value problem (27.1)-(27.3). Theorem 27.3. Suppose that the homogeneous single - impulse three-point boundary-value problem (27.4), (27.2), (27.3) has only the trivial solution. Then, for any piecewise-continuous function f (t) E CC1 [ a, b], there exists a piecewise -continuously differentiable solution of the semihomogeneous boundary -value problem (27.1)-(27.3) with discontinuity of the first kind at the point t = ti 1:

Green Function for a Three-Point Boundary-Value Problem 267

Section 27 s

b

(b ( t)L,Mt-'B2,b(b),b '(s)

-,b(t) M-'BO(b) 41-'(s)

^(t)^LiM1 -^ Bz^(b)i + E^^-^ (s) tt

^(t)IM '(B,Û1) + B O(b)) O-'(s) 0(t)[LMt-'(B1O(ti)+B2O(b))+Ellb-'(s)

-(b(t)M-'(B1(b (ti)+B2 (b )) O_'(s)

tit

0(t) Mf ' (Bi-D(t i )

+ Bz^(bU^^-^(s) -lb(t) Li [M-' (Bi-V(ty ) + B2-D(b ))+ E](b-, (s )

^(t)^M>-^ (Bi^(h ) + B20(b)) Li +E]0-'(s)

a a

tt

Tl

Fig. 5

x(t) =

xl(t),

t E [a, t1],

xi1(t),

tE[t1,T1], xil(t)=xi(tl),

x2(t),

t E [t1, b],

x1(t),

tE[a,T1],

xz(t),

t E [Ti , t1], x2(t) =x21(11),

x 1(t),

tE[t1,b],

for TIE( t 1, b) and

x(t) =

b t

268

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

for 't 1 E ( a, t I). This solution is given by the formula b

x(t) = JG(t,s)f(s)ds,

tE

[ a,b],

(27.23)

a

and admits a representation in the form r

t

t,

X11(t) = (1(t) f C2(s)fi(s)ds + JC^(s)fi(s)ds Ia

t

b + JC4(s)fl(s)ds + JC7(s)f2(s)ds , tE [ a , t1], r,

(27.24)

T,

*"(0 = fi(t)[ f C2(s)f1(s)ds + 5C5(s)fl(s)ds a

T,

t,

b

+ JC4(s)fl(s)ds + 5C7(s)f2(s)ds , t

tE [ t1,ti1],

(27.25)

tE [ T1, b] ,

(27.26)

T,

t,

T,

x2(t) = fi(t) JC3(s)fl(s)ds + JC6(s)fl(s)ds a t, t

b

+ JC9(s)f2(s)ds + JCs (s)f2(s) ds T, r 11

x1(t) = fi(t)[ JQ2(s)fi(s)ds + JQ1(s)fl(s)ds a

4

t

b

+ JQ4(s)f2(s)ds + JQ7(s)f2(s) ds , tE [ a , t1], T,

T,

xi(t)

= fi (t)I

(27.27)

t,

t

5Q 3(s)f1(s)ds + 5 Q6 (s)f2(s)ds a

t,

T,

b

+ JQ5(s).f2(s)ds + JQ8(s)f2(s)ds , t t, J

tE [ 'E1,

t1 ],

(27.28)

269

Green Function for a Three-Point Boundary- Value Problem

Section 27

xil(t) = t(t) 5Q3(s)fi(s)ds + JQ6 (s)f2(s)ds a

t,

(27.29)

+ f Q9(s)f2(s)ds + f Qs (s)f2(s) ds l , to [ t1, b]. t, t J

Proof. First, we consider the case where tit E (t 1, b ) and show that the functions 4(t), xl t(t), and x2(t) given by (27.24)-(27.26) satisfy the inhomogeneous differential equation (27.1) and the homogeneous conditions (27.2) and (27.3). In each of the intervals [ a,t], [t,t1] , [ t1,ti1], and [ti1,b] for xi (t), [a,t1], [t1,t], [t,til], and [,c1,b] for x111 (t), and [a,t1], [t1,-T 1], [til,t], and [t,b] for x2(t), the integrand in (27. 23) is continuously differentiable . Differentiating (27.23) with respect to t (regarded as a parameter), we get dxt(t) = f aG(t, s) fi(s)ds + G(t, s)Is-r_of (t) dt a at

+ f acct ' s) f (s)ds - G(t, s) I _ f (t) at

s- t+o

+ J'El aG(t, s) f (s ) ds + J aG (t, s)f2(s)ds t, at

at

b = f a^t's)f(s)ds + f (t), tE [ a,t1l, a at

f at f (s)ds

t a G(t, s) dxi t(t) = t' aG(t, s) fl (s)ds + dt a t,

f at

+ G(t, s)I _ _ f(t) + J aG(tat, s)ft(s)ds - G(t, s) I _ s- t o

+fa J,

G(t

r

, s) f2(s)ds

b

= f aG(t,s ) f(s)ds + f( t), at a

tE [ t1,T1],

s - t+0

270 Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

a G(t, s) T' a G(t, s) f,(s)ds + f (s) ds

dx2(t) dt

a

q

+ f aG(t's)f2(

t,

s)ds + G (t,s)l S_,_of2(t)

+ 1 aG(t,s)f2(s)ds - G(t ,s)I s_t_of2(t)

t at

b

_

aG(t,s ) f(s)ds + f2 (t), J a.

tE [ tit, b].

a

For the expression dz - A(t) x, taking (27.23) and the last relations into account, we get

j a t , s) f(s)ds + f(s) - A(t) j G(t, s)f(s)ds at a a _ ! I aG(t's) - A(t)G(t,s)]f(s)ds + fi(t), tE [ a,t1], at J a L

tE [ t1,t1], (27.30)

b ja G(t, s) f(s) ds + f2 (s) - A(t)! G(t, s) f (s) d s a at a f r aG( t, s ) - A(t)G(t, s)]f(s)ds + f2 (t), a t at

tE [ til , b]. (27.31)

In view of (27.10), relations (27.30) and (27.31) imply that, for t E [ a, t 1 ] and t E [ t 1, T 1 ] , the functions xi (t) and xl 1(t) satisfy the inhomogeneous differential equation dx = A(t)x + f(t), dt

tE [ a,'t1],

and, for tE [ti1, b ], the function x2(t) satisfies the equation dx = A(t)x + f2(t), dt

tE [ i1, b].

Section 28

Semihomogeneous Linear Two-Point Boundary-Value Problem 271

Let us show that functions (27.24)-(27.26) and (27.27)-(27.29) also satisfy the three-point boundary conditions (27.3) and the jump condition (27.2). Indeed , substituting (27.23) into (27.3) and (27.2) and taking conditions (27.11) and (27.13) into account, we get b b b Bp JG(a,s)f(s)ds + B1 JG(t1,s)f(s)ds + B2 f G(b, s)f(s)ds a

a

b

f [BOG(a, s) + B1G(t1 , s) + B2G(b, s)] f (s) ds = 0, a

b b Cf G(ti1-0, s)f(s)ds + Df G('c1+0,s)f(s)ds a

a

b = J[CG('tl-0, s)+DG('tl+0,s)]f(s)ds = 0.

Thus, we have shown that, for 't 1 E ( t 1, b), the solution of the semihomogeneous boundary-value problem (27.1)-(27. 3) is given by relations (27.24)-(27.26). Similarly, for 't 1 E (a, t I), the solution of this boundary -value problem is given by formulas (27.27)-(27.29). The theorem is proved. It is easy to see that, with the use of formulas (27.24)-(27.26) and (27 .27)-(27.29), the solution of the semihomogeneous impulsive boundary-value problem (27.1)-(27.3) can independently be found in explicit form on the groups of intervals [a , t 1 ] , [ t 1,'r 1 ] , ['tl,b] and [a ,ti1], [til,t1]. [ t1,b], respectively.

28. Semihomogeneous Linear Two-Point Boundary-Value Problem with m-Impulse Influence

Consider a linear system of differential equations with impulse influence at m fixed points T1,T2,... ,TmE (a,b)

dx = A(t)x + f(t), XE R", t * ri, dt Cix('ri - 0) + Dix('ci + 0) = 0,

tE [a,b] ,

i=1,2,...,m ,

t = ii, i = 1, 2, ... , m,

(28.1)

(28.2)

272

Collocation Method for Boundary-Value Problems with Impulses Chapter 4

with two-point boundary conditions B]X(a,b) + B2x{b) = 0,

(28.3)

where C,, £>,-, B ) t and B2 are ( n x n ) constant matrices such that detD, = 0, i* = 1, 2,...,m. We assume that A(r) is an ( « x n ) matrix continuous for fe [a,b] and / ( / ) is a piecewise-continuous vector function with discontinuity of the first kind at t = T |, x 2 > ... , xm of the form '/,(/), f2(t),

f(t) =

te[a,xxl re[T,,T 2 ],

(28.4)

./m+iC). *e[Tm,l»]. We consider the problem of finding a piecewise-continuously differentiable function

*
"1" 1.

_-2.5 10~12_

"1"

"1.0"

1.

.10.

"1"

-4.2-10" 13 "

1.

.-3.010"12.

.-1.5-10" .

«i2)

"1"

"-1.4-10" 13 "

1.

10

4 2)

"1"

"-2.110 - 1 3 "

1.

.-1.5-10 - 1 2 .

».

-1.0.

"1"

" -510-13 "

1.

.-7.910"14.

' 12

1.

"1"

1.

1.0

"1"

« ( o 2)

1.0 '

"1"

approximate value

(2)

b?

"

1.0

_

1.

-3 10 - 1 2 .

"1"

-2.4 10 -13 "

1.

. 1.4-10-12 .

['}

with the conditions x(0)

=

X(2K),

(31.69) X(0) = x(2n), and, for t = n, x(x + 0)

E

x(x-0)

2

JKT-0)

0

\-E\ i(T + 0)

Rewriting (31.68) as a system offirst-orderdifferential equations, we get *i,i(0 *i(0 =

/G[0,7t),

'1.2(0.

*(0 = '2.1(0

/e(jt,2jt],

x2(t) = ■X2.2W

(31.70)

Section 31

307

Equations with Piecewise -Continuous Right -Hand Side Table 3 m=3

aal) ail)

initial value

approximate value

0 0.011

[ -2 . 1 .10-12

0.01

]

a31)

bil) b21)

b31)

[ 0.98

r 3 . 7.10-12

]

IL-3.2.10-12

0 . 01 -0.01

- 9 . 3 . 10-13

[ -0.01 ,

[ -4 . 8.10-12

1 .01

1 .0

1- 1.01] 0.01

[ -1.0 ] 1 . 6.10-12

[ -0.01 ]

[ _1.1.10-11

0 . 01

[ -1 . 2 2.10-12

[-0.01 ]

a(22)

[ 1.0 1.6.10-12

1 : 0.01

[ -1.8.10-13

0.01 1 1 0.01 0.01

bit)

b22)

-1 . 5.10-13 [ -4.6.10-13 1.0

1 -0.01 0.0

1-2.8-10 -12

1 0.01

[ 2.0.10-12

1.6.10-12

0.01

1.0.10-13

1 : 0.01

[ -4.1.10-13

b32)

x1,2

t E [0, t),

xi1-x12-x1 , 1-2sin2t

m=

x2,2

z2(t)

3.0.10-12

0.01

a(32)

2.2.10-12

xl(t) _

[ -3.1.10-12

0.01

ail)

[1.0]

0.011 1 0. 01

1 0.01

1.0

0.99

1.0

0.991

ao)

4 .2- lo-' 3

1 1.01 a21)

initial approximate value value

t E (7n , 27c].

-x2, 1 sin t - x2 1 -1- 2 sin 2t

Conditions (31.69) and (31.70) take the form

(0)]

x1,1(0) x2,1(20 [x1,2

- [x22 (271 )

x2 2 (7G + 0) - xl, 1(7E - 0)

[ x2,20 +0)- x1,2(71 -0)

2

- [OJ.

308

Collocation Method for Boundary-Value Problems with Impulses

Chapter 4

The approximating trigonometric polynomials are as follows: M

x lm (t) = am +

xyn(t) =

(ak) cos kt+bk') sin kt), tE [ 0, 76),

ap2) + 7. (ak2)cOskt+bk2) bksin kt

tE (76, 276).

The unknown coefficients calculated for the cases m = 2 and m = 3 are presented in Tables 2 and 3. For comparison, the exact solution of the problem is as follows: 1(t) _ scot+cost cos t - sin t]'

t e [0, 7t),

x(0 = X2 (t) = rl+sint]

L

t E [76, 276).

Cost

32. Construction of Solutions of Two- Impulse Systems

Example 32.1. On the interval t E [ 0, 1 . 5 ], we consider the elementary semihomogeneous two-point two-impulse boundary - value problem 1 t 1 t dx e dt 1+t2 -1 + [1]'

t # TI = 0.5,

t]x

1

1 1

t * '62 = 1, (32.1)

l x (__o)+ I 1 ' ] X(1+0) = 0, 2

2

(32.2) 1 1 1 1 1 x(1-0) + 10 2 2 0

L

0 1 0 2 1x(0) + 20 81

1 x(1+0) = 0, 10

6 1 x (2 )

0.

(32.3)

Section 32

Construction of Solutions of Two-Pulse Systems

309

Let us construct the Green function for the corresponding homogeneous boundaryvalue problem (32.1)-(32.3). The normalized fundamental matrix of the homogeneous system of differential equations (32.1) has the form 1

*(0 =

f

.-'

1.

(32.4)

»

It follows from (30.11 ) that, in the case of the example considered , we have p = 1 and m = 2. Therefore, according to (30. 12), the domain of definition of the required Green function is decomposed into N1 = 12 subdomains. By using formula (28.9), we obtain

fi(t)( E- K-1 B2fi(2)L2L1 )(D-1(s),

-fi(t) K-1B2fi(2) L2L1

b(t) K

05 t5 $S.1;

(D-1(S),

oStSI5s51;

-1 B2fi(2)L2fi-1(s),

-fi(t)K-1 B2fi

(

2) fi

05t
-1(S),

OWL, (K-1B2fi(2)L2L1-E) C[0,T], fx(t) := f( t,x(t)), tE [ 0, T]. We introduce the mappings t J : C[0,TI ---> C[0,T], J x(t) = jx(s)ds, tE [ 0, TI,

0 P: C[O,T]-3 CPT[ R"], Px(t ) = x(t)- T[x (T)-x(0)], tE [ O,T], Q: C[O,TI - C[ O,T], Q := I-P. It is easy to verify that P is a projector that maps the space C[ 0, T] = C(C[O, T], R" ) of all continuous vector functions into the subspace

CPT[R" ] of T-periodic functions. Then, according to the basic idea of the method, the boundary -value problem (1) is equivalent to the system

x = z+P.ifx, Q.Ofx=0. In the last system (the first equation of which is integral, and the second of which is determining), the unknown is the pair (x(.), z), and x(t), x(0) = z, is the required T-periodic solution. ALL Basic Assumptions. Let us formulate the key ideas of the method. It was established in [Sam!], [Sam2], [SaRI], [SaR4], and [SaR6] that the method "works" under the following assumptions: (H1.1) The function f (t, x) = (f1(t, x), ... , f"(t, x)) is defined, continuous, and Tperiodic in t in the domain ( t, x) E R X D, where D is a domain in R.

Appendix

319

(H 1.2) The function I f (t, x) I = (If, (t, x) 1, ... , I f"(t, x ) 1) is bounded in its domain of definition by the vector M E R+ and satisfies the Lipschitz condition in x nxn: with matrix KE R+

If(t,x)I " replaced by "< is a necessary condition for the subdomain D2 to contain a point that may serve as the initial value for a periodic solution of problem (1). A1.2. Improvement of Applicability Conditions. Later, the numerical-analytic method of periodic successive approximations (for brevity, called below the "method") was developed in various directions. Part of the work was devoted the extension of its original scheme to more general classes of boundary-value problems; a survey of these investigations is given in the forthcoming parts of the present paper. However, it should be noted that some new generalizations of the method substantially differ from its original version and can be regarded as independent techniques inspired by [SaR1], [SaR4], and [SaR6]. We discuss this question in more detail in the final part of this survey. A considerably lesser part of the work was devoted to the extension of the field of applicability of the classical version of the method, i.e., to finding more exact convergence estimates, weakening conditions (H1.1)-(H1.4), etc. The importance of such investigations is also explained by the fact that the corresponding results can also be applied to other types of boundary-value problems. Below, we consider some of these results in more detail. A1.2.1. A Priori Estimates. The improvement of the a priori estimates appearing in the theorems of Sec A1.1 is, first of all, based on the more precise determination of the constant q in inequality (6). (It is clear that the greater this constant, the wider the field of applicability of the method.) This problem was studied in numerous works. In the list below, we outline the evolution of the problem with necessary comments. (1). 1965: Samoilenko ([Sam!], [Sam2]), q = 3.1.

Appendix

323

(2). 1976: Samoilenko and M. Ronto ([SaR4], [SaR6]), q = tt. (3). 1982: Samoilenko and Laptinsky [SaLa], q = 3.416... . (The proof has a gap, which can be eliminated with the use of the Krein-Rutman theorem [KrR].) (4). 1985: Zabreiko and Evkhuta ([EvZl], [EvZ2]); E. Trofimchuk [Tro2], q = 3.416... . (The a priori estimates obtained are rather complicated because they require the computation of A m (I - A)-1, where A is a certain integral operator.)

(5). 1992: Kwapisz [Kwa3], q = 10 = 3.1622. (6). 1996: M. Ronto and Meszi ros [RoM], q = 10• 3 (7). 1996: M. Ronto, A. Ronto , and S. Trofimchuk [RRT], q = 3.416... (A method is suggested that allows one to obtain simple a priori estimates.) (8). 1996: M. Ronto, A. Ronto, and S. Trofimchuk [RRT], q = tin. (This is a hypothesis; see also E. Trofimchuk [Tro2], where, in particular, it is proved that q < 2n.) Note that the results of [Saml], [Sam2], [SaR4], [SaR6], [Kwa3], and [RoM] also contain simple a priori estimates for the method. Problem 1. Find an analog of estimate (6) under the condition

If(t,x)- f(t,x")I s K( t)Ix -x"I,

(15)

where the components K. (t), i,j = 1, 2,... , n, of the matrix K(t) are nonnegative summable functions. The second condition important for us is assumption (H1.3). Clearly, it would be the optimal solution if one succeeds in discarding this condition completely. One can also try to diminish the quantity

M

T 2

M.

An obvious improvement is attained if we replace the first inequality in (4) by

If(t,x)I 5 M(t).

(16)

324

Appendix

Then we can choose 0 as follows:

p =

max 11- t I f M(s) ds + t J M(s) ds . Tt :E[0,T] \ T 0

The estimate from Lemma 3.1 in [SaRI, p. 13], which has the form t

f

f(t)

I +T

T f f(s)ds]dt

2(t- r)(1 - t-ti) max If(t)I V to[ ti,ti+T], T

tc[T,T +T]

(17)

was improved by M. Ronto and Meszaros [RoM] for ti = 0 as follows:

f [f(t) -Tf f(s) ds dt

]

2a1(t) I mo ]f(t)-t mm f(t)I

Here, a 1 ( t) = 2t (1 - t), 1 a 1(t) " Y " a 2 - - . ~ y - an\ =

z = [ ~Y~

z{ah,...,ait,),

which depends on k* numbers a (| , a, 2 ,..., a ^ e R [the indices 'i.t2.••-.'*+ specify the position of a "+1" enrry « ^« s«n'e5 o,, a 2 , . . . , on] and satisfies the system of k~ equations in k* variables

'-^Û J y;(Jf ^

z(a .

, a . 2 ,..., s + )))rf, = 0,

i = 1, 2 , . . . , n,

o determines the initial value for a T-periodic solution of system (I). Thus, the symmetry condition (19) allows one to reduce the dimension of the determining system (3) from n to k~, which obviously simplifies the scheme of the method. Let us consider the following example: Example 1. Let us investigate a r-periodic solution of the second-order equation

* " = / ( ' . x, A where f{-t, -x, y)= -f(t, x, xf). We rewrite this equation as the system dx d-t=y< (20) ^=fU,x,y). at It is easy to see that the right-hand side of (20) satisfies condition (19) with the matrix Q = diag (-1,1). Therefore, any vector of the form z = (0,a) is the initial value of a T-periodic solution of system (20) whenever hypotheses of the type (H1.1MH1.4) hold and the number a satisfies the scalar determining equation T

±]y(s,(0,a))ds 0

T

= j.]y(s,z(a))ds 0

= 0.

(21)

329

Appendix

Other interesting examples that demonstrate how property (19) can be used to substantially simplify the investigation of determining equations can be found in [Hall, Section 7]. It is worth noting that these arguments can also be used for the investigation of the characteristic equation of the averaging method (see [Hall, Section 7]).

A2. Relation to Other Investigations It is natural to guess that the numerical-analytic method should have certain common features with some of the other numerous approaches available for the investigation of boundary-value problems for ordinary differential equations (see, e.g., Conti [Con], Mawhin [Maw3], [Maw4], Capietto, Mawhin, and F. Zanolin [CMZ1], Boichuk, Zhuravlev, and Samoilenko [BZS], Kiguradze [Kig3], Farkas [Far2], and Rouche and Mawhin [RouM]). Indeed, the equations of the method and even its basic techniques are often closely related to equations and techniques of other methods proposed long before it (the Lyapunov-Schmidt method [Maw4, p. 27] or the Cesari-Hale method [Hall], [CeH], [Cesl]-[Ces3], which appeared almost simultaneously). However, the different objectives pursued by the authors resulted in essential differences in the results obtained. A2.1. Lyapunov-Schmidt Equation. Let us show how the main equations (2), (3) can be obtained by the direct application of the Lyapunov-Schmidt scheme to the periodic boundary-value problem (1). Following Mawhin [Maw4], we first recall how the Lyapunov-Schmidt equations are derived. It is reasonable to use the Lyapunov-Schmidt method for the investigation of the equation Ax

=

Nx,

(22)

where A , N : X - Z are continuous operators in Banach spaces X and Z, the operator A is linear and irreversible , and N is a nonlinear operator satisfying the Lipschitz condition with sufficiently small Lipschitz constant . ( In the case where A is invertible , by virtue of the Banach fixed -point theorem , equation ( 22) has a unique solution and can be studied by the method of successive approximations.) For the operator A, we require that there exist projectors P : X -*X, P (X) = ker A, and Q : Z -* Z, Q (Z) = Im A. We set X = kerA®X,, Z = ImA®Zt,

(23)

and denote the restriction A : Xt - Im A by A Then, according to the Banach theorem on the inverse mapping, the inverse operator A i t : Im A -* X t exists. Let x = k + v be the decomposition of an XE X corresponding to (23). Then

330

Appendix (22) a Av = N(k+v) p [Av=QN(k+v), (I-Q)N(k+v)=0] [v = Ai1QN(k+v), (I-Q)N(k+v)=0]. (24)

We fix an arbitrary k E ker A. By the assumption of the smallness of the Lipschitz constant for N, the first equation in this system has a solution v = v (k) (this solution can be found by the method of successive approximations). It only remains to find solutions k* of the bifurcation equation (I-Q)N(k+v(k)) = 0 to obtain all solutions k* + v (k") of the equation under consideration . Note that the reasoning presented above can obviously be given the local character by restricting the operators A and N to a certain neighborhood in X. In the special case of problem ( 1), it is natural to set X = {x(t)E C1 [0, T] : x(0) = x(T)j, Z = C[0, T],

Ax = t, (Nx)(t) = f (t,x(t)). d Then, obviously, T kerA = {x0E R"}, ImA = {v(t)eC1OTI: T f v(t)dt = 0

0 and, therefore, we can choose (Px)(t) = x(0),

(Qx)(t) = x(t)_

T

((I-P)x)( t) = x(t)-x(0),

T

T

T

f x(t)dt, ((I-Q)x)(t) = f x(t)dt. 0 0

Rewriting the Lyapunov-Schmidt equations (24) in this notation , we arrive exactly at equations (2), (3). Furthermore, according to the Lyapunov-Schmidt scheme, we should first solve equation (2) with respect to x (t, z) by iterations , fixing z e R"= kerA and assuming that the nonlinear Nemytsky operator f (t, x) satisfies the Lipschitz condition with sufficiently small constant . The next step of the method is to study the bifurcation equation. This completely coincides with the procedure used by the numerical -analytic method.

331

Appendix

A22. Cesari-Hale Method for Weakly Nonlinear Differential Equations. Let us return to problem (1). Assume that conditions (H1.1MH1.3) are satisfied. Instead of inequality (HI.4), we use the following more restrictive condition: (H2.1):

TXmax(K)0;

(H3.2) If(t,xi,Y1)- f(t,x 2,Y2) I < K 1Ix1 -x21 - K2IY1-Y2I; K1,K2>0; 5 5 T2M (H3.3) b-a > 2 , c < -6 TM < 6TM - 0, the function Am(z) defined by (41) satisfies the inequalities

min Am(Z) dm.

ZE/,

Then equation (37) has a T -periodic solution x = x(t) such that

I x(t), dx(t) e D dt )

for t r= R t and its initial data satisfy the conditions Idx(0)I 5TM X(O)C= 11, 1 dt 6 Consider some special cases of equation (37). Assume that the right -hand side of equation (37) is a polynomial in x and y, i.e., (42)

f(t'x,y) = Y,py(t)x'y', 1, i

where the functions p, (t) are summable and bounded on [0, T]. For such f(t, x, y), the following statement is true: Corollary S. Let f (t, x, y) be given by a finite sum of the form ( 42) and assume that conditions (H3.1)- (H3.4) are satisfied. Then the initial data

+T4

2

X(O) = zE I1 =

a

2

, b-T4 ,

J

r 56M 56M1 dd0) = Sf t f(s,x*(s,z),dxdt'z)1 dsE c+-,d-- = IZ 0 of the T -periodic solutions of equation (37) are either all isolated or completely fill the interval Ii I. It is clear that, in the first case, the initial data of T-periodic solutions are isolated points of the rectangle D 1 = II x IZ, whereas, in the second case, they fill a certain curve lying in D 1. The last statement is in good agreement with the known property of periodic solutions established for analytical autonomous systems on a plane (see [BaL]).

Appendix

345

Corollary 6. Suppose that conditions (H3.1)-(H3.4) are satisfied for equation (37) and the function f (t, x, y) is odd in t and x for all (t, x, y) E R t x D, i.e.,

f(-t, -x, y) = -f(t, x, y). (43) Then the differential equation (37) has a unique T -periodic odd solution x = x(t). This solution has the form x(t) = x *(t, z) =

lim Xm(t, Z),

where Xm(t, z) is defined by (38) and its initial data are the following: X(O) = 0,

ddto)

_ -SJ fl s, x*(s,0),dxds' 0) Ids.

Note that, generally speaking, the periodic solution whose existence is guaranteed by Corollary 6 is nontrivial. Corollary 6 complements some results of Pliss [Plil] and Heinbockel and Struble [HeS] concerning periodic solutions of systems with certain symmetry. Problem 3. Weaken conditions (H3.1)-(H3.4) so that the oddness condition (43) still guarantees the existence of a T-periodic solution of equation (37). It is easy to verify that the operator (L2 f)(t) can be rewritten in the form

(L2f)(t) = J f(i)(t - ti)dti - T J f ( ti)(t 2T - ti dti. (44) 0 0 / By using representation (44), Chornyi [Choi] , [Cho2] weakened conditions (H3.3) and (H3.4) as follows: 2 (H3.3*) b-a ? TTg , c 0 such that

T2 Max I f(x)I a >_ IxISa + max I(Ag)(t)j. 32 te[0,T]

(60)

Example 4. Let us apply inequality (60) to the investigation of the existence of a periodic solution of equation (45). We establish that a T-periodic solution exists if one can find a such that 2

w

2

2

2

2

Twa+ T 0 and the function (p(s, x) is continuous and T-periodic in s. [Here and above, we have assumed that the continuous functions g(-), h (), and f () are T-periodic in the first variable.] Remark 6. In [Zav], an equation of the following form was investigated: x '(t) = f (t, x, Bx1 (86) where the operator B possesses the same properties as the operator A in (84 ), and the continuous function f is assumed to be T-periodic in the first variable . It is obvious that, by setting (Ax)(t) = f ( t, x(t), (Bx )( t)), we can make the considerations more transparent. When studying equation (84), we assume that jAx-Ay6 5 Lix-ylo, where L = { L ;j >- 0 : i, j = 1 , 2, ... , n }. Then the T-periodic boundary-value problem for equation (84) is equivalent to the integral equation x(t, z) = z + (Nx)(t, z). (87) Here, the operator x H z + Nx defined by the formula

(Nx)(t) = J (Ax)(s)ds--t f (Ax)(s)ds 0 0 maps the space CPT(R, D) into itself whenever z belongs to the set Dp, where

380

Appendix (3 = T M, M =

sup

2 x E CPT(1R, D)

(88)

I Ax 10.

Moreover, since

INx-NyIO 0. Then the matrices Li (e) = Li + A (e) and numbers ki (e) = ki + e satisfy the conditions of the first part of the proof. Therefore,

m

r(L(e)) = r Yki( e )Li( e) . i=t As is known, the spectral radius of a finite-dimensional matrix continuously depends on the parameter. Therefore, passing to the limit as e - 0 in the last equality, we complete the proof of the lemma. The last statement can be generalized as follows:

Lemma 3. Let P = (KiLL)m. and Q = (LLKi)m. , where Li and K; are m x m matrices with nonnegative components, be m2 x m2 matrices. Then the spectral radii of the matrices P and Q are determined by the relations

m

r(P) = r

E

m

E

LJKi . KJLJ r(Q) = r i=t i=t

Proof. It suffices to repeat the reasoning used in the proof of Lemma 2 with the vectors a and b defined as follows:

Appendix

384 (i) For the matrix P, m

b > K,L, = µb,

a = (bL 1, bL2,... , bLm).

i=1

Then aP=µa. (ii) For the matrix Q,

KJI b = vb,

a = col (K1b, K2b,... ,Kmb).

t=t Then Qa = va. The following statement is very helpful in the proof of the convergence of various sequences of functions: Lemma 4. Suppose that there exist sequences of vectors {am}m 2:1 C 1[8+ and {bm }m,1 C 1R

and nonnegative continuous functions (p 1 and (p 2 such that, for a

certain sequence of functions {xm: M E NJ C C([0, T], Rn), the following inequality holds for all m, beginning with a certain number mo >- 1: Ixm+l(t)-xm ( t)I < am(p1 ( t)+bmcp2(t), 0 Om(z),

z€

as21 ,

(116)

where T Om(z)

=

T

J L(t) (1-S)-1Sml f(t, z)I dt. 0

398

Appendix

If, moreover, the Leray-Schauder degree deg(Am, a521, 0) * 0, then there exists at least one solution x = x *(t) of the periodic boundary-value problem (112) such that its initial value x *(0) E 521. Proof. As shown in Lemma 17 and Corollary 16 in [RoT2], it follows from the conditions of the theorem that the parametrized integral equation (107) corresponding to the boundary-value problem (112) (see (2)) has a unique solution for every z E 0 p, and a single-valued continuous determining function A(z) of the form (108) is given on Up. In view of our assumptions, the linear deformation of a finite-dimensional vector field Am(Z) of the form (115) into A(z), A(k,Z) = Am(Z)+ ?[A(Z)- Am(Z)], a,E[0,1], is nondegenerate on a 01. Indeed, otherwise, there exist zo E a S21 and ?O E [0,1] such that Am(ZO) = - A,0 [A(ZO)-A.(ZO)]Since , according to Lemma 17 in [RoT2],

IA(Z)-Dm(Z)I < 0m(Z),

we have IAm(z0)I a f Klelds+ lelo0.2a2 (,-n 0

(123)

c> f ale(s 0

)Ids+Ieio6.6a'

404

Appendix

Moreover, deg (D0 (x,y),

an, 0) = ind (A0(x,y), (x1,0)) 0 1 = sign det

-acosx1 0

)

# 0.

Consequently, all conditions of Theorem 21 are satisfied. Therefore, by virtue of this theorem, equation (121) has at least one periodic solution with the initial value (t = 0) lying in the indicated rectangle. Let us consider a particular case of equation (121). We set e(t)= n cos nt. Then e(t) = sin nt , Ii 1o = 1, and e = 0. One can verify that inequality (123) holds whenever

a < 4.81269..., c = 0.6a + 2 6-a it Therefore, with the periodic perturbation e (t) = it cos nt, there exists at least one periodic solution x(t, a) of equation (121) for all a < 4.81269.... This solution is such that

a Iat

< 6.6a + 2, Ix(0,(x)I < 2•

Remark 8. Let us compare the estimates obtained above with those obtained by Granas, Guenther, and Lee [GGL, Theorem 10.3]. In [GGL], an equation of the form

z

d2

+ g (x) d + a sinx = e(t)

was considered. Condition (B) in [GGL, Theorem 10.3] has the form 1 a+ J Ie(t)Idt

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