Generalized solutions of functional differential equations

Generalized Solutions of Functional Differential Equations

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GENERALIZED SOLUTIONS OF FUNCTIONAL DIFFERENTIAL

EQUATIONS JOSEPH WIENER The University of Texas-Pan American USA

World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

GENERALIZED SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orbyanymeans, electronic or mechanical, including photocopying, recording orany information storage and retrieval system now known or to be invented, without written permission from the Publisher.

ISBN 981-02-1207-0

For copying of material in this volume, please pay a copying fee through the Copyright Clearance Centre, Inc., 27 Congress Street, Salem, MA 01970.

Printed in Singapore.

Contents Preface

j

x

Chapter "I Differential Equations with Piecewise JL Continuous Arguments 1.1 1.2 1.3 1.4 1.5 1.6

Linear Retarded EPCA with Constant Coefficients Some Generalizations EPCA of Advanced, Mixed, and Neutral Types Asymptotic Behavior of Linear EPCA with Variable Coefficients Stability as a Function of Delay EPCA and Impulsive Equations

1 4 16 28 51 68 72

Chapter *\ Oscillatory and Periodic Solutions of *— Differential Equations with Piecewise Continuous Arguments 2.1 2.2

Differential Inequalities with Piecewise Continuous Arguments Oscillatory Properties of First-Order Linear Functional Differential Equations

V

81 82 91

CONTENTS

vi

2.3 2.4

Oscillatory and Periodic Solutions of Delay EPCA Differential Equations Alternately of Retarded and Advanced Type

114

2.5

Oscillations in Systems of DifFerential Equations with Piecewise Continuous Arguments

137

A Piecewise Constant Analogue of a Famous FDE

157

2.6

Chapter O *J

Partial DifFerential Equations with Piecewise Continuous Delay

107

163

3.1

Boundary-Value Problems for Partial Differential Equations with Piecewise Constant Delay

164

3.2

Initial-Value Problems for Partial Differential Equations with Piecewise Constant Delay

178

3.3 3.4

A Wave Equation with Discontinuous Time Delay Bounded Solutions of Retarded Nonlinear Hyper­ bolic Equations

Chapter A Reducible Functional Differential » Equations

190 201

213

4.1 4.2 4.3

Differential Equations with Involutions Linear Equations Bounded Solutions for Differential Equations with Reflection of the Argument

213 222

4.4 4.5

Equations with Rotation of the Argument Boundary-Value Problems for Differential Equations with Reflection of the Argument

241

4.6

Partial Differential Equations with Involutions

265

235

249

CONTENTS

vn

Chapter J " Analytic and Distributional Solutions \J of Functional Differential Equations 5.1

Holomorphic Equations

Solutions

of

Nonlinear

Neutral

5.2

Holomorphic Solutions of Nonlinear Advanced Equations

5.3 5.4 5.5 5.6

Analytic and Entire Solutions of Linear Systems Finite-Order Distributional Solutions Infinite-Order Distributional Solutions An Integral Equation in the Space of Tempered Distributions

Chapter #2 Coexistence of Analytic and \3 Distributional Solutions for Linear Differential Equations 6.1 6.2 6.3 6.4 6.5 6.6

Distributional, Rational, and Polynomial Solutions of Linear ODE Application to Orthogonal Polynomials Interesting Properties of Laguerre's Equation The Hypergeometric and Other Equations The Confluent Hypergeometric Equation Infinite-Order Distributional Solutions Revisited

Open Problems Bibliography Author Index Subject Index

271 272 275 279 292 309 321

325 326 330 340 347 359 364

379 381 399 403

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Preface

Functional differential equations (FDE) with delay provide a mathe­ matical model for a physical or biological system in which the rate of change of the system depends upon its past history. Although the gen­ eral theory and basic results for FDE have by now been thoroughly investigated, the literature devoted to this area of research continues to grow very rapidly. The number of interesting works is very large, so that our knowledge of FDE has been substantially enlarged in re­ cent years. Naturally, new important problems and directions arise continually in this intensively developing field. This book addresses the need for the study of generalized solutions to broad classes of FDE. In the first three chapters we concentrate on differential equations with piecewise continuous arguments (EPCA), the exploration of which has been initiated in our papers a few years ago. These equations arise in an attempt to extend the theory of FDE with continuous arguments to differential equations with discontinu­ ous arguments. This task is also of considerable applied interest since EPCA include, as particular cases, impulsive and loaded equations of control theory and are similar to those found in some biomedical mod­ els. A typical EPCA contains arguments that are constant on certain intervals. A solution is defined as a continuous, sectionally smooth function that satisfies the equation within these intervals. Continuity of a solution at a point joining any two consecutive intervals leads to recursion relations for the solution at such points. Hence, the solutions are determined by a finite set of initial data, rather than by an ini-

IX

X

PREFACE

tial function as in the case of general FDE. Therefore, underlying each EPCA is a dynamical system governed by a difference equation of a dis­ crete argument which describes its stability, oscillation, and periodic properties. It is not surprising then that recent work on EPCA has caused a new surge in the study of difference equations. Of significant interest is the exploration of partial differential equations (PDE) with piecewise continuous delays. Boundary and intial-value problems for some EPCA with partial derivatives are considered and the behavior of their solutions investigated. The results are also extended to equations with positive definite operators in Hilbert spaces. This topic is of great theoretical, computational, and applied value since it opens the possi­ bility of approximating complicated problems of mathematical physics by simpler EPCA. It is well known that profound and close links exist between func­ tional and functional differential equations. Thus the study of the first often enables one to predict properties of differential equations of neu­ tral type. On the other hand, some methods for the latter in the special case when the argument deviation vanishes at individual points have been used to investigate functional equations. Functional equations are directly related to difference equations of a discrete argument, and bordering on difference equations are impulsive FDE with impacts and switching and loaded equations (that is, those including values of the unknown solution for given constant values of the argument). The argument deviations of the EPCA considered in the book vanish at countable sets of points, and it would be interesting to investigate the relationship between EPCA and functional equations. Another deserv­ ing direction of future research is the exploration of hybrid systems consisting of EPCA and functional equations. Futhermore, EPCA are intrinsically closer to difference rather than to differential equations. Equations with piecewise constant delay can be used to approximate differential equations that contain discrete delays. It would be useful to draw a detailed comparison of the qualitative and asymptotic properties of differential equations with continuous arguments and their EPCA approximations, which has been widely used for ordinary differential equations and their difference approximations. Since the arguments of

PREFACE

XI

an EPCA have intervals of constancy we must relinquish smoothness of the solutions, but we still retain their continuity. This enables us to derive a homogeneous difference equation for the values of a solu­ tion at the endpoints of the intervals of constancy and to employ it in the study of the original EPCA, thus revealing remarkable asymptotic, oscillatory, and periodic properties of this type of FDE. Of course, it is possible to further generalize the definition of a solution for an EPCA, by abandoning its continuity, and to include in the framework of EPCA the impulsive functional differential equations. However, we do not pursue this goal in the field with abundant literature. In the last two chapters we turn from mildly weakened solutions of EPCA to generalized-function solutions of ordinary differential and functional differential equations. The unifying theme of the book is the development of theoretically meaningful and potentially applicable gen­ eralized concepts of solutions for important classes of FDE. A common feature of these equations is that their arguments have a fixed point. Thus, the argument of a typical EPCA is the greatest-integer function, and in the second part of the book the focus is on FDE with linearly transformed arguments. Hence, it is natural to pose the initial-value problem for such equations not on an interval but at a number of in­ dividual points. Contrary to general functional differential equations, EPCA of all types (retarded, advanced, mixed, neutral) have two-sided solutions, and FDE with linearly transformed arguments possess, under certain conditions, analytic or entire solutions. Some methods in the theory of entire solutions are applied to prove stability theorems for lin­ ear EPCA with variable coefficients. Integral transformations establish close connections between entire and generalized functions (distribu­ tions). Therefore, a unified approach may be used in the study of both distributional and entire solutions to some classes of linear ordinary and functional differential equations. Recently there has been considerable interest in problems concern­ ing the existence of solutions to differential and functional differential equations in various spaces of generalized functions. Many important areas in mathematics and theoretical physics employ the methods of distribution theory. Generalized functions are continuous linear func-

Xli

PREFACE

tionals on spaces of infinitely smooth functions with certain conditions of decay at infinity. They provide a suitable framework where major analytical operations such as differentiation can be performed. Fur­ thermore, the importance of the class of generalized functions stems from the fact that it includes the set of regular distributions repre­ sented by locally integrable functions. There is an abundance of sin­ gular distributions, and the Dirac delta function is one of them. It is well known that normal linear homogeneous systems of ordinary dif­ ferential equations (ODE) with infinitely smooth coefficients have no singular distributional solutions. However, these solutions may appear in the case of equations whose coefficients have singularities. We de­ velop the methods of study and establish some major results for linear ODE in the space of finite-order distributions (finite linear combina­ tions of the delta function and its derivatives). An existence criterion of such solutions for any linear ODE is found. Necessary and sufficient conditions are discovered for the simultaneous existence of solutions to linear ODE in the form of rational functions and finite-order distribu­ tions. The results are also used in the study of polynomial solutions to some important classical equations. Then distributional solutions of certain classes of ODE and FDE are presented as infinite series of the delta function and its derivatives. Existence and nonexistence the­ orems in spaces of infinite-order distributions are obtained for linear equations with polynomial coefficients and used to explore their entire solutions. We emphasize and investigate the conditions when linear FDE with polynomial coefficients and linearly transformed arguments have entire solutions of zero order. This is a remarkable dissimilar­ ity between the behavior of FDE and ODE since first-order algebraic ODE have no entire transcendental solutions of order less than i. An equally striking phenomenon is the existence of distributional solutions for linear homogeneous FDE without singularities in the coefficients. In other words, distributional solutions to linear homogeneous FDE may be originated either by singularities of their coefficients or by argument deviations. Recent studies have shown that nonexistence of infinite or­ der distributional solutions for linear time-dependent delay equations with real analytic coefficients implies nonexistence of small solutions

PREFACE

xin

(approaching zero faster than any exponential as t tends to infinity), which is important in the qualitative theory of FDE. It would be nice to extend the results on distributional and entire solutions and their interplay to partial differential equations (PDE). As a first step in this direction, one could take a linear PDE in two independent variables with polynomial coefficients that admits separation of variables, then consider a series whose terms are products of distributional solutions of the ordinary differential equations arising after separation. A special role is played by the chapter on differential equations with periodic transformations of the argument. These FDE can be reduced to ordinary differential equations and are very important in a number of biological models. Reducible FDE naturally appear in the construc­ tion of Liapunov functional for retarded differential equations. They represent a rich source of analytic solutions and provide an insight into the structure of solutions for more general FDE, especially equations with linearly transformed arguments. This is a major reason we decided to include the chapter on reducible FDE: to create a smooth transition from mildly generalized solutions of EPCA to distributional solutions of equations with arguments proportional to t and their relationship with analytic solutions. Although it is an older topic, interesting papers continue to appear in the field of reducible FDE. I was very fortunate to collaborate and to have fruitful discussions on the subject with many colleagues and friends, notably, A. R. Aftabizadeh, K. L. Cooke, L. Debnath, J. K. Hale, and S. M. Shah. We acknowledge with gratitude the generous financial support provided by the U. S. Army Research Office and the National Aeronautics and Space Administration during our work on the book. My thanks are due to Mr. Jose Gonzalez for typesetting the book using AM<S-lbTffi.. A special word of gratitude goes to the staff of World Scientific for their cooperation. Finally, a list of new research topics and open problems is included. University of Texas-Pan American Edinburg 1992

Joseph Wiener

CHAPTER 1

Differential Equations with Piecewise Continuous Arguments The general theory and basic results for functional differential equa­ tions (FDE) have by now been thoroughly explored and are available in the famous book of Hale [115] and subsequent articles by many au­ thors. Nevertheless, there is still need to extend the theory of FDE with continuous arguments to equations with discontinuous arguments. This task is also of considerable applied interest, since FDE with delay pro­ vide a mathematical model for a physical or biological system in which the rate of change of the system depends upon its past history. In this chapter, we shall describe some of the work that has been done over the last few years on the differential equations that we call equations with piecewise continuous arguments, or EPCA. Our attention was directed to these equations by an article of Myshkis [203], in which it was ob­ served that a substantial theory did not exist for differential equations with lagging arguments that are piecewise constant or piecewise con­ tinuous. The study of EPCA has been initiated by Wiener [288, 289], Cooke and Wiener [51], and Shah and Wiener [244]. A brief survey of the present status of this research has been given in [55]. A typical EPCA is of the form x'(t) t(t), x(h(t))), At) -== f(t,:f(t,x(t),x(h(t))), where the argument h{t) has intervals of constancy. For example, in [51] equations with h(t) = [t], [t — n], t — n[t] were investigated, where n is a positive integer and [•] denotes the greatest-integer function. Note that h(t) is discontinuous in these cases, and although the equation fits within the general paradigm of delay differential or functional differenl

2

1. PIECEWISE CONTINUOUS ARGUMENTS

tial equations, the delays are discontinuous functions. Also note that the equation is nonautonomous, since the delays vary with t. More­ over, as we show below, the solutions are determined by a finite set of initial data, rather than by an initial function, as in the case of gen­ eral FDE. In fact, EPCA have the structure of continuous dynamical systems within intervals of certain lengths. Continuity of a solution at a point joining any two consecutive intervals then implies recursion relations for the solution at such points. Therefore, EPCA represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equations. An equation in which x'(t) is given by a function of x evaluated at t and at arguments [t],..., [t — N], where TV is a non-negative integer, may be called of retarded or delay type. If the arguments are t and [t + 1 ] , . . . , [t + N], the equation is of advanced type. If both these types of arguments appear in the equation, it may be called of mixed type. If the derivative of highest order appears at t and at another point, the equation is generally said to be of neutral type. All types of EPCA share similar characteristics. First of all, it is natural to pose the initial-value problem for such equations not on an interval but at a number of individual points. Secondly, for ordinary differential equations with a continuous vector field the solution exists to the right and left of the initial f-value. For retarded FDE, this is not necessarily the case [115]. Furthermore, it appears that advanced equations, in general, lose their margin of smoothness, and the method of successive integration shows that after several steps to the right from the initial interval the solution may even not exist. However, two-sided solutions do exist for all types of EPCA. Finally, the problems for EPCA studied so far are closely related to ordinary difference equations and indeed have stimulated new work on these. It is important to note that EPCA provide the simplest examples of differential equations capable of displaying chaotic behavior. For instance, following Ladas [155], one can see that the unique solution of the initial-value problem x'(t) = 3x([t])-x2([t)),

x(0)

= Co

1. PIECEWISE CONTINUOUS ARGUMENTS

3

where [t] is the greatest-integer function, has the property that x(n + l) = 4x(n) - x 2 ( n ) ,

n = 0,l,....

If we choose Co = 4 sin2(7r/9), then the unique solution of this difference equation is x(n) = 4 s i n 2 ( 2 » 0 , which has period three. By the well-known result [171] which states that "period three implies chaos," the solution of the above differen­ tial equation exhibits chaos. Furthermore, the equation of Carvalho and Cooke x'(t) = ax(t)(l - *([*])) is analogous to the famous logistic differential equation, but t in one argument has been replaced by [ 0 and H(t) = 0 for t < 0. If we admit distributional derivatives, then differentiating the latter relation gives oo

x"(t) = ax'(i) + Y. (oo*(0 + a\x{} - 1))(*(* - *') - K* ~ { ~ l))i i=—oo

where 6 is the delta function. This impulsive equation contains the val­ ues of the unknown solution for the integral values of t. We introduce: Definition 1.1. A solution of Eq. (1.1) on [0, oo) is a function x(t) that satisfies the conditions: (i) x(t) is continuous on [0, oo). (ii) The derivative x'(t) exists at each point t G [0, oo), with the possi­ ble exception of the points [t] £ [0, oo) where one-sided derivatives exist. (iii) Eq. (1.1) is satisfied on each interval [n, n + 1) C [0, oo) with integral end-points. Denote m0(t) = eat + (eat-l)a-1a0, &0 = m 0 (l),

m^t) = {eat - l)aTxau &i=mi(l),

(1.2)

5

1.1. LINEAR RETARDED EPCA

and let Aj and A2 be the roots of the equation A2 - 60A - 61 = 0.

(1.3)

T h e o r e m 1.1. Problem (1.1) has on [0,00) a unique solution x(t) = m0({t})c[t] + mi({ A2. Formula (1.4) was obtained with the implicit assumption a ^ 0. If a = 0, then x„(t) = cn + (a0c„ + aicn_i)(< - n ) , which is the limiting case of (1.4) as a —► 0. The uniqueness of solu­ tion (1.4) on [0, oo) follows from its continuity and from the uniqueness of the problem xn(n) = cn for (1.6) on each interval [n,n + 1]. ■ COROLLARY 1.1. The solution of (1.1) cannot grow to infinity faster than exponentially as t —> +oo.

PROOF. Since eaW < e'"' we conclude from (1.4) that it remains to evaluate the coefficients Cm. From (1.5) we observe that if Ai ^ A2, then |Cm| < km1, where k is some constant and m = max(|Ai|, |A2|). And if Ai = A2, then from (1.8) it follows that \c,A < k(t + l)m'. ■ In ordinary differential equations with a continuous vector field the solution exists to the right and left of the initial t-value. For retarded functional differential equations, this is not necessarily the case [115]. Since the solution of (1.1) on [0,oo) involves only the group eat it can be extended backwards on (—oo,0].

7

1.1. LINEAR RETARDED EPCA

T h e o r e m 1.2. If a\ ^ 0, the solution of (1.1) has a unique backward continuation on (—oo,0] given by formulas (1.4) and (1.5). PROOF. If X-n(t) denotes the solution of (1.1) on [—n, —n + 1) sat­ isfying the conditions x(—n) = C-n,x(—n — 1) = c_„_i, then from the equation x'-Jt)

= ax.n(t)

+ a0c_n + aic_ n _i

it follows t h a t x_ n (f) = ea(t+n)c - a-\a0c-n

+ aic_n_i),

where c = (1 + a _ 1 a 0 )c_„ + a _ 1 a i c _ „ _ i . Therefore, z_„(i) = m0(t + n)c_n + m ^ f + n)c_„_i.

(1.9)

This result proves (1.4) for t < 0. Since x _ n ( - n + 1) = x _ n + i ( - n + 1) = c_„ +1 , we obtain the recursion relation c_ n + i = 60C-„ + &ic_„_i.

(1.10)

The formula c_„ = A - " gives Eq. (1.3) for A and the general solution c_„ = &iAj~n + ^ A ^ " for (1.10). In particular, \ylh

+ A ^ 1 ^ = c_i,

Aj"2A;i + Aj" 2 ^ = c_ 2 ,

and _ A?(c_i A22c_ c_2) A 2 (c-i -- A 2) 1 Ai — - A A2 Ai2 _ A|(AiCA|(Aic 22-C- c_i) -1) fc2 = Ai --A A22 Ai-

8

1. PIECEWISE CONTINUOUS ARGUMENTS

Since a\ ^ 0 we have b\ ^ 0, hence, we can find c_2 = bj (CQ — &o c -i) from (1.10) and substitute in the latter equations. Taking into account bo — Ai + A2 and 61 = —A1A2 we obtain the formula _ Ar" + 1 (c 0 - A 2 c-i) + (A l C -! - c0)A2-"+1 AJ

— A2

which together with (1.9) proves the theorem. If a\ = 0, we formulate: T h e o r e m 1.3. The problem x'(t) = ax(t) + a0x([t\), has on [0,oo) a unique

x(0) = Co

(1.11)

solution x(t) = m0({t})b0t]co .

(1.12)

T h e o r e m 1.4. If bo ^ 0, the solution o / ( l . l l ) has a unique backward continuation on (—00,0] given by formula (1.12). REMARK 1. If b0 = 0, then x(t) = m0(t)c0 on [0,1] and x(t) = 0 on [l,oo). In this case, x(t) coincides on [l,oo) with the trivial solution of Eq. (1.11). Of course, fusion of solutions is impossible for ordinary differential equations with uniqueness conditions. T h e o r e m 1.5. Ifbo^O and mo({*o}) ^ 0, then Eq. (1.11) with the initial condition x(to) = #0 has on (—00,00) a unique solution mQ({t})mQ-\{t0})b0t]-[t°]xo,

x(t) =

where {t} is the fractional part of t. The last theorem reveals an important fact t h a t the initial-value prob­ lem for Eq (111) may be posed at any point, not necessarily integral. A similar proposition is also true for Eq. (1.1). T h e o r e m 1.6. If a\-^Q

and

Aimo({*0}) + m i ( { * 0 } ) ^ 0 ,

* == 1,2

where A,- are the roots of (1.3), then the problem x(t0) = x0,

x(t0 - 1) = x_i

for Eq. (1.1) has a unique solution on (—00,00).

(1.13)

1.1. LINEAR RETARDED EPCA

9

PROOF. With the notations fa = m0({t0}), A = mi({( 0 }), we obtain from (1.4) the equations fa%] + Pl%-1] = XO ,

Po%-X]+Pic[to_2]

= x-i.

(1.14)

Let \M

P[to] ~

x[«o]

A l

-A

2

'

_ A ^ " 1 - A2A['O]

%,] -

Al

-A2

"

Then from (1.5) we have %]

=

P[0,

(1.28)

where B0 = eA + (eA - l)A-lA*,

Bx = {eA - / ) > * - % .

Looking for a nonzero solution c„ = Xnk, with a constant vector k, we conclude that A satisfies the equation det(A2J - \B0 - Bx) = 0

(1.29)

which has 2r nontrivial solutions if det £?i ^ 0. Assuming that these roots are simple we write the general solution of Eq. (1.28) C = AJfci + A5fc2 + • • • + A^rfc2r, with constant vectors kj each of which depends on the corresponding value Xj and contains one arbitrary scalar factor. These factors can be found from the initial conditions (1.26). If some of the Xj are multiple zeros of Eq. (1.29), then the expression for c„ also includes products of exponentials by polynomials of n. T h e o r e m 1.15. Assume that in Eq. (1.26) the matrices A, eA — I, and A\ are nonsingular. Then problem (1.26) has a unique solution on [0,oo); and this solution cannot grow to infinity faster than expo­ nentially.

1.2. SOME GENERALIZATIONS

19

T h e o r e m 1.16. The solution x = 0 of Eq. (1.26) is asymptotically stable ast—> +oo iff all zeros of Eq. (1.29) are located in the open unit disk. We note that the unknowns c„ in Eq. (1.22) can be computed also by means of the so called branching continued fractions [25], without re­ sorting to the characteristic equation (1.23). The instrument of matrix continued fractions [219] may be used to find the vectors c„ in (1.28). Eq. (1.29) is often encountered in various problems of quantum me­ chanics and theory of small oscillations. Some sufficient conditions for all zeros of Eq. (1.29) to lie in the open unit disk may be found in [212]. Let B0 = (bf),

Bi = ($>),

i,j =

l,2,...,r.

Then |A| < 1 for all zeros of (1.29) if

E(K 0) | + | # | ) < 1 ,

i = l,2,...,r.

Consider in a Banach space E the equation x'(t) = Ax(t) + Bx([t])

(1.30)

with linear constant operators A: 1>(A) —* E and B: 1>(B) —► E, their domains T>(A) C ^>{B) C E, and T>{A) is everywhere dense in E. We introduce the definition of a solution to Eq. (1.30) that modifies Definition 1.1 and the given in [144] for the equation x' = Ax.

(1.31)

Definition 1.2. A solution of (1.30) on [0,oo) is a function x(t) sat­ isfying the conditions: (i) x(t) is continuous on [0, oo) and its values lie in the domain "D(A) for all* € [0,oo); (ii) At each point t € [0, oo) there exists a strong derivative x'(t), with the possible exception of the points [t] € [0, oo) where one-sided derivatives exist, (iii) Eq. (1.30) is satisfied on each interval [n,n + 1) C [0,oo) with integral end-points.

20

1. PIECEWISE CONTINUOUS ARGUMENTS

The Cauchy problem on [0, oo) is to find a solution of the equation on [0, oo) satisfying the initial condition x(0) = co G T>(A).

(1.32)

Let £ ( E , E) be the space of bounded linear operators from E into E. It is easy to establish the existence and uniqueness of solution to prob­ lem (1.30), (1.32) as well as its exponential growth and backward con­ tinuation. Theorem 1.17. If A,B G £ ( E , E) and A is bijective, then prob­ lem (1.30), (1.32) on [0, oo) has a unique solution x(t) =

V({t})V®(l)c0,

(1.33)

where V(t) = eAt + (eAt -

I)A~lB.

This solution cannot grow to infinity faster than exponentially. If, in addition, there exists a bounded inverse of the operator V(l), then the solution o/(1.30), (1.32) has a unique backward continuation on (—oo, 0] given by formula (1.33). REMARK 2. It is well known that if the spectrum of the operator A G £ ( E , E ) lies in the open left halfplane, then for any bounded function f(t) G E: sup \f(t)\ < oo, 0(l + ( l - « r W . The reason for this is the change of the free term as t passes through the integral values. T h e o r e m 1.18. If A, B e £ ( E , E ) , then the equation x'(t) = Ax(t) + Bx(t - [t])

(1.34)

with the initial condition (1.32) has a unique solution on [0,oo): x(t) = x0(t) = e ^ + B > V [t]

x(t) = eAtc0 + £ /

k

eA^-s^Bx0(s

0< t 1. (1.35)

PROOF. Relation (1.34) represents an interesting example of an equa­ tion with unbounded delay [t] that admits a pointwise initial condition. Let xn(t) be the solution of (1.34) on [n,n + 1) satisfying x(n) = c„. Then x'n(t) = Axn{t) + Bxn(t - n),

n>0

and for xo(t) this gives the first part of (1.35). For n > 1 we have 0 < t — n < 1, hence x'n(t) = Axn(t) + Bx0(t - n),

xn(n) = cn .

From here, xn(t) = e^-")c„ + J* e^-^Bxois

- n) ds

and x„-i(*) = c il{ '- B+1) c„- 1 + / ^ e^^Bxois

-n + 1) ds.

Since x„_i(n) = x n (n) we put t = n to get c„ = eAcn_1 + I" eA^n-s^Bx0(s -n + 1) ds. Jn—1

(1.36)

22

1. PIECEWISE CONTINUOUS ARGUMENTS

Applying this formula successively to (1.36) yields (1.35).

■

T h e o r e m 1.19. In the conditions of Theorem 1.18 solution (1.35) has a unique backward continuation on (—oo,0]: x(t) = eAtc0-

[k+XeA^-^Bx0(s-k)ds

£ fc=-i ■'*

+ [teA^Bx0(S-[t])ds j\t\

(1.37)

P R O O F . For the solution x_„(i) of (1.34) on [—n, —n + 1), we have the equation

x'_n(t) = Ax.n(t)

+ Bx0(t + n),

for which we put £_„(—n) — c_„. Then x_„(*) = e^ +n >c_„ + f

eA^-^Bx0{s

+ n) ds.

Similarly, on [—n + 1, — n + 2) this gives s- B+ i( 0 and a domain £1 (of the indicated type) such that the resolvent set of A contains Cl and for all AGO, \(A-

-xiy

■ - 1 + |A|'

We state the following: Theorem 1.21. Problem (1.30), (1.32) has on [0, oo) a unique solu­ tion given by formula (1.38),. if A is a closed abstract elliptic operator with the domain T)(A) dense in E, and the operator B: 1)(B) —* 1>(A) where T>(B) D T>(A).

25

1.2. SOME GENERALIZATIONS

Now we consider in a Banach space E the equation

*'((A) —► E, having the same do­ main T)(A) dense in E. Definition 1.3. The function x{t) is called a solution of the initialvalue problem for (1.40), if the following conditions are satisfied: (i) The function x(t) is continuous on (-co, oo) and its values lie in the domain 2)(A) for all t G (—00,00). (ii) At each point t G [0,00) there exists a strong derivative x'(t), with the possible exception of the points [t] where one-sided derivatives exist, (iii) On (—00,0], x(t) coincides with a given continuous function <j>(t) G D(A) and satisfies Eq. (1.40) on each interval [n, n + 1). The solution xo(t) of (1.40) on [0,1) satisfies the equation

4(*) = ( E A) *(*},

*o(0) = #0).

(1.41)

We also employ the homogeneous equation x'(t) = A0x(t)

(1.42)

corresponding to (1-40) for t > 1. Denote fk(t) = A1x0(t-k)

+ '£Ai(t-ik),

k = 1,2,...

(1.43)

i=2

(k\eH,

t>0

with positive constants a, b and \(j>\ = sup \(t)\ for —r < t < 0 [115]. Similar results hold for (1.40). Theorem 1.24. If in the conditions of Theorem 1.22 the functions fk(t) are bounded in T)(A) uniformly relative to t and k: \fk(t)\\.

This gives oo

\x(t)\ < Mxeui + M2eut £ e~"k + M3eu\ and proves the assertion since the series converges, and any value of t > 0 can be placed in a segment [n, n + 1]. ■ Theorem 1.25. If the operators Ak G £ ( E , E ) and the function <j>(t) is bounded on (—oo,0], then solution (1.44) satisfies (1.48). 3. EPCA of Advanced, Mixed, and Neutral Types We now describe how an initial-value problem may be posed and solved for the following linear equation of mixed type, since it provides a simple framework for understanding more complicated problems: x'(t) = Ax(t)+

£

AjX([t + j]),

(1.49)

j=-N

x{j) = Cj,

-N<j 0 satisfying the conditions x(n + j) = cn+j,

-N<j 1. These relations yield inequalities opposite to (1.68) and (1.69) and prove the theorem. ■ COROLLARY 1.2. The solution x = 0 of Eq. (1.11) is stable (asymp­ totically stable) if and only if the inequalities -a(ea + l) e° - 1 (strict inequalities) take place.

_

Theorem 1.29. In each interval (n,n + 1) with integral endpoints the solution of Eq. (1.64) with the condition x(0) = CQ ^ 0 has precisely one zero 1 a0 + aie a tn = n + - In ■ a a + ao + a\

0

(1.71)

leads to ao >

aea

a

ea-l'

ea - 1

By adding inequalities (1.72), it is easy to see that (1.71) is a conse­ quence of (1-72). If a > 0,

a + ao + ai < 0,

then (a + ao + a\)eu < ao + a\ea < a + ao + ai and aea

ao < - ^ y ,

a 0l


1.3. EPCA OF ADVANCED, MIXED, AND NEUTRAL TYPES

39

then solution (1.74) with CQ / 0 has precisely one zero in each interval (n, n + 1) and lim x(t) = 0. 2. If aea ea - 1 < ao < —a, then x(t) has no zeros in (0, oo) and lim x(t) = 0. 3. For a(ea + 1) ea — 1 the solution has precisely one zero in each interval (n, n + 1) and is unbounded on [0,oo). 4. For ao > —a the solution has no zeros in (0, oo) and is unbounded. The case ao = — a yields x(t) = CQ. If aea a0 = ea-l' then n:(0

(1.87)

42

1. PIECEWISE CONTINUOUS ARGUMENTS

Assuming that all roots of (1.56) are simple the general solution of (1.87) is given by 2Nr

cn=£A;^+p„,

n>0

(1.88)

i=i

where the kj are constant vectors each of which depends on the corre­ sponding value Xj and contains one arbitrary scalar factor, and p n is a particular solution of (1.87). If some roots of (1.56) are multiple, the components of the vectors kj are polynomials in n of certain degree. To find p n , we use the variation of parameters method, according to which 2Nr

Pn=E

X]qj(n),

(1.89)

3=1

with unknown vectors qj(n). Let

Aqj(n) =qj(n +

l)-qj(n).

Then these differences satisfy the system of equations 2Nr

£ A f 'Afc(n) = 0,

i = l,...,2iVr-l

j=l 2Nr

r

+2JV A (n) qj E A? j=\

:=

/»,

from which it follows that n-l

.T)(m)

»(») = £ (-i)'g/™,

«>i-

x^m^ m=o Here 2)(m) denotes Vandermonde's determinant S(m) = detCAf*), i , i = l,...,2JVr and 2)j(m) is obtained from 2)(m) by crossing out the last row and the j-th. column. Simple computations show that 2)(m) = (A 1 A 2 ...A 27 , r ) m det(A;.), % ( m ) = (-iy(A 1 A 2 ...A 2Afr ) m A7 m det(A / t ), (k,l = l,...,2Nr-l; kf^l).

1.3. EPCA OF ADVANCED, MIXED, AND NEUTRAL TYPES

43

Hence, «,■(») = Yi{X) £ A 7 m / m , Vj(X) = ^ f l . (1.90) det(A') m=o Substituting (1.90) in (1.89) yields a particular solution of (1.87) and its general solution (1.88). Then successively letting n = 0 , 1 , . . . ,N — 1 in (1.87) and using the initial conditions Cj = x(j) from (1.85), we de­ termine the values CJV, c^+i, ■ ■., C2N-\- Inserting into (1.88) the vectors Co, c i , . . . , C2N-1 produces a system of equations which uniquely deter­ mines the unknowns kj. ■ T h e o r e m 1.33. All solutions of Eq. (1.85) are bounded on [0,oo) if f(t) is bounded on [0,oo) and inequalities (1.58) hold true. PROOF. From the boundedness of f(t) it follows that the vectors / „ are uniformly bounded for n > 0. This is also true for the sums of the terms Xfkj in (1.88) and for Sjn = £

X]-mfm.

(1.91)

m=0

Therefore, pn and c„ possess the same property and since the matrices Mj(t — n) — Mj(t — [t]) are bounded, the proof is a result of (1.86). ■ T h e o r e m 1.34. All solutions of Eq. (1.85) tend to zero as t —► +oo if the roots of (1.56) satisfy (1.58) and lim f(t) = 0. PROOF. To show that c„ —► 0, we observe that X"kj —* 0, / „ —► 0 and it remains to evaluate (1.91). We write M

Sjn

=

n-l

E Xj

fm+

m=0

2-,

Xj

fm,

m=M+l

where M is a natural number such that | / m | < e for a sufficiently small e > 0 and m > M. Since fm is bounded we have | / m | < L and M

n-l

|Si»|i_l,

i=

From the latter relation, cm- = Al~lcni, that

2,...,N.

and from the first it follows

c„i = (A + A0)cno + £

AiAi~lcnx-

i=l

Hence, cni = A^B'^A

+ A0)cn0,

i > 1.

The general solution of (1.94) is x

n(*)

=

£

N c — 2_, A

A.iCni,

:'=0

with an arbitrary vector c. For t — n, we get c = 5c„o and xn(t) = E(t - n)cn0.

(1.95)

Therefore, on [n,n + 1) there exists a unique solution of Eq. (1.92) with the initial condition x(n) = c„o given by (1.95). The requirement xn(n + 1) = xn+i(n + 1) implies c„o = £ , (l)c„_io from which cn0 = £"(l)c 0 . Together with (1.95) this result proves the theorem. The unique­ ness of solution (1.93) on [0,oo) follows from its continuity and from the uniqueness of the problem x(n) = c„o for (1.92) on each interval [n, n + 1). It is easy to see that the restriction det A ^ 0 is required not in the proof of existence and uniqueness but only in deriving for­ mula (1.93). ■

46

1. PIECEWISE CONTINUOUS ARGUMENTS

T h e o r e m 1.37. // the matrix B is nonsingular, problem (1.92) on [0,oo) has a unique solution x(t) = V0(t)c0,

0< t 1

k=[t]

where

Vk(t) = eA^

+ fk eA(*-*)C ds

and C = B-i(A +

A0)-A.

COROLLARY 1.4. The solution of (1.92) cannot grow to infinity faster than exponentially PROOF.

Since ||T4_i(fc)|| < m and ||V[t)(*)|| < m, we have \x(t)\<mt+l\co\.

where m = (1 + ||C||)eW.

■

T h e o r e m 1.38. //, in addition to the hypotheses of Theorem 1.36, the matrix E(l) is nonsingular, then the solution of (1.92) has a unique backward continuation on (—oo,0] given by formula (1.93). P R O O F . If x_n(t) denotes the solution of (1.92) on [—n, — n +1) with the initial condition x_„(—n) = c_„o, then

x-„(t) = E(t + n)c_n0. Since z _ „ ( - n + 1) = x_„ + i(-n + 1) = c_„+i]0 we get the relation c_n0 = ^ - 1 ( l ) c _ „ + i | 0 which proves the theorem.

1.3. EPCA OF ADVANCED, MIXED, AND NEUTRAL TYPES

47

Theorem 1.39. / / the matrices A, B, E(\), and E{{t§}) are nonsingular, then Eq. (1.92) with the condition x{t§) = xo has on (—00,00) a unique solution E({t})EU-M(l)E-l({t0})x0.

x(t) =

PROOF. From (1.93) we have c0 = £H'°](l)a;( [*\

(1.97)

1. PIECEWISE CONTINUOUS ARGUMENTS

48

PROOF. Let xn(t) be the solution of (1.96) on [n,n + 1). Then *'„(*) = Axn{t) + £ AiX®(t - n),

n> 0

i=0

and for n = 0 this gives the first part of (1.97). For n > 1 we have 0 < £ — n < 1 and the equation x'n(t) = Axn(t) +

f(t-n).

Its solution that satisfies the condition xn(n) = c„o is xn(t) = e^-")c„ 0 + / ' e^('- s )/( S - n) ds.

(1.98)

./n

Since x„_i(n) = a;n(n) we change n to n — 1 and put t = n to get Cno = e"cn_i>0 + / " , e j4( "~ s) /(s - n + 1) ds. Applying this formula to (1.98) successively yields (1.97). Eq. (1.96) generalizes (1.34) ■ T h e o r e m 1.41. In the conditions of Theorem 1.40 solution (1.97) has a unique backward continuation on — oo < t < 0: x(t) = eAte0~

[t]

...

£ f k=-xJk

eA^^f(s-k)ds + f[t]eA^f{s-[t])ds.

(1.99)

PROOF. For the solution x_n(f) of (1.96) on [—n,—n + 1) we have the equation x'_n(t) = Ax_n(t) + f(t + n) with the initial condition x-„(—n) = c_„o, whence *_„(*) = eA{t+n)c_nQ + f_n eA^f(s

+ n) ds.

Since £_„(—n + 1) = £_„+i(—n + 1) = C-n+\fi it follows that c_n0 = e-AC-n+lfi - /_~"+ e-^ ( s + n )/(s + n) ds. This formula leads to (1.99).

■

1.3. EPCA OF ADVANCED, MIXED, AND NEUTRAL TYPES

49

Theorem 1.42. The solution of (1.96) is bounded on [0,oo) if the equation x'(t) = Ax(t) is asymptotically stable as t —► oo. If the latter equation is stable, the solution of (1.96) cannot grow to infinity faster than a linear function. Now we consider the equation *'(*) = E AM* - *[*]) + £ Aax'(t - i[t}). i=0

(1.100)

i=0

which is a generalization of (1.40). Definition 1.4. The function x(t) is called a solution of the initialvalue problem for (1.100), if the following conditions are satisfied: (i) x(t) is continuous on (—00,00). (ii) The derivative x'(t) exists at each point t 6 [0,oo), with the possible exception of the points [t] where one-sided derivatives exist, (iii) On (—00,0], x(t) coincides with a given continuously differentiable function <j>{t) and satisfies Eq. (1.100) on each interval [n,n + 1). Theorem 1.43. If the matrices N

A0 = I-A0i,

Ai = J - £ A,-i i=0

are nonsingular, then the initial-value problem for (1.100) has a unique solution x(t) = x0(t) = e B V(0), [t]

0 < t\

e^g^ds (1.101)

50

1. PIECEWISE CONTINUOUS ARGUMENTS

where A - A0 lA00, B = EJI 0 Ax yAm> gk(s) = A0 lfk(s), and N

fk(t) = A10x0(t -k) + Anx'0(t -k) + J2 Ai0(t>{t - ik) «=2 N

+ J2Aa'(t-ik),

k(t - in) + J2 Aii4>'(t ~ in),

n>l

t'=2

i=2

which also can be written x'n(t) = A00xn(t) + A0lx'n(t) + /„(*). Hence, x'n(t) = Axn(t) + gn(t). The formula xn{t) = e^-") C „ + fn eA^~^gn(s) ds

(1.102)

yields the solution of this equation with the condition xn(n) Therefore, the relation x n _i(n) = c„ gives c„ = eAcn_i + £_{ e^-^gn^s)

= c„.

ds.

Continuing the procedure leads to the formula cn = eAln-»Cl + ± [k

eA^-^gk^{s)

ds,

n>l

which together with (1.102) proves the theorem since c\ =

XQ(1).

■

1.4. LINEAR EPCA WITH VARIABLE COEFFICIENTS

51

Theorem 1.44. / / in the conditions of Theorem 1.43 the functions {t) and '{t) are bounded on (—oo,0], the solution (1.101) satisfies the inequality \x(t)\ < Leut\(p\i,

t>0

where L and cu are some positive constants and \(t) and '(t) for t < 0. Hence, \gk(t)\ < K, and we also use the estimate Ije^'l) < Meut to obtain

\x(t)\
1.

The inequality that follows from here, \x(t)\ < (M^

+ M2eui £ e~»k + M3e«") \ 0 . Since the matrices 5/v„(n+l) are nonsingular, for all n > 0, there exists a unique solution cn(n > N) provided that the values c_w,..., c^-i are prescribed. Substituting the vectors c„ in (1.114) yields the solution of (1.112). ■

57

1.4. LINEAR EPCA WITH VARIABLE COEFFICIENTS

For the scalar equation x'{t) = a(t)x{t) + a_i(t)x([t - 1]) + a0(t)x([t]) + ai(t)x([t + 1]), x(-l) = c_i, x(0) = c0 with coefficients continuous on [0,oo) we can indicate a simple algo­ rithm of computing the solution. According to (1.113) and (1.114), we have xn(t) = B_i„(t)c^i + B0n(t)cn + Bin(t)cn+i,

(1.115)

with

U(t) = exp(j\(s)dsy The coefficients c„ satisfy the equation (Bln(n + 1) - l)c„ +1 + B0n(n + l)c n + B_ l n (n + l ) c _ i = 0 ,

n > 0.

Denote d

~^

+ l)

B_i„(n+1) = 1-BUn + iy rn =

, Mn

. + 1} =

£ 0 n (n + 1 ) l - B l B ( n + l)'

Cn+l

Then from the relation c„+i = d0(n + l)c,, + d.-i(n + l)c„_i it follows that r„ == do(n + l) +

—i(n +

l)

rn-\

which yields r 0 = 4(1) +

Mi) r-i

M2) j /.» , n = do(2) + — ^ = do(2) + '■(-

d

,

,M

-i(2) , M i )

d0(l) +

58

1. PIECEWISE CONTINUOUS ARGUMENTS

and continuing this procedure leads to the continued-fraction expansion rn = d0(n + 1) +

; . do(n) +

•■■ , . 7 4 ( 1 ) + c 0 /c_!

and to the formula c„ = r_ 1 r 0 ---r„_ic_i,

n> 1

for the coefficients of solution (1.115) Along with the neutral equation x'(t) = A(t)x(t) + A0(t)x([t]) + Ai(t)a:'([*]).

x

(°) = °o (1.116)

(0 < t < oo) we also consider (1.104). Theorem 1.50. If the coefficients of Eg. (1.116) are continuous on 0 < t < oo and i/ie matrices 5 , = I — A\(i) are nonsingular for each i > 0, then there exists a unigue solution of problem (1.116) given by the formula x(t) = U(t){u-i([t])

+ ^]U-l(s)(A0(s)

+ Ai(s)P[t])dsy[t],

(1.117)

where

c[t]=

hu(i)(u-\i-i) i=[t]

\

+ j ! _ i U-\S)(A0(s)

+ A^P^)

ds)c0

and Pi = Br\A(i)

+ A0(i)).

PROOF. On the interval n < < < n + 1 we have the equation x'(t) = A(t)x(t) + A0(t)x(n) +

Ai(t)x'(n).

For t = n this gives x'(n) = Pnx(n), hence x'(t) = A(t)x{t) + (A0(t) +

AitfPJxin).

(1.118)

1.4. LINEAR EPCA WITH VARIABLE COEFFICIENTS

59

Its solution xn(t) satisfying the condition x(n) = c„ is given by the expression *»(*) = Un(t) ( / + fn U-l(s)(A0(s)

+ A1(s)Pn) ds} c„,

Un(t) being the solution of (1.106) such that Un(n) = I. We obtain from x„_i(n) = cn the relation

c„ = Un^(n) (i + £_t U'l^iAois)

+ Al{s)Pn-l) ds) c_,.

To prove (1.118) it remains to observe that Ui(t) = U{t)U~l{i).

■

Theorem 1.51. If A{t), Ao(t), Ai(t) are continuous on [0,oo) and the matrices B{ have uniformly bounded inverses for all i > 0: \\Brl\\<m, then the solution (1.117) satisfies the estimate \x(t)\<elt+1M(b(t)

+

l)W\co\,

where a(t) = max||A(s)||, a{(t) = max||A;(s)||, 0 < s < t, b(t) = a0(t) + mai(t)(a(t) + aQ(t)).

(i = 0,1)

The proof is analogous to that given for Theorem 1.47. In particular, if A(t), A0(t), and A^t) are bounded on [0,oo), then \x(t)\ < Meut, with some constants M > 0, w > 0. In this case, the conclusion about the solution growth is the same as for Eq. (1.116) with constant coeffi­ cients. Now we shall investigate the stability properties of linear EPCA with variable coefficients. Theorem 1.52. The solution of the scalar equation x'(t) = ax(t) + aQ(t)x([t]) + ai(t)x([t + 1]),

x(0) = c0 (1.119)

tends to zero as t —► +oo if the following hypotheses are satisfied: (i) a < 0 is constant, ao(t),a\(t) G C[0, oo);

60

1. PIECEWISE CONTINUOUS ARGUMENTS

(ii) sup |ai(i)| = ql < 1 on [0, oo) and, starting with some to > 0, a(ea + l-qi) ea- 1

aqx a < m0' < Mo' < -a - e— - 1'

(1.120)

where mo = inf ao(t), MQ = supao(tf), t > to. PROOF. The solution of Eq. (1.119) is given by (1.115), with BQ„(t) = ea((-") + fn ea^ao{s)

ds,

Bln(t) = £ e'^a^s)

ds

B-m{t) = 0 and n < t < n + 1. Since the functions Bon(t) and B\n{t) are bounded on [0,oo), it remains to show that lim c„ = 0. From xn(n + 1) = c n+ i we get (1 - Bln(n + l))c n + 1 = B0n(n + l)c„, that is, cn _ -Bo,n-i(") cn_i 1 - 5i,„_i(n) and c„ = rir 2 ---r„_ic 0 , where ea +

„

f

e«i-°)aQ(s)

dS

Ji—l

l-l'_xea^-^a1(s)ds By virtue of (i) and the first part of (ii), 1- f

e^-^a^s)

ds > 1 - f

\ai{s)\ds > 1 - qx.

1.4. LINEAR EPCA WITH VARIABLE COEFFICIENTS

61

Therefore, 1 — B1>1-_1(i) ^ 0 for all i > 1. Let j be a natural number such that j > to, then due to (1.120) we have

£„,,_,« = «• + £ , 1 - «i,

02

j=-N

x(j) = ch

-N <j

+00 if the following hypotheses are satisfied: The coefficient a is constant, the functions aj(t) G C[0,oo). The functions aj(t)(—N < j < N — 1) are bounded on [0,oo). The function a^(t) is positive, monotonically increasing, and lim^oo aN(t) = 00.

Consider the equation x'(t) = Ax(t) + A^(t)x([t

- 1]) + A0(t)x([t]) + Ai^xdt

+ 1]), (1.128)

the coefficients of which are r x r-matrices and x is an r-vector with norms r

\x\ = ma* N ,

P l l = max. £ \aki\-

T h e o r e m 1.56. All solutions of (1.128) are exponentially asymptot­ ically stable if \\eA\\ < I, the matrices Aj(t) are continuous on [0,00), Aj(t) —> 0 as t —> +00 for j = 0, ±1 and

PROOF.

II fn+1 eA(n+\-s)Ars•) d J K l (1.129) n > 0 II n !! The solution xn{t) of (1.128) on [ n , n + l ) is given by (1.115),

with Bjn(t) = £eA^-^Aj(s)ds,

j = ±l

B0n(t) = e^'-O + fn eA^A0(s)

ds.

The condition xn(n + 1 ) = c n+ i implies the equations (7 - Bln(n + l))c„+i = 5 0 n (n + l)c„ + 5_i„(n + l)c„_i,

n > 0

from which the unknown vectors c„ may be found successively. Indeed, the matrices I — B\n(n + 1) have inverses for all n > 0, since (1.129) is equivalent to ||5i„(n + 1)|| < 1. Furthermore, since | | ( / - B i . ( n + l ) ) - , l | < E I | B i . ( n + l)||' »'=0

1.4. LINEAR EPCA WITH VARIABLE COEFFICIENTS

and |jT+V k.

The application of (1.131) to (1.130) successively yields 7

AT.

+ htM2(cf>)}e-^-Kh)1

80

1. PIECEWISE CONTINUOUS ARGUMENTS

(b). Assume the additional hypothesis that cj> is differentiable on [—r, 0] and satisfies the consistency condition