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 [<]. As a result, the equation has solu­ tions that display complicated dynamics [38]. It seems likely that other simple nonlinear EPCA may display other interesting behavior. The numerical approximation of differential equations can give rise to EPCA in a natural way, although it is unusual to take this point of view. For example, the simple Euler scheme for a differential equation x'(t) = f(x(t)) has the form xn+i — xn = hf(xn), where xn = x(nh) and h is the step size. This is equivalent to the EPCA

x'(t) = f(x([t/h]h)). Impulsive differential equations and loaded equations of control the­ ory fit within the general paradigm of EPCA. Another potential ap­ plication of EPCA is the stabilization of hybrid control systems with feedback delay. By a hybrid system we mean one with a continuous plant and with discrete (sampled) controller. Some of these systems may be described by EPCA [49]. EPCA have only been researched for a few years. In each of the areas — existence, asymptotic behavior, periodic and oscillating solu­ tions, approximation, application to control theory, biomedical models, and problems of mathematical physics — there appears to be ample opportunity for extending the known results.

1. PIECEWISE CONTINUOUS ARGUMENTS

4

1. Linear Retarded EPCA with Constant Coefficients Cooke and Wiener [51] considered the scalar initial-value problem x'{t) = ax(t) + aQx([t]) + aix([t - 1]), x(-l) = c_i, x(0) = c0

(1.1)

with constant coefficients and two delays, [t] and [t — 1], where [■] desig­ nates the greatest-integer function. The initial conditions are posed at t = — 1 and t = 0, and the solution is sought for t > 0. This equation is very closely related to impulsive and loaded equations, that is, those including values of the unknown solution for given constant values of the argument. Indeed, write Eq. (1.1) as oo

x'(t) = ax(t) + J2 (a0x(i) + aix(i - l))(H(t - i) - H(t - i - 1)), where H(t) = 1 for t > 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({<})c[4_1]!

(1.4)

where {t} is the fractional part of t and c

A [ / +1] (c 0 -A 2 c_ 1 ) + (A lC _ 1 -co)A^ +1]

w"

•

A T ^

(L5)

PROOF. Assuming that xn{t) is a solution of Eq. (1.1) on the inter­ val n < t < n + 1, with the conditions x{n) = cn,x(n — 1) = c„_i, we have x'n(t) = ax(t) + a0cn + aic„_i .

(1.6)

The general solution of this equation on the given interval is xn(t) = e a(t_n) c - a~1(a0cn + aic„_i), with an arbitrary constant c. Putting t = n here gives cn = c- a~1(a0cn + aic„_i). Hence, c = (1 + a^ao)^

+ a _1 aic„_i

and xn(t) = m0(t - n)cn + mi(t - n)c„_i. If xn-i(t) designates the solution of Eq. (1.1) on [n — l,n) satisfying the conditions x(n — 1) = c n _i,x(n — 2) = c„_2, then xn_i(t) = m0(t -n + l)c„_i +mi(t-n

+ l)c„_2-

Since x„_i(n) = xn(n), we obtain the recursion relation c„ = 6oc„-i + &ic„_2,

n > 1.

(1.7)

We look for a particular solution of this difference equation in the form c„ = A". Then A" = 60An_1 + M " ~ 2 ,

6

1. PIECEWISE CONTINUOUS ARGUMENTS

and A satisfies (1.3). If the roots Ai and A2 of (1.3) are different, the general solution of (1.7) is cn = AriAJ + M ? , with arbitrary constants k\ and fc2. In fact, it satisfies (1.7) for all integral n. In particular, for n = —1 and n = 0 this formula gives Aj"1^! + A j 1 ^ = c_i,

fci

+ fo = co

whence _ AI(CQ Ai(co-- • A22c_i) - AA ' Ai — Ai2 2 _ A2(Aic. (Aic_i CQ) -1 -- Co) Ai -- AA22 ■ AiThese results establish (1.5). If Ai = A2 = A, then c„ = A"tc0(n + 1) - Xe-m), (1.8) which is the limiting case of (1.5) as Ai —> 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[
C

[*o-i]=^o]Co

+

%]c-1'

C

[i0-2] = P[t0-lfO + % - l ] C - l

and substitute these expressions in (1.14): (PoP[t0+l] + AP N )C0 + ( # % + ! ] + ^1^ 0 ]) C -1 = *0 (PoP[to] + PiP[to-i])co

+ (Poq[ta] + A « f o _ i ] ) c - l = * - i •

This system with the unknowns Co and c_i has a unique solution if its determinant A = (A1A2)['°-1](A1A2/?02 + (A! + X2)PoPi +P\) is different from zero. The condition a\ ^ 0 gives AiA2 ^ 0, and A ^ 0 iff (AiA, + Pi){hPa + Pi) ± 0 which is equivalent to (1.13). ■ The theory of continued fractions [139] also provides a useful instru­ ment in the study of (1.1). In this manner, we can easily compute the coefficients c„. Let r n = c n /c„_i, then from (1.7) 7-1 =

-

+

^

1. PIECEWISE CONTINUOUS ARGUMENTS

10

After this we find 7ri"2 — = hO0+H

O+0-\ ==bo bo+

+ ^3 b0 H '3 = 6

"1

h

r

,

co/cco/c-i•1 "1

= bOO = + 0+

h

bo+ b0+

b

;

co/c-i •1 C 0/C-

This procedure leads to the representation of r„ by a finite continued fraction ,

O

r„ = &o +

i

6i

+ bo +

0!

6i

b0 + c 0 /c_i

Then cn = r\r2 ■ ■ ■ rnco . As t —* oo, the continued fraction for'M r m becomes infinite,

h h b0 + ————....——... (1.15) bo + b0 + o0 + and questions arise about its convergence. It is well known [139] that the continued fraction 1 1 1 . . a0H ■ ; ; = [ao; ai, a 2 , 0 3 , . . . J a i + «2 + 0,3 + with positive components an (n > 1) converges iff the series £ a „ di­ verges. The fraction (1.15) can be changed to [bo;bo/bi,bo,bo/bi,...]. Therefore, if bo > 0, bi > 0, then (1.15) converges. If bo < 0, bi > 0, then (1.15) also converges since it can be transformed into the contin­ ued fraction —[|bo|; |bo|/bi, |bo|, |bo|/bi,...]. In the case bo > 0, bi < 0 we employ a theorem [44] which states that the continued fraction , , 00 +

a\

0,2

bi+ b2+

a,i

b,+

1.1. LINEAR RETARDED EPCA

11

with positive terms b{ (i > 1) converges if bt — |a,| > 1, for any * > 1. Regarding (1.15) it means that this fraction converges if b0 + &i > 1. For 60 < 0, 6i < 0 we change (1.15) to

- (N+

h

h

&i

N N++ N N++

N N ++

■ ■ )

and conclude that it converges if - 6 0 + h > 1. Combining the last two inequalities gives a sufficient condition N + 61 > I for the convergence of (1.15) when 61 < 0. From here it follows that 62) + 4&i>(l + 6 1 ) 2 .

(1.16)

Thus we proved: T h e o r e m 1.7. The continued fraction (1.15) and c„/c„_i converge to the same limit if either of the following hypotheses is satisfied: (i) h>0, b0 ± 0 ( i i ) 6 i < 0 , |&oj + 61 > 1. T h e o r e m 1.8. In the conditions of Theorem 1.7, the continued frac­ tion (1.15) converges to the root of Eq. (1.3) which has the greater absolute value. PROOF. Inequality (1.16) guarantees that the roots of Eq. (1.3) are real and have distinct absolute values. We derived (1.15) from the difference equation (1.7) as a result of the limiting process in the finite continued fraction for c„/c„_i. On the other hand, (1.15) arises also as a formal expansion in a continued fraction of a root of Eq. (1.3). We have 2 A (bpX + h) A2 +h = b + hh (boX h) A =: 0 = A A b0+J

='» r

- +T

12

1. PIECEW1SE CONTINUOUS ARGUMENTS

Continuing this procedure leads to

"o +

h + oo +

If 6i > 0, bo > 0, this fraction converges to the positive root of Eq. (1.3) which is greater than the absolute value of the other (negative) root. For 6i > 0, bQ < 0 we change (1.17) to

N+N+

N+

In this case (1.15) converges to the negative root of Eq. (1.3) the abso­ lute value of which is greater than the other (positive) root. Let D =

b

Kb

where bi < 0, and let D

l/2

_N

1

2

2/

2/>0.

Then

-(\b0\/2

y

+ D^) bi

and Dm

_N 2

h

1

' M

+ Z )l/2

2

From here we obtain h

D1'2--

b\

N+ N+

2

&i

N+

and the formula ^1,2 =

|

( ±

¥ +

N+

N+

1.1. LINEAR RETARDED EPCA

13

for the roots of Eq. (1.3). If &o > 0, the root of greater absolute value is Ai = | +

D^

where 2^1/2 _ ^o

*>i

"2

h

h

b0+ b0+'"b0+'

' '

and Ai is the limit of continued fraction (1.15). When b0 < 0, the root of greater absolute value is A2 = | -

D^

where 1/2

_

b0

6i

2

-b0+

&t

&i

-b0+'

and A2 is the limit of (1.15).

' -b0+'

_

(h

U

h

\

60+'"/'

■

T h e o r e m 1.9. The solution x = 0 of Eq. (1.1) is asymptotically sta­ ble as t —> +00 i/f £/ie moduli of the roots of Eq. (1.3) satisfy the in­ equalities \Xi\ < 1,

|A2| < 1.

(1.18)

PROOF follows from (1.4) and (1.5) since eaW is bounded, and Cm —» 0 as i -> +00 iff (1.18) holds. ■ It is easy to see that if (1.18) and the hypotheses of Theorem 1.7 are fulfilled, the series £c„ converges absolutely. By resorting to for­ mula (1.5) we can relax the conditions of this proposition. T h e o r e m 1.10. The series E|c„| converges if the roots of Eq. (1.3) satisfy (1.18).

14

1. PIECEWISE CONTINUOUS ARGUMENTS

T h e o r e m 1.11. If the solution x = 0 of (1.1) is asymptotically as t —► + o o , then - a ( 2 + ea)

stable

a(2 - ea)

a-19)

hi < ^ i PROOF. Prom (1.18) it follows t h a t

N = |AiA a |
M = |Ai + A 2 |<2,

which together with (1.2) gives (1.19).

■

T h e o r e m 1.12. The solution x = 0 of Eq. (1-1) is asymptotically stable as t —> + o o iff either of the following hypotheses is satisfied: ,.v (i)

a(2 + ea) aea - \ « _ l ' -< -a -0 < c ea_V a 1 a 2 a a(e + a - ( e - l ) a 0 ) a(e + 1)


4(?3T) ,.., w

..... (m) v '

ao;

ae a a(2 - e a ) e -l ea-l ' a(e a + a - 1 ( e a - l ) a 0 ) 2 < aj < —a — ao; 4(e a - 1) a

a(2 + ea) a(2-ea — a ■— < a 0 < —a -^, e - 1 e - 1 ' a a(e a + ar\e° - l ) a 0 ) 2 fll < c» _ 1 < 4 ( e a _ !)

PROOF. According to (1.18), the necessary and sufficient condition for asymptotic stability of Eq. (1.1) takes the form

b0 ± M + 46, < 2 .

1.1. LINEAR RETARDED EPCA

15

First, assume that the roots of Eq. (1.3) are real, then bo + M+4b~i < 2,

b0 - \fb~l + 4&i > - 2 .

From here we obtain the following system of inequalities: - 2 < 60 < 2, bx < 1 + b0,

- 1 < &i < 1, &i < 1 - &0,

n >d -

bl

If - 2 < b0 < 0, then b2 —7<&i<&o + l. 4 Hence, if - 2 < ea + a _ 1 (e a - l)a 0 < 0, then _ ( e - + a - V - l ) f l o ) ' < a - 1 ( e a _ 1 ) a i < e« + j

+ a-i(ea

_

1 K

which is equivalent to (i). In the case 0 < bo < 2 we have bl - f4 < 6i < 1 - &o, that is, 0 < e a + a-1(ea-l)a0 <2 is combined with (e a + a - 1 ( e a - l ) a 0 ) 2 4

< a _ 1 (e a - l)ai < 1 - ea - a _ 1 (e a - l)a 0

This gives (ii). It remains to consider the case when the roots of Eq. (1.3) are complex. Then b20 + 46j < 0,

b0 ± ij-bl

From here, 1 < 6, < - ? ,

- 4&i < 2 .

16

1. PIECEWISE CONTINUOUS ARGUMENTS

and with the notations (1.2), - K a

(e-l)aK

^ — ^

.

To obtain (iii), we associate with these inequalities the first of the conditions (1.19). ■ 2. Some Generalizations Let xn(t) be a solution of the equation N

x'(t) = ax(t) + £ aix([t - »']),

aN ^ 0

(1.20)

i=0

with constant coefficients on the interval n < t < n + 1. If the initial conditions for Eq. (1.20) are x(n — i) = c„_,-, 0 < i < N, then we have the equation N

x'„(t) = ax„(t) + Yl a-iCn-i the general solution of which is .

,

N

x„(t) = eaV-n>c - £ a-laicn-i,

a^O.

i=0

For t = n this gives N 2_^ ® >'=0

Cn — C

Qi^R—i

and N

x

n(t) = $] m,-(* - n)cn_i, i=0 at i /„at at

m0(<) = e + (e - il \j „—1 o"1^, mi(t) = (eat-l)a-1ai,

t = l,...,JV.

(1.21)

1.2. SOME GENERALIZATIONS

17

Since x„(n + 1) = xn+1(n + 1) = c„ +1 , we put t = n + 1 in (1.21) and obtain the equation N Cn+1 = £ hCn-i, i=0

(1.22)

where 6,- = m,(l). Its particular solution is sought as c„ = A"; then

\N+l = £

fc^-'.

(1.23)

i=0

If all roots Ai, \%,... ,\N+I of Eq. (1.23) are simple, the general solution of (1-22) is given by c„ = M ? + hK + ■■■ + kN+1\nN+1,

(1.24)

with arbitrary constant coefficients, and the initial-value problem for Eq. (1.20) may be posed at any N + 1 consecutive points. Thus we consider the existence and uniqueness of the solution to Eq. (1.20) for t > 0 satisfying the conditions x(i) = a,

i = 0,-l,...,-N.

(1.25)

Then letting n = 0,-1,...,—TV and c„ = x(n) in (1.24) we get a system of equations with Vandermonde's determinant det(A]) which is different from zero. Hence, the unknowns &,- are uniquely determined by (1.25). If some roots of Eq. (1.23) are multiple, the general solution of Eq. (1.22) contains products of exponential functions by polynomials of certain degree. The limiting case of(1.21)asa—>0 gives the solution of Eq. (1.20) when a — 0. We proved: T h e o r e m 1.13. Problem (1.20), (1.25) has on [0, oo) a unique solu­ tion. This solution cannot grow to infinity faster than exponentially as t —* +oo. T h e o r e m 1.14. The solution x = 0 of Eq. (1.20) is asymptotically stable as t —> +oo if and only if |A,|<1,

i = l , 2 , . . . , i V + l.

18

1. PIECEWISE CONTINUOUS ARGUMENTS

For the problem x'(t) = Ax(t) + A0x{[t\) + AlX([t - 1]), x(-l) = c_i, x(0) = co

(1.26)

in which A, AQ, A\ are r x r-matrices and x is an r-vector, let xn{t) designate the solution of Eq. (1.26) on [n, n+1) satisfying the conditions x{n — 1) = c n _i, x(n) = c„. Then, if A is nonsingular, xn(t) = ( e ^ - " ) + (e^"") - t)A-lM)

cn + (gMf-n) _ / ) A - % c _ i

(1.27)

Taking into account x n (n + 1) = x„ + i(n + 1) = c„ + i, we obtain the recursion relation C+i = B0c„ + Bicn-i,

n>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
all solutions of the nonhomogeneous equation x'(t) = Ax(t) + f(t) are bounded on [0, oo). In general, this is not true for Eq. (1.30) which is a nonhomogeneous equation with a constant free term on each inter­ val [n,n + 1). For instance, the solution x(t) = (1 + (1 - e-W)6)(l + (1 - e - 1 ) ^ of the scalar problem x'(t) = -x(t) + (1 + 6)x([i\),

x(0) = 1, 6 > 0

1.2. SOME GENERALIZATIONS

21

is unbounded on [ 0, oo) since

|*(*)l>(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

-k + l)ds

+ t eA{t-s)Bx0{s J[t]

- [t]) ds,

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(<) = e^(' +n - l ) C _ n+ i + / '

e^'- s »Bi 0 (s + n - 1) ds.

J—n+l

Since x_„(—n + 1) = x_„+i(—n + 1) = c_„+i it follows that c_„+1 - e V „ + /_~"+1 e ^ - n + 1 - s ) £ x 0 ( s + n) ds, whence c_n = e~Ac-n+i - J " This formula leads to (1.37).

eA<>~n~s)Bx0(s + n) ds.

■

The Cauchy problem (1.31), (1.32) is correctly posed on [0,oo) if for any Co G *D(A) it has a unique solution, and this solution depends continuously on the initial data in the sense that if x„(0) —> 0, x n (0) G T>(A), then xn(t) —» 0 for the corresponding solution at every t G [0, oo). If the Cauchy problem for (1.31) is correct, its solution is given by the formula x(t) = U(t)co,

c0 G T)(A),

23

1.2. SOME GENERALIZATIONS

where U(t) is a semigroup of operators strongly continuous for t > 0. For many applications it is necessary to extend the concept of solution of the Cauchy problem. A weakened solution of (1.31) on [0,oo) is a function x(t) which is continuous on [0,oo), strongly continuously differentiable on (0, oo) and satisfies the equation there. By a weakened Cauchy problem on [0, oo) we mean the problem of finding a weakened solution satisfying the initial condition x(0) = c$. Here the element Co may not already lie in the domain of the operator A. Thus, the de­ mands on the behavior of the solution at zero are relaxed. On the other hand, we require the continuity of the derivative of the solution for t > 0. However, for a correct Cauchy problem this requirement is automatically satisfied [144]. Theorem 1.20. Suppose that Eq. (1.30) with linear constant opera­ tors A and B satisfies the following hypotheses: (i) The operator A is closed and has at least one regular point, the domain T^>{A) is dense in E. (ii) The weakened Cauchy problem for (1.31) is correct on ^(A). (iii) T>(B) D V(A) and Bx G D(A), for any x G T>(A). Then on [0,oo) problem (1.30), (1-32) has a unique solution x(t) = {u(t-[t))

+

J^U(t-s)Bds^ x n

(u(l) + JkCl

k=[t] V

U k

(

- s)Bds)

~

°o ■ (1-38) i

P R O O F . If xn{t) is a weakened solution of (1.30) on [n,n + 1) then, by virtue of (i) and (ii), it can be represented in the form

xn(t) = U{t - n)cn + fn U(t - s)Bcn ds,

(1.39)

where c„ = x(n). If cn G D(A), the first term in the right of (1.39) is a solution of (1.31). Since the term Bc„ is constant the integral in (1.39) really yields a weakened solution of (1.30). Furthermore, from (iii) we have Bc„ G 1){A). Hence, this integral gives a particular solution of (1.30), and the assumption c„ G T>(A) enables us to maintain

24

1. PIECEWISE CONTINUOUS ARGUMENTS

that (1.39) is a solution of (1.30) in the given interval. If c„_i G 'D(A), then a;„_i(0 = U(t-n

+ l)c„_i + /

U(t-

s)Bc„_i ds,

Jn—\

xn-i(n-

1) = c„_i

is a solution of (1.30) for n — 1 < t < n. The recursion relation c„ = (u(l) + £_x U(n - s)B ds) c„_! which follows from the requirement x„_i(n) = xn{n) leads to (1.38). Since Co G ^(A) and zoW is the solution of (1.30) on the interval [0,1], then c\ = £o(l) £ 1)(A). Now we consider the solution x\(t) on [1,2] and see that ci = ii(2) G D(^l)- Continuing the solution to the right we conclude that cn G U(-A) for any n. ■ If the correctness of the Cauchy problem is not known beforehand, we may resort to certain restrictions on the resolvent of the operator A in order to prove the existence and uniqueness of the solution. There are various conditions of this kind. We follow the approach suggested in [198]. Let a be any real number, 9 G [0,7r/2], and let O = fi(o;, 6) be the domain that contains the semiaxis (a, +oo) and is bounded by two rays from the point (a, 0) forming angles 6 and — 6 with the negative direction of the real axis. The operator A is called an abstract elliptic operator, if there exists a constant fi > 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

*'(<) = E A-:c(* - *[*])

(1.40)

t=0

with linear constant operators A,-: 1>(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
k + 1).

Theorem 1.22. Suppose that Eq. (1.40) satisfies the following hy­ potheses: (i) The Cauchy problem for (1.41) is correct. (ii) The Cauchy problem for (1.42) is uniformly correct. (iii) The values /,-(*) G ?>(A), the functions fi(t) and A0fi(t) are con­ tinuous.

1. PIECEWISE CONTINUOUS ARGUMENTS

26

Then the initial-value problem for (1.40) has a unique solution

where U(t) is the semigroup operator generated by (1.42)

PROOF. For t E [n, n

+ I), Eq. (1.40) takes the form N

xl(t) = Aox(t)

+ Alxo(t - n) + C Ai4(t - in), i=2

which.also can be written as ~ ' ( t= ) Aox(t)

+ fn(t).

(1.45)

By virtue of (ii) and (iii) and Theorem 6.5 [144], the formula

yields the solution of (1.45) with the condition xn(n) = cn. On the interval n - 1 5 t < n we have xn-l(t) = U(t

-n

+ l)cn-1 + /

t

n-1

U(t - S)fn-1(s) ds,

where x n - ~ ( n- 1) = cn-1. Therefore, the relation xndl(n) = c, gives Cn

+ R1U(n - S)fn-1(s) d ~ .

= U(l)~n-1

F'rom here,

+

Jn

n- 1

U(n - s)fn-~(s)ds.

The property U(tl)U(t2) = U(tl

+ tz),

0

I tl,

t2

< 00

27

1.2. SOME GENERALIZATIONS

implies C = f/(2)c„_2 + / " _ 1 U(n - s)/„_ 2 (s) ds + f , t/(n - s)/„_i(s) ds. Continuing this procedure leads to the formula " - 1 /-t+i

c„ = £/(n - l)ci + E A tf(" - «)/*(«) <*s,

n> 1

which together with (1.46) proves the theorem. Since xo(t) is the solu­ tion of (1.41) where (0) G D(A), then Cj = x 0 (l) G T>(A). ■ Theorem 1.23. If the operators Ai G £ ( E , E ) , the initial-value prob­ lem for (1.40) /ias a unique solution x(t) = eA^-^eA<j>(0) + £

/ i + 1 e ^ C - ^ / ^ s ) ds

+ j£]C4>('-')/w(s)«fo, where A = Eilo-^"' z 0 (s) = eAs{0).

anc

,>

x (147)

^ ^ e functions /t(s) ore defined by (1.43) ioi£/i

It is well known that the solution of the scalar differential-difference equation x'(t) = Ax(t) + Bx(t - T), x{t) = <j){t), te[-r,0] satisfies the exponential estimate \x(t)\ < a\(f>\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)\
(k
+ l),

A; = 1,2,...

28

1. PIECEWISE CONTINUOUS ARGUMENTS

then solution (1.44) of the initial-value problem for Eq. (1.40) satisfies the inequality \x(t)\ < Le"\

t>0

(1.48)

where L and LJ are some positive constants. PROOF.

The estimate \\U(t)\\ < Mew\

t>0,

(M,w>0)

holds for the semigroup U(t) generated by a uniformly correct Cauchy problem. Therefore, it follows from (1.44) that /* + 1 «*"('-> ds

|x(*)| < M|zo(l)|e-"e"" + CM £

+ CMj[t\f\e^'sUs,

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
(1.50)

where [•] designates the greatest-integer function, A and Aj are constant r X ^-matrices, and x and Cj are r-vectors. In contrast to the situation

1.3. EPCA OF ADVANCED, MIXED, AND NEUTRAL TYPES

29

with general functional differential equations, the fact that (1-49) con­ tains both retarded and advanced arguments does not pose particular difficulties. Let M0(t) = eAt + (eAt - / ) A - % , At

Mj{t) = (e

- I)A-%,

B1 = Mi(l)-I,

(1.51)

j = ±1,±2,...,±N Bj=Mj(l),

j^l.

(1.52) (1.53)

Theorem 1.26. If the matrices A and B±N are nonsingular, then problem (1.49), (1.50) has a unique solution on [0,oo). This solution cannot grow to infinity faster than exponentially. PROOF. Let xn(t) be the solution of (1.49), (1.50) on the interval [n, n + 1 ) . If we let c„ = x(n), for integer n, then we have the equation N

x'„(t) = Axn(t) + £ AjCn+j j=-N

with the solution x„(f) = eyl(<-")cn + ( e ^ - " ) - 7 ) E

A-lAjCn+h

j=-N

which can be written, by virtue of (1.51), (1.52), as xn(t)=

£

Mj{t-n)cn+j.

(1.54)

j=-N

Prom (1.54) we can see that it suffices to know the constants c„ in order to determine x(t). Taking into account xn(n +1) = xn+i(n +1) = c„ + i, we obtain N

c„+i = £

Mj(l)cn+j,

n > 0.

j=-N

With the notations (1.53), this equation takes the form £ j=-N

BjCn+j

= 0.

(1.55)

30

1. PIECEWISE CONTINUOUS ARGUMENTS

Its particular solution is sought as cn = X"k, where k is a nonzero vector. Then det I £ B-N+j\>\

= 0.

(1.56)

Eq. (1.56) has 2Nr nontrivial solutions if det B-N ^ 0 and det BN ^ 0. Assuming that these roots are simple we can write the general solution of (1.55) 2Nr Cn

=

2—i "j * i i j=l

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 initial conditions (1.50). Letting n = — N,..., N — 1 and cn = x(n) in (1.50), we get a system of equations with Vandermonde's determinant det(A') which is different from zero. Hence, the unknown vectors kj are uniquely determined by initial conditions (1.50). If some roots of (l.56) are multiple, then m

Cn = £ A?py(n),

m < 2Nr

(1.57)

J=I

where the components of the vectors Pj(n) are polynomials of degree not exceeding m — 1. Substituting the vectors c„ in (1-54) yields the solution of problem (1-49) o n n < K n + l. This concludes the proof of the assertion, which generalizes Theorem 2.4 in [54]. ■ REMARK 3. Another way to interpret Theorem 1.26 is as follows. We see that the values cn = x(n) satisfy an autonomous difference equation, and so there is an underlying discrete dynamical system that determines the solution for real t. REMARK 4. The requirement that B-N be nonsingular is nonessential. If B-N is singular, solution (1.54) on [0,oo) depends on, at most, 27V - 1 initial conditions x(j) = cjy -(N - 1) < j < N - 1. If B-x is nonsingular, the solution of problem (1.49), (1.50) has a unique backward continuation on (—oo,0). Prom formulas (1.54) and (1.57) it follows that the solution x = 0 of Eq. (1.49) is globally

1.3. EPCA OF ADVANCED, MIXED, AND NEUTRAL TYPES

31

asymptotically stable as t —► +00 if and only if the roots of (1.56) satisfy the inequalities j = l,...,2Nr.

IAJ-|<1,

(1.58)

Consider the scalar initial-value problem x'(t) = / (x(t), {x([t + j])}f=_N), x

c

(j) = h

(0 < t < 00)

~N<j

(1.59)


where / is continuous in the space of its variables. Denote » = {N}?=-N. For the study of (1.59), it is useful to consider the ordinary differential equation with parameters f

=/(*,**)•

(1-60)

If f(x,fi) is continuous and different from zero on a set fl, then on fl there exists a general integral F(x,fi) = t + g(fi), with an arbitrary function g(fi). Assume that the solutions of (1.60) can be extended over all t > 0, and let x(t) be a solution of Eq. (1.59) on t > 0 satisfying the conditions x(n + j) = cn+j,

-N<j
Let xn(t) be the restriction of x(t) to the interval [n,n + 1]. Then we have the equation dxn _ f(xn,cNn), dt

n
+l

where CNn

If ft =

CJVTU

{cn+j},

-N<j<

then F(x„,cNn) = t + g(cNn).

N.

(1.61)

32

1. PIECEWISE CONTINUOUS ARGUMENTS

For t = n this gives F(cn,cNn)

= n + g(cNn)

and F(xn,cNn)

- F(cn,cNn)

= t - n.

(1.62)

Now xn+l(n + l+j)

= cn+1+j,

-N<j
Since xn(n + 1) = xn+i(n + 1), we put t = n + 1 in (1.62) and get F(cn+i,cNn)

- F(cn,cNn)

= 1.

This equation can be written as

f

/•c„+i

Jc„

dx

,, .. , = 1 , f[X,CNn)

(1-63)

which also follows directly by integrating (1.61) from t = n to t — n+1. If the difference equation (1.63) has a unique solution cn+N = h(cn-N,cn-N+1,...

,c„ + w_i),

n>0

with respect to cn+pi, initial conditions (1.59) allow the successive deter­ mination of the values c n+ # for all n > 0. Then we substitute the values c n(n > —N) in (1.62) and find xn(t), the solution of problem (1.59) on n < t < n + 1. We may always suppose / ( x , c ^ n ) ^ 0, x G (c„,c n+ i) since the assumption f(c,c^„) = 0 for some c 6 (c„,c n+ i) leads to the conclusion that the constant function xn(t) = c is a solution of Eq. (1.59) on the interval n < t < n + 1. However, this solution does not satisfy the condition xn(n) — c„ because c ^ cn. On the other hand, if /(c„,c^„) = 0, then xn(t) — c„ is a solution of Eq. (1.59) on (n, n + 1). By virtue of the condition xn(n + 1) = xn+\(n + 1) = c n + i, we have c„ = c„ + i. But in this case Eq. (1.63) is impossible. This concludes the proof of the following: Theorem 1.27. Assume that (1.60) satisfies existence and unique­ ness conditions everywhere and its solutions can be extended over [0, oo). 7/(1.63) has a unique solution with respect to cn+N, then on [0, oo) there exists a unique solution of problem (1.59).

33

1.3. EPCA OF ADVANCED, MIXED, AND NEUTRAL TYPES

Thus, for the scalar Eq. (1.49) with constant coefficients a and a,j, Eq. (1.63) takes the form /•Cfi+i

Jcn

dx

ax + E/L-jv aj°n+j

that is, N

N

ea(acn + £

a,jCn+j) - {acn+x + £

j=-N

a,jCn+j) = 0,

a ^ 0.

j=-N

For N >2, the coefficient of cn+N is (e" — l)ajv. For iV = 1, we have ea(acn + a_ic„_i + a 0 c n + aic n+ i) - (ac n+ i + a_ic„_! + a0c„ + aic n + i) = 0, and here the coefficient of c„+i is (ea — l)ai — a. Therefore, the hy­ potheses a,N ^ 0, N > 2, and ai ^ a/(e a — 1), N = 1, will figure in the existence-uniqueness theorem for the scalar version of (1.49). The scalar initial-value problem x'(t) = ax(t) + aox([t]) + alx([t + l\),

x(0) = c0

(1.64)

with constant coefficients was studied by Shah and Wiener [244]. It has on [0, oo) a unique solution x(t) = (m<,({0) + Am1({i}))AWc0,

(1.65)

a

if a\ ^ a/(e — 1), where {t} is the fractional part of t. The functions mo(i) and mi(t) are defined in (1.2), and

\--taA_

i-6r

with fe0, h from (1.2). The solution of (1.64) has a unique backward continuation on (—oo,0] given by formula (1.65) if bo ^ 0, that is, a0 ± -aea/(ea - 1). If b0 # 0,&i ^ 1, and m0({t0}) + Ami({t0}) ± 0, then the problem x(t0) — Xo for Eq. (1.64) has a unique solution on (—oo, oo). If b\ = 1, two possibilities may occur: the case bo ^ 0 implies x(t) = 0, and for bo = 0 problem (1.64) has infinitely many solutions. The solution x = 0 of Eq. (1.64) is stable (respectively, asymptotically stable) as t —► +oo, if and only if |A| < 1 (respectively, |A| < 1).

1. PIECEWISE CONTINUOUS ARGUMENTS

34

T h e o r e m 1.28. The solution x = 0 of Eq. (1.64) is stable (respec­ tively, asymptotically stable) as t —> +oo, if and only if (a + a0 + ai) la{ - a0 - ° ^ —j- J > 0

(1.66)

(respectively, > 0). PROOF. The inequality |A| < 1 can be written as -1 < If 1 - by > 0, then

bo 1 -h

< 1.

hh---11 < 60 b0 < 1 ---61 &i and 6i + 6 o < l , 6i-6o
< 1 - ea,

that is, a + a0 + ay<0,

(1.68)

and a _1 (ai - a0)(ea - 1) < eu + 1 which is equivalent to a(ea + l) ay-ao-^—^KO. If 1 - by < 0, then 6i+&o>l,

6I-&0>1-

(1.69)

1.3. EPCA OF ADVANCED, MIXED, AND NEUTRAL TYPES

35

These inequalities imply 61 > 1. Therefore, we consider only 1, a - l oi(c" - 1) - ea - a -1 ao(e° - 1) > 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\


(- + ?ri)(«'-Hrr) >a

< 1J0 >

// (1.70) is not satisfied and ao ^ —aea/(ea — 1), CQ ^ 0, then solu­ tion (1.65) has no zeros in [0,00). PROOF. For A ^ 0,Co ^ 0, the equation x(t) = 0 is equivalent to m0({t}) + Ami({<}) = 0. Hence, mp({t}) _ 6p w»i({*>) ~ 61 " 1 and eaW + g-1ao(e°^> - 1) ea + g-1a0(ea - 1) a ^ a i ^ W - 1) ~ a~lai(ea - 1) - 1 '

36

1. PIECEWISE CONTINUOUS ARGUMENTS

It follows from here that a{t)

_

a

o + aie" a + a 0 + ai

If a > 0, then

1 < a ° ± aiC" < e« a + ao + ai

and, by virtue of b\ ^ 1, the equality sign on the left must be omitted. The case a + ao + ar>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

<^-y.

,

(1.73)

Again, the inequality a + ao + ai < 0 follows from (1.73). The case a < 0, together with b\ ^ 1, gives e° <

ao + aie " < 1, a + ao + ai

and assumption (1.71) yields (1.72). And if a + ao + a\ < 0, then we obtain (1.73). Hypothesis (1.70) can also be written as &o(&i — 1) > 0 which is equivalent to A < 0. ■

1.3. EPCA OF ADVANCED, MIXED, AND NEUTRAL TYPES

37

COROLLARY 1.3. In each interval (n,n + 1) the solution

«04*(1+s)_a)(,(1+s)_a)\

(,74)

of (1.11) satisfying the condition x(0) = CQ ^ 0 has precisely one zero + . 1 I a° t„ = n-\—a In -a + ao if a0 < - ^

T

.

(1.75)

7/(1.75) is noi satisfied, solution (1.74) has no zeros in [0,oo). From the last two theorems we obtain the following decomposition of the space (a,ao,ax).

(0„ + _£L)( 01 - ao _^±I>) >0 ,

(,76)

then solution (1.65) with CQ ^ 0 has precisely one zero in each interval (n, n + 1) and lim^ + 0 O x(<) = 0. In fact, the solution possesses the required properties iff —1 < A < 0, that is A(A+1) < 0. With the notations (1.2), we have b0(b0-bi+l) < 0 which yields (1.76). 2. If / a>ea \ (a + a0 + a{) \a0 + ^ — j j < 0, (1.77) solution (1.65) with CQ ^ 0 has no zeros in (0, oo) and lim x(t) = 0. The properties take place iff 0 < A < 1. Hence, we consider the inequality A(A — 1 ) < 0 leading to (1.77). 3. If

«.-^)(«.-°»-^)«>. then solution (1.65) with c$ ^ 0 has precisely one zero in each interval (n, n + 1) and is unbounded on [0, oo).

38

1. PIECEWISE CONTINUOUS ARGUMENTS

The proof follows from the inequality A < — 1 which is equivalent to (61 - l)(6i - 60 - 1) < 0 and gives (1.78). 4. If (a + a0 + 01) (ai - ^ - y ) < 0,

(1.79)

solution (1.65) with CQ ^ 0 has no zeros in (0,oo) and is unbounded. Inequality (1.79) results from A > 1 which can be written as (6x - 1)(60 + 61 - 1) < 0. 5. If a(ea + 1 ) „ aea , n — — = 0, a + ——? 0, 0 ea — 1 ea — 1 then solution (1.65) with c$ ^ 0 has precisely one zero in each interval (n, n + 1) and is periodic of period 2 since, in this case, A = — 1. 6. If a(ea + l) aea n 01 - 00 — — = 0, a + ——= 0, 0 ea — 1 ea — 1 then ai = a/(e" — 1), and problem (1.64) has infinitely many solutions. 7. If a a+ + a0 + + cn en = — 0, 0, a
x(t) = (m0({t}) + mi({t})) c0 = a - 1 (ea{t\a + a0 + en) - a0 - ai) c0. 8. Finally, if ae

"

«

a

then x(tf) = mo(t)co, for 0 < t < 1, and x(tf) = 0, for £ > 1. For Eq. (1.11) we have the following decomposition of the plane (a,a0). 1. If a(ea + 1) aea ea - 1
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:(<) = mo(t)co on [0,1), and x(t) = 0on [l,oo). For a(ea + 1) ea — 1 the solution x(£) is periodic of period 2. Theorem 1.30. 77ie solution x = 0 of the equation x'(t) = ax(t) + a0x([i\) + axx{[t + 1]) + a2x([t + 2]),

(1.80)

a2^0 is asymptotically stable if (ao + ^ Y ) a

2

<0

(1.81)

and either of the following hypotheses is satisfied: (i)

(a + a0 + ax + a2) (ax - ^ — r ) < 0, a

/

(")

(

ai

\ /

- ^ T l J |-"o + a i - o 2 -

a(e a + l ) \

n

e a _ 1 1<0,

40

1. PIECEWISE CONTINUOUS ARGUMENTS

where the first factors in (i) and (ii) retain the sign ofa2. PROOF. The characteristic equation (1.56) corresponding to (1.80) is M 2 + (6i - 1)A + b0 = 0,

(1.82)

with bj = m ; (l) and m.j(t) from (1.21). It follows from (1.81) that D2 = (1 - 6j)2 - Ab0b2 > 0, hence the roots A1|2 = (1 - h ± D)/(2b2) of (1.82) are real. If a2 > 0, the inequalities |Aj| < 1 are equivalent to l-b1

+ D<2b2

(1.83)

and 1 - bi - D > -252.

(1-84)

Prom (1.83) we obtain &I + 2 & 2 > 1 ,

60 + 6i + 6 2 > l ,

and (1.84) yields 6i - 262 < 1,

-bo +

bl-b2
The inequalities b\ + 2b2 < bo + bi + b2 and —bo + b\ — b2 < b\ — 2b2 contradict to bo < 0 and b2 > 0 which are consequences of (1.81) and a2 > 0. Therefore, it remains to consider only b0 + bi + b2>l,

-bo +

h-b2
b2> l-bo-bu

b2>-l-b0

that is, + bi.

If 1 — bo — b\ > — 1 — bo + &i, then b\ < 1. In this case, a sufficient condition of asymptotic stability is b0 + bi + b2>l,

i>i < 1

which, in terms of the coefficients aj, coincides with hypothesis (i). If 1 — &o — ^i < —1 — bo + bi, then &i > 1, and (ii) follows from the inequalities &! > 1,

- 6 0 + h - b2 < 1.

The case 6Q > 0, b2 < 0 is treated similarly.

■

41

1.3. EPCA OF ADVANCED, MIXED, AND NEUTRAL TYPES

Theorem 1.31. The solution x = 0 of the equation x'(t) = ax(t) + a_ix([t - 1]) + a0x([t]) + axx{[t + 1]) is asymptotically stable if

( ai - e^l)

a l

~ <°

and either of the following hypotheses is satisfied: (i)

(Q + ao + a! + a_i) f a 0 + /

(")

aea \ (

a

a

-_

J < 0, a(ea + l)\

a

n

( ° + ^ r i j p - i - - i + e . _ i j <°-

where the first factors in (i) and (ii) retain the sign of a\ — a(e a — 1) _ 1 . Theorem 1.32. The problem x'(t) = Ax(t)+ x{j) = ch

£

A,-*([t + ;]) + /(<),

j =

(1.85)

-N,-(N-l),...,N-l

has a unique solution on [0, oo) if the matrices A and B±N are nonsingular and the vector f(t) is locally integrable on [0, oo). PROOF. The general solution of Eq. (1.85) o n n < £ < n + l i s xn(t)=

£

Mj(t-n)cn+j

j=-N

+ fe^t-^f(s)dS,

(1.86)

"

where the matrices Mj(t) are defined in (1.51) and (1.52). The con­ dition x„(n + 1) = x n + i(n + 1) = c n+ i leads to the nonhomogeneous difference equation £

BjCn+j

= fn,

j=-N

with n+i

fn^-f

/■n+l eA{n+l-s)f{s)ds Jn

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»|
\Xj\n-m

m=M+l

< L ( l - | A i | ) - 1 | A i r ^ + e\XA(l -\XA)-K

44

1. PIECEWISE CONTINUOUS ARGUMENTS

This proves that Sj„ and pn vanish as n —► oo.

■

Theorem 1.35. The solutions of (1.85) cannot grow faster than ex­ ponentially if f(t) has the same property. Consider the initial-value problem x'(t) = Ax(t) + £ AiZ®{[t]), x(0) = co (1.92) «=o with constant r x r-matrices A and A,- and r-vector x. If Ai = 0, for all i > 1, then (1.92) is a delay equation. For A ^ 0 and Ai — 0, (i > 2), it is of neutral type. And if A,- ^ 0, for some i > 2, (1.92) is an advanced equation. T h e o r e m 1.36. / / the matrices A and B = I — E ^ i AjA' -1 are nonsingular, then problem (1.92) on [0, oo) has a unique solution x(t) = E{{t})E®(l)co,

(1.93)

where E(t) = I + {eAt - I)S and S = A-lB-\A

+ AQ).

PROOF. Assuming that xn(t) is a solution of Eq. (1.92) on the interval n < t < n + 1 with the initial conditions x«(n+) = cm-,

t = 0,...,JV

we have x'n(t) = Axn(t) + £ AiCni.

(1.94)

i=0

Since the general solution eA^~n'c of the homogeneous equation corre­ sponding to (1.94) is analytic, we may differentiate (1.94) and put t = n.

1.3. EPCA OF ADVANCED, MIXED, AND NEUTRAL TYPES

45

This gives N

Cnl = Acn0 + J2 AiCni, i=0

cni = Acn>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

*(*) = V[n(t) II ^_i(fc)c0,

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;( [*<,]) and x(t) =

E({t})EU-M(l)x([t0}).

Furthermore, from (1.95) it follows that x([t0]) =

E-\{t0})x0.

It remains to substitute this result in the previous relation.

■

Theorem 1.40. Suppose in the equation x'(t) = Ax{t) + £ AiX(i)(t

- [t])

(1.96)

i=0

x (,) (0) = a,

i=

0,...,N-l

the matrix AN is nonsingular if N > 1, and I — A\ is nonsingular if N = 1. Then on [0,oo) there exists a unique solution x(t) = x0(t), !(*) = eAtco + £

0< t <1

/ , eA{t-s)f(* -k + l)ds + f[t]eA{-t-s)f{s-[t\)ds,

where xo(t) is the solution of the equation x'0(t) = Ax0(t) + £ xtf\0) = ck,

k=

Akxtf\t), 0,...,N-l

and

/(r) = E^4fc)(0-

t>\

(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<1 r

x(t) = eA^x0(l) + E /. , + CeA^g[t](s)ds, 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
+ l,

k = 1,2,....

The solution x0(t) of (1.100) on [0,1) satisfies the equation x'0(t) = Bxo(t), a;o(0) = <^>(0), from which we obtain the first part of (1.101). For t G [n,n + 1), Eq. (1.100) takes the form PROOF.

x'n(t) = A00xn(t) + A0ix'n(t) + AiQx0(t -n) + Anx'0(t - n) N

N

+ £ Ai0(j>(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<0.

P R O O F . We observe that the functions gk(t) in (1.101) are bounded since they involve the values of xo(t) and x'0(t) only on 0 < t < 1 and the values of <j>(t) and '(t) for t < 0. Hence, \gk(t)\ < K, and we also use the estimate Ije^'l) < Meut to obtain

\x(t)\ <

M\x0(l)\e-Ueut

+ KM\^i[gjlie^Us +

J^Usy

t> 1.

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

+ M2eui £ e~»k + M3e«") \
proves the assertion. Furthermore, it is easy to see that if, in addi­ tion, x'(t) — Ax(t) is asymptotically stable as t —> oo, then (1.101) is bounded on [0,oo). ■ 4. Asymptotic Behavior of Linear EPCA with Variable Coefficients Along with the equation x'(t)=A(t)x(t)

+ B(t)x([t}),

0
(1.103)

we also consider x'(t) = A(t)x{t). (1.104) If A(t) is a bounded linear operator strongly continuous in t, the Cauchy problem x(0) = c0 G E

(1.105)

52

1. PIECEWISE CONTINUOUS ARGUMENTS

for (1.104) has a unique solution. The value of the solution x(t) to (1.104), (1.105) at the moment t is denoted as x(t) = U(t)c0. The operator U(t) can be considered as the solution of the Cauchy problem U'(t) = A(t)U(t),

U(0) = I

(1.106)

for the differential equation in the space £ ( E , E) of bounded operators acting in a Banach space E. For each t there exists a bounded inverse operator U~l(t). The solution of the generalized Cauchy problem x(s) = c0,

s e [0, oo)

for (1.104) is represented in the form x(t) = U(t)U-\s)c0

= U(t,s)c0,

where U(t, s) is the evolution operator. Theorem 1.45. If A(t),B(t) G £(E, E) and are strongly contin­ uous on 0 < t < oo, then there exists a unique solution of prob­ lem (1.103), (1.105) given by formula x(t) = U(t) (u-\[t))+j^

U-\s)B(s)ds}

c[t],

(1.107)

where c[t] = II U(k) (u~\k k=[t]

V

- 1) + / * U-\s)B{s)

ds) co. (1.108)

*_i

/

PROOF. For n < t < n + 1 we have the equation x'(t) = A(t)x(t) + B(t)x(n). Its solution x„(t) satisfying the condition x(n) — c„ is given by the expression *»(*) = Un(t) (l + J*U;\s)B(s)ds)

c„,

1.4. LINEAR EPCA WITH VARIABLE COEFFICIENTS

53

Un(t) being the solution of Eq. (1.106) such that U(n) = I. From x n _i(n) = c„ we obtain the relation c„ = L7n_j(n) (/ + £_ x U-l^Bis)

ds) c„_!.

To prove (1.108), it remains to observe that £/,(*) = U{t)U~l(i).

■

We can restate this theorem also in the case of unbounded operators. Theorem 1.46. / / the closed operators A(t) and B(t) are strongly continuous for all 0 < t < oo and have the common constant domain D which is dense in E, the Cauchy problem for (1.104) is uniformly correct and B(t): D —► D, then (1.107) yields the unique solution of problem (1.103), (1.105) with c0 G V. Theorem 1.47. If A(t),B(t) G £(E,E) and are strongly continuous on 0 < t < oo, then solution (1.107) satisfies the estimate \x(t)\ < et<+1W<) (b(t) + 1 ) M [col,

(1-109)

where a(f) = max||A(s)||, PROOF.

b(t) = max||fi(s)||,

0 < s < t.

The operator Uk(t) =

U(t)U~\k)

is the solution of Eq. (1.106) with the condition Uk(k) = I. Therefore, it is the sum of the convergent in the operator sense series Uk(t) = I+i A(s) ds+ I A{s) ds f A(si) dsi + ■ ■ ■ Jk

Jk

Jk

whence

||trt(*)|| < e«-kw\ and turning to (1.107) we find \x(t)\<{e^^

+

b(t)J^-^ds)\c[t]\.

Hence,

\x(t)\<(b(t) +

iyv\c[t]\.

54

1. PIECEW1SE CONTINUOUS ARGUMENTS

Since \\U(k)U~\k-l)\\ = \\Uk-1(k)\\<ea{t\ \\U{k)U-\s)\\ < e (*- s H'), we obtain from (1.108) [t]

,

\c[t]\ < n{eait)

V

,

+ b(t)Jk_/k-^dsj\c0\

< eU«M(b(t) + l)W|q,|.

Finally, |x(t)| < ef'+1l-«(6(t) + l) lH1] |co|. In particular, if the operators J4(£) and S(t) are uniformly bounded on [0,oo), then \x(t)\ < Mexp(a* + iln(6 + l)) with some constant M, where a = sup||A(t)||,

& = sup||B(*)||,

0
In this case, the conclusion about the solution growth is the same as for Eq. (1.30). ■ T h e o r e m 1.48. Suppose the equation x'(t) = A(t)x(t) + f{t, x(t - [*]), x(t - 2[t]), ...,x(t-

N[t})) (1.110)

satisfies the following hypotheses: (i) The operator A(t) G £ ( E , E ) is strongly continuous on R+ = [0,oo). (ii) The mapping / : R + x E ^ —» E is continuous on the direct product o/R+ and N copies o/E. (iii) The function (p: R_ —► E is continuous on ]R_ = (—oo,0]. (iv) The solution of the equation x'0(t) = A(t)x0(t) + f(t, x0(t), x0{t),..., exists on 0 < t < 1 and is unique.

x0(t)),

x 0 (0) = # 0 )

1.4. LINEAR EPCA WITH VARIABLE COEFFICIENTS

55

Then the initial-value problem x(t) = <j>{t),t 6 R_ for (1.110) has a unique solution x(t) = U(t) (u-\l)x0(l)

Jkk+1 U-\s)fk(s)

+ g

ds

+ j{t]U-1(s)f[t](s)dS)j,

t>\

(1.111)

where U{t) is the solution of (1.106) and fk(s) = f{s,x0(s PROOF.

- k),{s - 2k),...,(s-

Nk)).

Eq. (1.110) on n < < < n + 1 assumes the form x'{t) = A(t)x(t) +

fn{t).

Its solution xn(t) that satisfies the condition x(n) = cn is x„(i) = Un(t) [cn + fn U-\s)fn{s) and Un(t) = U(t)U-\n).

dsj ,

Hence

xn(t) = U(t) (u-\n)cn

+ J* U-\s)fn(s)

ds) .

The same formula for xn_i(tf) together with x„_i(n) = x„(n) yields the result - 1 ) ^ ! + £_% l7- 1 (s)/„_ 1 (s) ds)

c„ = U(n) (u-\n which leads to (1.111).

■

For the equation of mixed type x'(t) = A(t)x(t) + £

Aj^xdt

+ j]),

( 0 < * < o o ) (1.112)

j=-N x{j)

= Cj,

-N<j
N>2

56

1. PIECEWISE CONTINUOUS ARGUMENTS

the coefficients of which are r x r-matrices and x is an r-vector, let B0n(t) = U(t) (u-\n)

+ fn U-\S)A0(s)

Bjn(t) = U(t) I* U-\S)Aj(S) Jn

ds,

ds) ,

j = ± 1 , . . . , ±N. (1.113)

T h e o r e m 1.49. Problem (1.112) has a unique solution on 0 < t < oo if A(t) and Aj(t) G C[0, oo), and the matrices Bffn{n + 1) are nonsin­ gular, for n = 0 , 1 , . . . . PROOF. On the interval n < t < n + l we have the equation N

x'(t) = A(t)x(t) + £

Aj(t)cn+j,

cn+j = x(n + j).

j=-N

Its solution xn{t) satisfying the condition x(n) = c„ is given by the expression xn(t) = U(t) (u-\n)cn

+ fnU-\s)

£

A^cn+jds)

.(1.114)

Hence, the relation xn(n + 1) = xn+\{n + 1) = cn+\ implies c n + i = U(n + 1) (u-\n)cn

+ £+1 U~\s)

£

Aj(s)cn+j ds\ .

With the notations (1.113), this difference equation takes the form BNn{n + l)cn+N -\ h B2n(n + l)c n + 2 + ( 5 l n ( n + 1) - I)cn+1 + B0„(n + l)c„ + 5_ l n (n + l)c„_i + --- + 5_ Arn (n + l)c„_vv = 0, n > 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

£„,,_,« = «• + £ , <*"->«<,(•) & < e- + Mo(e""1), a and D

/.v

, B i i(0

°' -

a(ea-l)

a

>

aqi(ea-l)

i

a(e' + l - g l ) ( e " - l )

e

^ 3 i j — = -a - »)•

Hence, |5 0 ,,_i(0l < 9(1 - gi),

|1 --B l t j_i(i)| > 1 - «i,

0
and

M < ?,

*' > i-

On the other hand, for i < j we use the inequalities M(ea - 1)L \Bo,i-i(i)| < e + —i =pu a

M = sup |o 0 (t)\,t G [0, oo)

and M < i = p. l-gi The boundedness of the function |ao(£)| on [0, oo) is a result of its continuity and of (1.120). To finish the proof, we write

M = tain M "n M < ^'"Y^'tal, :'=1

« > j +1

i=j

and here the second factor vanishes a s n - > oo. The method of proof also suggests the following:

62

1. PIECEWISE CONTINUOUS ARGUMENTS

Theorem 1.53. The solution of (1.119) tends to zero as t —► +oo if the coefficient a < 0 is constant, the functions ao(t), a\(t) G C[0,oo), &l(t) < 0 for all t G [0, oo) and, starting with some to > 0, a(ea + l) ^ _, —^ a —- <m0< M0< e —1

-a.

Theorem 1.54. The solution of the scalar equation N

x'(t) = ax(t) + £ ai(t)x([t - i]), x(i) = C{,

(a = const.)

(1.121)

i = 0 , . . . ,N

tends to zero ast —> oo if a < 0, a,(r) G CfiV, oo), and l i m / - ^ o,-(t) = 0, forO
a

% + f[t] e<*-) ( g a,(5) C[< _^
(1.122)

where c

w = ( ea+ i ea( '" )a °( s ) ds )vi] + g ( ( 1 1 e a ( 1 H i W ^ i . W ] 1 <>^-

Take a number <5 > 0 such that ea + (N+l)6

=

q
There exists a number T such that, for t > T, \ai{t)\<6,

i=

0,...,N.

Then |c [ ( ] |<(e a + 5)| C[t _ 1] | + ^ | c M _ 1 ] | .

(1.123)

1.4. LINEAR EPCA WITH VARIABLE COEFFICIENTS

63

To evaluate e™, we employ the method developed in [287] for the study of distributional solutions to linear functional differential equations. Denote M[t] = max \CJ\,

0 < j < [t].

From (1.123) it follows that |c w | < g M M ] ,

t > T.

Hence, Mjt] = Afp_i], and starting with some k, Mn = Mk,

n> k.

(1-124)

The application of (1.124) to (1.123) successively yields |c*+i| < qMk, \ck+2\ < qMk,..., \ck+N+i\ < qMk. Now we put [t] = k+N+2,..., to obtain \ck+N+2\ < q2Mk,

\ck+N+3\

k+2N+2 and use the latter inequalities < q2Mk,...,|cjb+2jv+2|

<

q2Mk.

Continuation of the iteration process shows that \ck+{j_l)N+i\

< q3Mk,

for all natural j and i = j , j' + 1 , . . . , j + N. This implies Cm —> 0 as t —> co, and the proof follows now from (1.122). ■ T h e o r e m 1.55. The solution of (1.112) tends to zero as t —+ +oo if the following hypotheses are satisfied: (i) All matrices Bjn(t) in (1.113) are continuous and their norms are uniformly bounded on [0, oo) for n > 0 and —N < j < N — 1. (ii) The matrices Bflfn(n + 1) are nonsingular for all n > 0 and the norms ||5^J,(n + 1)|| are monotonically decreasing to zero as n —> co. (iii) The norms of BN„(t)Bjfln(n + 1) are uniformly bounded for all n > 0 and n < t < n + 1.

64

1. PIECEWISE CONTINUOUS ARGUMENTS

PROOF. The solution of Eq. (1.112) on the interval n < t < n + 1 is given by the formula N

%n(t) = £

Bjn(t)cn+j,

xn(n) = c„.

j=-N

Since the functions ||-Bjn(OII are bounded for — N < j < N — 1, it remains to prove that |c„| —> 0 as n —► oo and that \Bjf„(t)cn+N\ —> 0 as 2 —* oo. The continuity of the solution implies x n (n + 1) = c n+ i and leads to the equation iV-l

£

% ( n + l)c„ + i ,

(1.125)

with S)i(n + 1) = B ^ ( n + 1)(/ - B l n (n + 1)), % ( n + 1) = - B ^ ( n + l)B j n (n + 1),

j£l.

-N <j
On the basis of the hypotheses, we conclude that

11^(^ + 1)11 < <5„ where 8n is monotonically decreasing to zero as n —> oo. Let 7„ = |c„|, then from (1.125) it follows that N-l

ln+N < 6„ J2 7n+jDenote Mn = max 7,-,

— JV < i < n.

Then j n + N < 2N8nMn+N_1. Starting with some natural k, 2N6n < 1,

j n + N < Mn+ft-u

(1.126)

1.4. LINEAR EPCA WITH VARIABLE COEFFICIENTS

65

and Mn+N = Mn+jv-i,

n> k.

Hence, Mn+N = Mk+N,

n>k.

(1.127)

The application of (1.127) to (1.126) successively yields lk+N+i < 2N6k+iMk+N,

jk+N+2 < 2N6k+2Mk+N, ■■■i

...

lk+ZN ^

2N6k+2NMk+N.

Since t h e sequence {<$„} is decreasing it follows that -yic+N+i+i < 2N6k+lMk+N,

i=

0,...,2N-l.

Now we p u t n = k + 2N +1, k + 2N + 2 , . . . , k + 4N in (1.126) and use the latter inequalities to obtain lk+3N+i < (2N6k+1)2Mk+N,

jk+3N+2

< (2N6k+l)2Mk+N,

• • ■,

... (2N6k+i)2Mk+N,

lk+5N <

that is, 7*+3^+i+i < (2N5k+1)2Mk+N,

i=

0,...,2N-l.

Continuation of the iteration process shows that lk+(2m+l)N+i+l

<

(2N8k+l)mMk+N,

for all natural m and i = 0 , . . . , 2./V — 1, and 2iV<5fc+i < 1 implies that 7fc+(2m+i)jv+i+i —> 0 as m —> oo. This proves that \cn\ —> 0 as n —» oo. By virtue of hypotheses (i) and (iii), the norms | | 5 j v n ( 0 ® i ( n + -Oil a r e uniformly bounded for — N < j < N—l, for all n > 0 and n < i < n + 1 . Therefore, multiplying (1.125) by B^n(t) shows that \BNn(t)cn+N\ —> 0 as t —► co, which concludes the proof. ■ COROLLARY

1.5. The solution of the scalar problem N

x'(t) = ax(t)+

£

aj(t)x([t+j]),

N>2

j=-N

x(j) = ch

-N <j


1. PIECEWISE CONTINUOUS ARGUMENTS

66

tends (i) (ii) (iii)

to zero as t —> +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<»+i-)Ai(s)da < e ,

j=0,±l

for sufficiently small e and large n, then

IK/ - «u.(n +1»- x |l < (1 - 0"1 and

|cn+i|
(1.130)

Inequality (1.130), in which qo

= (l-e)-\\\eA\\

+ e),

?1

= e(l - e)" 1 ,

holds for large n. Denote 7,- = |c,|,

Mn = max7i,

— 1 < i < n.

Then from (1.130), 7n+i < qM„, where q

= qo +

qi

= (l-e)-\\\eA\\

+ 2e).

For e < (1 - He^lD/3 we have q < 1 and 7„+i < Af„,

M n + i = M„.

Thus, starting with some natural k, Mn = Mk,

n> k.

The application of (1.131) to (1.130) successively yields 7
Continuation of this procedure shows that 7k+2m-i < qmMk,

i = 0,l

(1.131)

68

1. PIECEWISE CONTINUOUS ARGUMENTS

for all natural m, and proves the theorem since the functions Bj„(t) in (1.115) are bounded on [0,oo). ■ 5. Stability as a Function of Delay Cooke and Wiener considered in [20] stability of the null solution of equations of the form N

\

/r t I

x'(t) = ax(t) + £ aiX [ — ah- iah , (1.132) \\.ah\ ) t=0 where a and a, are real constants, h > 0, and a is a positive parameter. Special cases of (1.132) may arise in hybrid systems with a time delay h in applying feedback control. For the case h = a = 1, this equation coincides with (1.20), whose zero solution is asymptotically stable if and only if the roots of (1.23) have moduli less than one. We now direct attention to the way in which stability depends o n a l i , as well as on the coefficients. Let xn{i) denote a solution on the interval [nah, (n + l)ah). Then N

x'n(t) = axn(t) + J2 o.ix((n -

i)ah),

i=0

which has the solution xn(t) = xn{nah)ea{-t-nah^ + ( e a ( ( - na/l ) - 1) £ a - 1 a i x((n - i)ah). »=o Let x(nah) = cn. Then for continuity at (n + l)ah we obtain Let x(nah) = cn. Then for continuity at (n + l)ah we obtain N c n+1 = eaahcn + (eaah --l)Yl a~laiCn-ic n+1 = eaahcn + (eaah - 1) i=0 •£ a^atc^. i=0 Stability for this difference equation is governed by the characteristic Stability equation for this difference equation is governed by the characteristic equation /(A, ah, A) = XN+1 - b0{a)XN bN(a) = 0, (1.133) where b0(a) = eaah + (eaah - l j a " 1 ^ , bi(a) = (eaah - l)a-xai,

(i = l,...,N).

(1.134)

69

1.5. STABILITY AS A FUNCTION OF DELAY

The symbol f(X,ah,A) is used to indicate that / is a polynomial in A whose coefficients depend on ah and on the set of numbers A=

{a,a0,ai,...,aN}.

We now present a theorem on the stability of Eq. (1.132), which is anal­ ogous to the theorems of Datko [62], Hale, Infante and Tsen [116], and Silkowski [253], relating to stability of differential-difference equations. Theorem 1.57. Let h > 0 and assume a + X>,<0. (1.135) »=o Then there exists a maximal interval (0, ao), with 0 < ao < °°> such that all roots of f(X,ah,A) lie in |A| < 1 for all a € (0,aio)j and therefore the zero solution of (1.132) is asymptotically stable. PROOF. Since 60(0) = 1 and 6,-(0) = 0, then (1.133) has, at a = 0, an N-iold root at A = 0, and a simple root at A = 1. Consider the root A(a) such that A(0) = 1. This root depends continuously on a and is a differentiable function of a for small a. We have (N + 1 ) A " £ - bo(a)N\"-l£

- b'0(a)XN

- E KWX"-' - £ bt(a){N - O ^ - ' " 1 ^ = o. i=l

a a

1=1

Now b'Q(a) = h(a + a0)eaah,

b'^a) = haieaah

b'0(0) = h(a + a0),

b'i(0) = hai.

Setting a = 0, A = 1, we obtain I N dX = h a + E a»' da a=0 V i=0 By hypothesis (1.135), this is negative. Consequently, X(a) < 1 and X(a) is real for all sufficiently small positive a. Thus all roots of f(X,ah,A) = 0 lie in |A| < 1 for sufficiently small positive a. Since 6Q

1. PIECEWISE CONTINUOUS ARGUMENTS

70

and b{ depend continuously on a, so do the roots, and the existence of the maximal CXQ follows. ■ COROLLARY 1.6. Assume that (1.135) holds and also /(A, ah, A) ± 0,

for |A| = 1,

0 < a < oo.

(1.136)

Then (1.132) is asymptotically stable for every positive a. Indeed, since (1.135) holds, there exists «o such that all roots of the polynomial f(X,ah,A) = 0 lie inside the unit circle for 0 < a < a<). By (1.136), no root reaches or crosses the unit circle as a increases, and hence there is asymptotic stability for 0 < a < oo. COROLLARY

1.7. Assume that (1.135) holds and also

/(A, ah, A) ^ 0,

for |A| = 1,

0 < a < 1.

Then (1.132) is asymptotically stable for 0 < a < 1 and, in particular, for a = 1. For Eq. (1.132) with N = 0 and a = 1, that is, x'{t) = ax(t) + a^x I

t

h) (1.137) h we are able to precisely describe the stability region in the (a, ao) pa­ rameter space. This is compared with the stability region for the first order differential-difference equation with constant lag h. For (1.137) we have bQ = eah + (eak -

/(A, h,A) = X- b0,

l)a~la0.

The necessary and sufficient condition for asymptotic stability is

N < i. This condition is - 1
+ a0)-ao)

< 1

or - 2 < a-\eah

- l)(a + a 0 ) < 0.

1.5. STABILITY AS A FUNCTION OF DELAY

71

This is equivalent to 2a eah

_

l


(1.138)

The function F(a) =

-a-

2a

is readily seen to be increasing for a < 0 and decreasing for a > 0, with F(0) = —2/h. It is asymptotic to —a as a —► +oo, and to +a as a —v —oo, and lies below those lines. Therefore, the stability region defined by (1.138) has the appearance indicated in Fig. 1. We note that if a parameter is varied in such a way that the line a + ao = 0 is crossed, the root A = &o crosses + 1 , and nonoscillatory instability arises. If the lower boundary in Fig. 1 is crossed, the root A = &o crosses —1, and oscillatory instability arises. Observe that as h —> 0 + , the stability region expands to cover the region a + ao < 0, whereas as h —► +oo, the stability region reduces to a quarter plane. In fact, we have the following result. T h e o r e m 1.58. Eq. (1.137) is asymptotically stable iff (1.138) is sat­ isfied. The region of stability in the (a,ao)-plane decreases as h in­ creases. The region of stability for all delays h,Q < h < oo, is the set {(a,ao): a < 0,a < ao < —a}. The region in Fig. 1 may be compared with the stability region for the differential-difference equation x\t) = ax(t) + aox(t - h). This region is defined by g(a) < ao < —a, a = 1/h, where g is a certain function for which #(0) = ir/2h, g(l/h) — -1/h, and g(a) is asymptotic to the line ao = a as a —► —oo. This region is similar to that in Fig. 1, but for fixed h does not extend to the right of a = 1/h. The equation x'(t) — ax(t) + a0x I — ah)+aix(

—

ah-ahj.

72

1. PIECEWISE CONTINUOUS ARGUMENTS

was also investigated in [52] and the region of stability for all delays deduced from Corollary 1.6. 6. EPCA and Impulsive Equations Let {tn}, where n = 0 , 1 , 2 , . . . , be a strictly increasing sequence of real numbers such that limf„ = oo as n —► oo, and let g(t) be a function defined on 0 < t < +oo and constant on each interval [tn,tn+i): 9ni

' n < t < tn+l.

(1.139)

We shall call g(t) a step function. Assume, for simplicity, to = 0 and consider the initial-value problem x'(t) = f(x(t), x(g(t))),

x(0) = co,

(1.140)

where / is a continuous function of its variables. Definition 1.5. A solution of Eq. (1.140) on [0, oo) is a function x(i) that satisfies the following conditions: (i) x(t) is continuous on [0,oo); (ii) the derivative x'(t) exists at each point t £ [0, oo), with the pos­ sible exception of the points tn where one-sided derivatives exist; (iii) Eq. (1.140) is satisfied on each interval [tn,tn+i).

73

1.6. EPCA AND IMPULSIVE EQUATIONS

T h e o r e m 1.59. Assume that the ordinary differential equation with a parameter §-/(x,/i)

(1.141)

satisfies existence and uniqueness conditions in M? and its solutions can be extended over [0,oo), and let 0 < g(t)
for 0
oo.

(1-142)

Then on [0,oo) there exists a unique solution of problem (1.140). PROOF. On the interval 0 < t < t\ we have the equation x'(t) = f(x{t),x(g0)),

9o = g(0)

and since 0 < go < t < t\, then approaching t to zero gives go = 0 and produces the problem x'(t) = f(x(t),c0),

a;(0) = co.

Its unique solution xo(t) exists on [0, t\), and on the next interval [ti, £2) we have the problem x'(t) = /(*(<),*(5i)) s

x(t{) = x0(ti),

gx = g(h).

By virtue of (1.142), 0 < gi < h and x(pi) = xo(<7i)- Hence, x'(t) = f(x(t), x0(gi)),

x(ti) = *o(*r)»

and this problem has a unique solution on [
F(x,n) = t + C(n) of (1.141), with an arbitrary function C(/x). Let xn(t) be a solution of Eq. (1.140) on [tn,tn+i) satisfying the condition x(g„) = s„, where gn is defined in (1.139). Then we have the equation ^

= /(x„,4

(1.143)

74

1. PIECEWISE CONTINUOUS ARGUMENTS

If // = sn, then F(xn,sn)

= t + C(sn).

For t = tn this gives F(cn, s„) = tn + C(sn), where cn = x n ^ i n j .

Therefore, F(xn, sn) - F(cn, sn) = t - tn.

(1.144)

For t =
tn,

since Z n (*n+l) = Xn+l(tn+l)

= Cn+1-

This can be written as rcn+i

dx

J 77 ; = tn+i-tn, 1.145 Jen f(x, S„) and also follows directly by integrating (1.143) from t = t„ to t = tn+\. The above operations are justified since the solution of (1.141) can be extended over [0, oo), and 0 < g„ < tn, by virtue of (1.142). Eq. (1.145) is a recursion relation for the values c„ which we substitute in (1.144) to find the solution xn(t). We may always suppose f(x,sn) ^ 0 for x £ (c„, c n + i), since the assumption f(x, sn) ~ 0 for some c G (c„, c n+ i) leads to the conclusion that the constant function xn(t) = c is a solution of Eq. (1.140) on the interval tn < t < t n + i. However, this solution does not satisfy the condition xn(tn) = c„. And, if f(c„,s„) = 0 for a particular n, then xn(t) = c„ is a solution of Eq. (1.140) on the interval (tn, tn+1). ■ REMARK 5. For g(t) — [t], where [•] denotes the greatest-integer func­ tion, we have tn = g„ = n and c„ = sn = xn(n). If /(co,c 0 ) = 0, the only solution of problem (1.140) on [0,oo) is x(t) — CQ. If /(co,co) ^ 0 and f(cn,cn) = 0, for a particular n > 1, then the solution of (1.140)

1.6. EPCA AND IMPULSIVE EQUATIONS

75

equals c„ on [n,oo) but is not constant on [0, oo). On the other hand, the constant function x(t) = cn is a solution of Eq. (1.140) with the initial condition x(0) = c„ on [0, oo). This means that two solutions of Eq. (1.140) which are different on [0, oo) coincide on [n, oo). Needless to say, fusion of solutions is impossible for ordinary differential equations with uniqueness conditions. Theorem 1.60. The linear problem x'(t) = ax(t) + a0x(g(t)),

x(0) = c0

(1.146)

with constant coefficients and step function g(t) satisfying (1.142) on [0, oo), has a unique solution on [0, oo). PROOF. On the interval tn < t < tn+i we have the equation x'(t) = ax(t) + aosn, where sn = x(gn), the general solution of which is xn(t) = e a( '"' n) c - a_1a0SnLet Xn\tn)

=

Cn,

then cn — c- a_1aoS„ and Xn(t)

= ea^-^cn

+ (e"('-<») - l^aosn.

(1.147)

By virtue of t0 = 0, g0 = 0 and s0 = c0, we get the solution x0(t) = (eat + (eat - \)a-lao)cv.

(1.148)

of problem (1.146) on 0 < t < t\. Since s\ = x0(g{h)) and c% - x0( 1 and t„ < t < i„ + i, we have

xJt) - e^-^x^tn)

+ (e-C-'") -

^a-'aox^iigj,

76

1. PIECEWISE CONTINUOUS ARGUMENTS

where i < n is some non-negative integer. If ao = —a, the only solution of (1.146) is x(t) = c0. If ax0(h) + a0x0(gi) = 0,

a0 ^ - a ,

(1.149)

where xo(t) is given by (1.148), then the constant solution x(ti) and the solution that equals xo(t) on [0, t\] fuse on [tfi,^)- * EXAMPLE 1. The solution of the problem (1.11) is given by for­ mula (1.12). Note that if xo(l) = 0, then we define XQ(1) = 1. Con­ ditions (1.149) imply #o(l) = 0, that is, ao = —ae a /(e a — 1). In this case, the solution of Eq. (1.11) which equals xo(t) on [0,1] coincides on [l,oo) with the solution x(t) = 0. EXAMPLE 2. Consider the problem x'(t) = ax(t) + a0x(g(t)),

x(0) = c0

where 9

^

[0, \1,

0<*<2, 2
On the interval 0 < t < 2 we have x'0(t) = ax0(t) + a0co,

x0(0) — c0

whose solution is x0(t) = (eat + (eat -

l)a-la0)co.

On 2 < t < oo we get the problem x\(t) = axi(t) + a 0 x 0 (l),

xi(2) = x0(2~)

whose solution is xi(t) = (i 0 (2) + xo(l)a- 1 ao)e°('- 2 ) - a- 1 a 0 x 0 (l). The requirement XQ(2) + a _1 ao#o(l) = 0 means e2a + a-la0(e2a - 1) + a - 1 ^ 0 + (ea - lja^oo) = 0, (eu - l^a-'aof

+ (e2a + ea - l j o " 1 ^ + e2a - 0,

(1.150)

1.6. EPCA AND IMPULSIVE EQUATIONS

77

whence ae2a aQ = —a- — - . e —1

a0 = -a,

If ao = —a, then the only solution of (1.150) on [0,oo) is the constant function x(t) = Co- The case ao = - a e 2 a / ( e a - 1) leads to the solution x(t) = x0(t), 0 < t < 2

and

x(t) = -e3aco, 2 < t < oo.

Theorem 1.61. If the differential-difference equation x'{t) = f(x(t),x(t

- T), fi),

x(t) =
satisfies existence and uniqueness conditions in IR3 and its solution can be extended over (0,oo), then the problem X'(t) = f(x(t),x(t-T),x([t])),

X(t)=
t£[-T,0]

has a unique solution on (0, oo). Theorem 1.62. The solution of the problem with constant coeffi­ cients x'{t) = ax{t) + a0x([t]) + aix(t - 1), x{t) = ip(t), -1 < t < 0

(1.151)

on (0, oo) is given by the formula x(t) =
(1.152)

where y(t) is the solution of the problem y'(t) = ay(t) + aiy(t-l), y{t) = (a + a 0 )v(0) + a^t

1 < t < co - 1)

+ aai f e a ( '- s V(s - 1) ds,

0
PROOF. Eq. (1.151) on the interval [n,n + 1) takes the form x'(t) = ax(t) + a\x{t - 1) + a0cn,

c„ = x(n)

(1.153) (1.154)

78

1. PIECEWISE CONTINUOUS ARGUMENTS

which we differentiate to obtain (1.153), with y(t) = x'(i). Integrating y(t) yields (1.152), and on [0,1) we have the equation x'(t) = ax(t) + a0
x(Q) =
with the solution x(t)

= (eat + (eat - 1)0-^0)^(0) + ai ]* e°<'-V(* - 1) da,

the derivative of which coincides with (1.154).

■

It should be noted that in a recent paper Jayasree and Deo [131] de­ rived a variation of parameters formula and a Gronwall type integral in­ equality for linear EPCA with variable coefficients, and indicated at the possibility of extending these results to equations with both constant and piecewise constant delays. It would be also interesting to investi­ gate connections between EPCA and impulsive differential equations. It appears that these types of systems are very closely related. Indeed, if Definition 1.5 is modified to allow ordinary discontinuities of the so­ lution, with certain conditions on the jumps, and if g(t„) = tn, then Eq. (1.140) becomes an impulsive ODE. However, if an EPCA includes several arguments, it is capable of displaying a greater variety of behav­ ior than impulsive ODE. Even in the case of a single argument g(t) < t (for instance, g(t) = [Xt], 0 < A < 1), Eq. (1.140) is more intricate than an impulsive ODE. The literature on impulsive equations is extensive, and we refer the reader to the works of Halanay and Wexler [114], Kulev and Bainov [149], Lakshmikantham, Bainov and Simeonov [162], Milev and Bainov [192, 193], Samoilenko and Ilolov [236], Samoilenko and Perestyuk [238], and Simeonov and Bainov [254]. In conclusion, equations with piecewise constant delay can be used to approximate delay differential equations that contain discrete delays. For example, consider the scalar equation, with one constant delay r, x'(t) = -p(t)x(t-T),

t>0

(1.155)

with the initial condition x(t) = <j>(t),

-r
(1.156)

1.6. EPCA AND IMPULSIVE EQUATIONS

79

where <j> is a given function in C ( [ - T - , 0 ] , R ) , the space of continuous real functions, and p(t) is in C ([0, oo), R). Let k be any positive integer and let h = r/k. We may approximate this problem by the EPCA 2/« = - p ( % ( [ ^ - [ ^ ] / > )

(1-157)

with the initial condition y(nh) = (j>(nh),

n = -*,...,0.

(1.158)

Moreover, we find that y(nh) — c„ satisfies the difference equation c n+ i -cn

= - Uh

cn = (nh),

p(u) du\ c n _ t n = —k,..., 0.

It was proved by Gyori [108] that the solution of (1.157), (1.158) pro­ vides a uniformly good approximation to the solution of (1.155), (1.156) on any compact interval [0,T], T > 0. This result was extended by Cooke and Gyori [47] to convergence on t g (0,oo), in case the zero solution of (1.155) is exponentially stable. Specifically, the following theorem was proved. Let x(to,(f>) denote the solution of (1.155) with the initial condition set on [to — T, to]. Assume that there exist constants L and a such that for every to > 0 and continuous \x{toA){t)\ < LMe-^-^,

t>to

where ||<^|| denotes the supremum norm of <j>. T h e o r e m 1.63. (a). For every h = r/k (k > 1) and every (j> in C, there exist constants K, M\(4>, h) > 0, and Mi((t>) > 0 such that M\(,h) tends to zero as h —> 0, and such that the solutions x()(t) of (1.155), (1.156) andy(t) of (1.157), (1.158) satisfy

|*(M0 " VWWI < Wi(M for t >

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 <£'(CT) = - p ( O M - r ) . Then there exists a constant Mz((f) > 0 such that for t > 0 \x(4>)(t) ~ V(m)\ < {Mi(4>) + tM2()}he-("-Kh». Thus, if h is small enough, y(t) provides a uniform approximation on [0,oo) and has an exponential decay rate very close to that of x{t). This result generalizes easily to the case of any finite number of discrete delays T\, ..., Tm, and to systems of equations.

CHAPTER 2

Oscillatory and Periodic Solutions of Differential Equations with Piecewise Continuous Arguments

The oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments has been the subject of many in­ vestigations. Of particular importance, however, has been the study of oscillations which are caused by the argument deviations and which do not appear in the corresponding ordinary differential equation. Re­ search in this direction is extremely intensive, and here we refer only to the works of Arino and Gyori [18], Arino, Ladas, and Sficas [19], Bainov, Myshkis, and Zahariev [20], Grace [96], Graef, Grammatikopoulos, and Spikes [97, 98, 99], Gyori and Ladas [110], Kitamuraand Kusano [140], Kusano and Onose [153], Ladas [154], Ladas and Stavroulakis [160], and Lalli [164]. Although some oscillatory properties of equations with piecewise continuous arguments (EPCA) were mentioned in [244], [289], and [51, 54], the first consistent attempt in this direction was made by Aftabizadeh and Wiener [4], who applied the method of differen­ tial inequalities. This approach turned out to be very useful also for differential-difference equations and for equations with linear transfor­ mations of the argument. Therefore, in the first part of the chapter we develop the general theory of differential inequalities with piecewise continuous argument, which we employ in the next two sections to study oscillations of linear EPCA with variable coefficients. Then we explore a class of differential equations with the argument [t + | ] , which have stimulated considerable interest and have been investigated in several articles. In these equations, the argument deviation T(t) = t — [t + |] 81

82

2. OSCILLATORY AND PERIODIC SOLUTIONS

changes its sign in each interval n — \ 0 for n < t < n + \, which means that the equation is alternately of advanced and retarded type. This generates two essentially different conditions for oscillations in each interval [n — | , n + |] and leads to interesting properties of periodic solutions. Section 5 provides a detailed study of oscillations in systems of EPCA with constant coefficients. Finally, we discuss an interesting EPCA with unbounded delay analogous to the famous equations of the Kato-McLeod type. 1. Differential Inequalities with Piecewise Continuous Arguments It is well known that the theory of differential inequalities for initialvalue problems has been very useful in the theory of differential equa­ tions [163], [271]. Existence of extremal solutions for a variety of non­ linear differential equations is studied by a combined approach of the method of lower and upper solutions and the monotone iterative tech­ nique [161]. It is natural to extend this useful method to retarded differential equations. The study of EPCA by means of differential inequalities was carried out in [6]. Consider the equation x'(t) = f{x(t),x([t])),

x(0) = c0,

(2.1)

where / is continuous o n K x M and [■] is the greatest-integer function. Along with (2.1) we take the following ordinary differential equation with a parameter: dx — = /(z,/z).

(2.2)

Eq. (2.1) is a particular case of (1.59), and we can apply the argument that has led to Theorem 1.27. If f(x,fi) is different from zero on a set f2, then on Q there exists a general integral

F(x,n) = t + g((t), with an arbitrary function 0, and let xn(t) be a solution of Eq. (2.1) on

2.1. DIFFERENTIAL INEQUALITIES

83

the interval [n,n + 1), satisfying the contition x„(n) = cn. Then we have dx ~JT (2.3) dt = f(xn,Cn). If n = c„, then F(xn,cn)

= t + g(cn).

For t = n, this gives F(cn,cn) = n + g(cn). Therefore F(a 'nt

* \cni Cn) — t

Cn)

■ n

This can be written as fin

Jcn

dx

(2.4)

f{x,Cn)

At t = n + 1, we have fcn+i JCn

dx fix,

i

Cn)

(2.5)

Eq. (2.5) also follows directly by integrating (2.3) from n to n + 1. If Eq. (2.5) has a unique solution with respect to c„+i, then cn+i = H(cn),

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

cn = Hn(co),

n = 0,l,2,...,

or n

(2.6)

where H is the n-th iteration of the function H. Substitute (2.6) in (2.4), and we find xn(t,Co), the solution of (2.1) on [n,n + 1). Note that if (2.2) satisfies uniqueness conditions, then (2.4) has a unique solution with respect to xn. In this case (2.5) has a unique solution with respect to cn+\. We may always suppose f(x,cn) / 0 for i G (c„,c n+ i), since the assumption f(c,cn) = 0 for some c 6 (c n ,c n + i) leads to the conclusion that the constant function xn(t) = c is a solution of Eq. (2.1) on the

2. OSCILLATORY AND PERIODIC SOLUTIONS

84

interval (n,n + 1). However, this solution does not satisfy the condi­ tion xn(n) = cn, because c ^ c„. On the other hand, if /(c„, c„) = 0 for a particular n, then xn(t) = c„ is a solution of Eq. (2.1) on [n,n + 1). But in this case, x{t) = c„ is a unique solution on [0,oo) of Eq. (2.1) with the initial condition x(0) = c„. Hence, it satisfies the problem (2.1) only if c„ = Co. This concludes the proof of the following: Theorem 2.1. Assume that (2.2) satisfies existence and uniqueness conditions in R2 and its solutions can be extended over [0, oo). Then on [0, oo) there exists a unique solution of (2.1). Theorem 2.2. If f(x,fi) is a continuous function in R2 and the so­ lutions of (2.2) can be extended over [0,oo), then problem (2.1) has a solution on [0, oo). Definition 2 . 1 . A continuous function v(t) on [0, oo) is said to be an upper solution of x'(t) = f(t,x(t),x([t})),

x(0) = c0

if the derivative v'(t) exists at each point t 6 [0, oo), with the possible exception of the points [t] € [0, oo) where one-sided derivatives exist, and "*(<)>/(*,»(*). «([*])),

«(<>)> <*;

it is said to be a lower solution if the reversed inequalities hold. A fundamental result concerning upper and lower solutions is: Theorem 2.3. Consider the differential inequalities v'(t)
(2.7)

w'(t)>f{t,w(t),w([t})), t >0,

(2.8)

where / G C ([0, oo) x R x R, R). Suppose that f(t, x, y) is nondecreasing in y for each (t, x) G [0, oo) x R, and f(t,x,y)

- f(t,z,y)

< L(x - z)

whenever x > z.

Then v(Q) < w(0) implies that v(t) < w(t) for t > 0.

(2.9)

2.1. DIFFERENTIAL INEQUALITIES

85

PROOF. We demonstrate that vn(n) < w„(n) implies vn(t) < w„(t) for t 6 [n, n + 1). Let us first prove this for strict inequalities. To this end,suppose that v'n(t) < f(t,vn(t),vn(n)),

(2.10)

w'n(t) > f(t,wn(t),wn(n)), t£ (n,n + l),

(2.11)

and vn(n) < wn(n). We shall prove that vn(t) < wn(t) on [n,n + 1). If this is false, there exists a t „ £ (n,n + 1) such that vn(tn) = wn(tn) and vn(t) < wn(t),

t e [n,tn).

For small h < 0, we have vn{tn + h)- vn(tn) < wn(tn + h)-

wn(tn),

or vn(t„ + h) -v„(tn) h

wn(tn + h)- w„(t„) h

which implies that v'n(tn) > w'n(tn). This yields, from (2.10) and (2.11), and the nondecreasing characteristic of / , f(tn,Vn(tn),Vn(n))

>

f(tn,Vn(tn),Vn(n)),

which is a contradiction. So vn(t) < wn(t) on [n, n + 1). Define zn(t) = wn(t) + ee2Lt,

te[n,n

+ l),

where e > 0 is sufficiently small. Then zn(t) > wn(t) and z'n(t) = w'n(t) + 2Lee2Lt > f(t, wn(t), wn(n)) + 2Lee2U. Then using (2.9), we have z'n(t)>f(t,zn(t),vn(n)), v'n(t) < f(t,vn(t),vn(n)), t £ (n,n + l),

2. OSCILLATORY AND PERIODIC SOLUTIONS

86

and vn(n) < zn(n). Hence, it follows from the previous argument that vn(t) < zn(t), t e [n,n + 1). Letting e —► 0, we obtain vn(t) < wn(t), t G [n, n + 1), n = 0 , 1 , 2 , . . . . Therefore v(t) < w(t)

for t > 0.

■

If we know the existence of upper and lower solutions w, v such that v(t) < w(t), t > 0, we can prove the existence of a solution in the closed set fi = {(x,y): v(t) <x,y<

w(t),t > 0}.

T h e o r e m 2.4. Letv(i) andw(t) be lower and upper solutions of (2.1) such that v(t) < w(t) on [0, oo) and f G C(fi). Assume that (2.2) satis­ fies existence conditions on £1. Then there exists a solution x(t) of (2.1) such that v(t) < x(t) < w(t) on [0, oo). PROOF. Let P: [0,oo) x M -> R be defined by P(t,x) = max{w(f),min(x,u;(<))}. Then f(P(t,x(t)),P([t],x([t]))) defines a continuous extension of / to R x l which is bounded, since / is bounded on £1. Therefore by The­ orem 2.2 x'{t) = f(P(t, x(t)), P(t, x([t)))),

x(0) = co

has a solution on [0, oo), because Eq. (2.2) has a solution on fi defined on [0,oo). For e > 0 consider w(t) = w(t) + e(l + t) and v(t) = v(t) - e(l + t) We have v(0) < c0 < w(0). We wish to show that v(t) < x(t) < w(t) on [0,oo). To do this, we need to show that vn(t) < xn(t) < Wn(t) on [n,n + 1), for all n = 0 , 1 , 2 , . . . , whenever v„(n) < xn(n) < wn(n). If this is not true,

87

2.1. DIFFERENTIAL INEQUALITIES

then there exists tn 6 (n, n + 1) such that vn(t) < x„(t) < wn(t) on [n,tn) and xn(tn) = wn(t„). Then xn(t„) > wn(tn) and so P{t„,xn(tn))

= wn(tn).

Since wn(t) is an upper solution we have w'n(t)> f(wn(t),wn(n)),

te[n,n

+ l),

or <(*») >

f(P(tn,xn(tn)),P(tn,xn(n)))=x'n(tn).

Since u4(t n ) > u^(i„), we have Wn(tn) > x'n(tn). This contradicts xn(t) < wn(t) for £ G [n,i„). Consequently, u„(i) < xn(t) < w„(t) on [n,n + l), and letting e —► 0, we get vn(t) < xn(t) < wn(t) on [n,n + l). This proves that v(t) < x(t) < w(t) for t G [0, oo). ■ We apply now the monotone method to prove the existence of minimal and maximal solutions for the delay differential equation *'(*) = f(t, x(t),x([t})),

x(0) = co,

(2.12)

where / G C([0, co) x R x E,M), This constructive method yields monotone sequences that converge to solutions of (2.12). Since each member of these sequences happens to be a solution of a linear delay differential equation which can be computed explicitly, the advantage and importance of the technique need no special emphasis. For more details of the monotone iterative technique, the reader should refer to [161]. We need the following lemmas: LEMMA

2.1. The linear delay differential equation x'(t) = ax(t) + bx([t]) + f(t),

x(0) = c0

(2.13)

has on [0, co) a unique solution given by x(t) - c0A(* - [t])\f + X(t - [t])S + j * ea^f(s)

ds,

(2.14)

88

2. OSCILLATORY AND PERIODIC SOLUTIONS

where [t] [t] 1=1

J, 1

~

f(t) is continuous on [0, oo), and \(t) = eat + -(eat - 1),

Ai = A(l), and £ = 0.

Note that (2.14) is obtained with the implicit assumption a =£ 0. If a = 0, then the unique solution of (2.13) is given by x(t) = (1 + b{t - [i])) {S + o ( l + &)W) + f[t] f(s) ds, where

5 = E(i + 6 ) { M / , / W ^ which is the limiting case of (2.14) as a —> 0. LEMMA 2.2. Suppose that x G C([0,oo),M), and the derivative x'(t) exists at each point t G [0,oo), with the possible exception of the points [t] G [0, oo) ; where one-sided derivatives exist. Assume that

x'(t) < Mx(t) + Nx([t\),

x(0) < 0,

(2.15)

where M and N are constants such that N

Thenx(t)

* ^nrT^'

0 < a < 1.

(2.16)

< 0 on [0,oo).

PROOF. Fori G [n,n+l),n = 0 , 1 , 2 , . . . , consider according to (2.15), x'n(t) < Mxn(t) + Nxn(n),

x„(n) < 0.

From this and (2.16) we find that x„(t) < 0,t G [n, n + 1). This implies, because of continuity, x(t) < 0 for t > 0. ■

89

2.1. DIFFERENTIAL INEQUALITIES

T h e o r e m 2.5. Let a(t) and fi(t) be lower and upper solutions, re­ spectively, of (2.12) such that a(t) < fi(t) on I = [0,oo). Suppose that f(t, u, v) - f(t, x, y) > M(u -x)+

N(v -y),

t> 0, (2.17)

for a(t) < x(t) < u(t) < (3(t), a(t) < y(t) < v{t) < f3(t), and N > -^—~eaM -

eaM

_ l

for each

0 < a < 1.

J

Then there exist monotone sequences {vm(t)}, {wm(t)} withvo(t) = a(t), u>o(t) = (3(t) such that vm(t) —> v(t),wm(t) —> w(t) as m —> oo monotonically on I, and v(t),w(t) are minimal and maximal solutions of (2.12). PROOF. For any n G C([0,oo),IR) such that a(t) < n(t) < f3(t), we consider the linear delay differential equation x'(t) = f(t, n(t), V([t])) + M(x(t) - n(t)) + N(x([t}) - n([t])) (2.18) with x(0) = Co. It is clear that for every such 77, by Lemma 2.1, there exists a unique solution x(t) of (2.18) on I. Define a mapping T by Tn = x where x is the unique solution of (2.18). This mapping will be used to define the sequences {^m(*)} an( ^ {wm(t)}- Let us prove that

(i) a < Ta, P > TP; (ii) T is a monotone operator on the segment [a,(3] = {xe C ( I , R ) : a < x < /?}. To prove (i), set Ta = V\, where v\ is the unique solution of (2.18) with n = a. Let p(t) = Vl(t) - a(t). On each unit interval [n, n + 1), n = 0 , 1 , 2 , . . . , we have Pn(t) = vln(t) -

an(t),

where vin(t) is the solution of (2.18) on [n, n + 1) with rj = an and vi„(n) — c„. Then from p'n(t) = v'ln(t) - a'n(t) and the fact that an(t)

90

2. OSCILLATORY AND PERIODIC SOLUTIONS

is the lower solution of (2.12), i.e., a'n(t) Mpn(t) + NPn(n),

Pn(n)

> 0.

From Lemma 2.2, we obtain pn(t) > 0 for t € [n, n +1), n = 0 , 1 , 2 , . . . , and thus p(t) > 0 on [0,oo), which shows that Ta > a. Similarly, we can show that (5 > Tj3. To prove (ii), let 77,77 e C(l,R) such that Suppose that x = Tr) and x = Trj. Set p(t) = x(t) — x(t), so that p'n(t) =f(t,rjn(t),rjn(n)) - /(<,77„(i),%(n)) + M(xn(t) - rjn(t)) - M(xn(t) - r,n(t)) + N(xn{n) - 77„(n)) - N(xn(n) - r)n(n)), with pn(n) > 0. Then from (2.17) we have p'n(t) > Mpn(t) + NPn(n),

pn(n) > 0.

This implies that, by Lemma 2.2, p„(t) > 0 on < 6 [n, n + 1) for n = 0 , 1 , 2 , . . . , and thus p(t) > 0, t > 0. Therefore Trj > TT), proving (ii). We can now define the sequences vm = Tvm-\,wm = Tu/ m _i and conclude from the previous arguments that a(t) = v0(t) < 7Ji(t) < • • • < vm(t) m(t)<---<wl(t)<wo(t) = 0(t) for t > 0. It then follows that Jijgo vm(t) = v(t),

liu^ wm(t) = w(t)

on I.

It is easy to show that v(t) and w(t) are continuous on I and are the solutions of (2.1) in view of the fact that vm and wm satisfy v'm(t) = f(t,vm^(t),vm^([t])) + M(vm(t) - 7Jm_i(<)) + % ( W ) - «m-i([<])), vm(0) = c0

2.2. OSCILLATORY PROPERTIES

91

and

«*!»(<) = f(t,Mm-l(t),U>m-l(\t]))+M(wm(t)

+ N(wm([t)) - u>m_i([<])),

-

Wm^(t))

wm(0) = co.

To prove that v and w are minimal and maximal solutions of (2.12) we need to show that if u is any solution of (2.12) such that a(t) < u(t) < P(t) on I, then a(t) < v(t) < u(t) < w(t) < f3(t) on I. To do this, suppose that for some m we have vm(t) < u(t) < wm(t) on I, and set p(t) = u(t) -

vm+l(t),

so that p'(t) = f(t,u(t),u([t}))

-

f(t,vm(t),vm([t}))

- M(vm+l(t)

- vm(t)) - N(vm+l([t\)

- vm([t])),

or p'(t)>Mp(t) + Np([t]). Since p(0) = 0, then p(t) > 0, which implies u(t) > vm+i(t) on I. Similarly, u(t) < wm+i(t), and hence vm+\(t) < u(t) < wm+i(t) on I. This proves by induction that vm(t) < u(t) < wm(t) on I for all m. Taking the limit as m —> oo, we conclude that v(t) < u(t) < w(t) on I, and the proof is complete. ■ 2. Oscillatory Properties of First-Order Linear Functional Differential Equations Aftabizadeh and Wiener have studied in [4] the oscillatory properties of solutions of the first-order linear EPCA of the type x'(t) + a(t)x(t) + p(t)x([t]) = 0

(2.19)

x'(t) + a(t)x{t) + q(t)x([t + 1]) = 0,

(2.20)

and where a(t), p(t), and q(t) are continuous on [0, co), and [•] is the greatest-integer function. Sufficient conditions under which Eqs. (2.19) and (2.20) have oscillatory solutions are given, and we emphasize the

92

2. OSCILLATORY AND PERIODIC SOLUTIONS

fact that these conditions are the "best possible" in the sense that when a,p, and q are constants, the conditions reduce to a aea p > e»-l and q* < — e « - l ' which are necessary and sufficient. As it is customary, a solution is said to be oscillatory if it has arbitrarily large zeros. Theorem 2.6. Consider the delay differential inequality x'{i) + a(t)x(t) + p(t)x([t]) < 0

(2.21)

where a(t) and p(t) are continuous on [0,oo). Assume that Jin^sup /

p(t) exp I I a(s) ds) dt > 1.

(2.22)

Then (2.21) has no eventually positive solution. P R O O F . We prove that the existence of an eventually positive solution leads to a contradiction. To this end, suppose that x(t) is a solution of (2.21) such that

x(t) > 0 for t > n, where n is a sufficiently large integer. Prom (2.21) we have y\t) + p(t) exp ( ^ a(s) ds) y([t]) < 0,

t > n,

where y(t) = x(t) exp I / a(s) ds\ ,

t > n.

For t e[n,n + 1), (2.23) becomes y'(t) + p(t) exp U a(s) ds) y(n) < 0 ,

n
< n + 1.

Taking the integral from n to n + 1 we have y(n + 1) < y(n) (l - £+1p(t)

exp (£ a(s) ds) dt) .

(2.23)

93

2.2. OSCILLATORY PROPERTIES

Since y(t) > 0 for t > n, then 1 - J"

p(t) exp (f a(s) ds) dt > 0,

or Jim^sup j This contradicts (2.22). tion. ■

p(t)exp I / a(s)ds\

dt < 1.

So (2.21) has no eventually positive solu­

T h e o r e m 2.7. / / condition (2.22) is satisfied, then the delay differ­ ential inequality x'(t) + a(t)x(t) + p(t)x([t]) > 0

(2.24)

has no eventually negative solution. P R O O F is the same as in Theorem 2.6. ■ From Theorems 2.6 and 2.7 it follows that the delay differential equa­ tion (2.19) has no eventually positive or eventually negative solutions and therefore we are led to the following conclusion. COROLLARY 2.1. Subject to condition (2.22), the delay differential equation (2.19) has oscillatory solutions only.

Note that when a(t) and p(t) are constants, i.e. x'(t) + ax{t) +px([t\) = 0

(2.19')

the condition (2.22) reduces to

P > ^31

(2-25)

which is a sharp condition. More precisely, we have: T h e o r e m 2.8. Every solution o/(2.19') is nonoscillatory if and only if

p< ^n-

( 2 - 26 )

94

2. OSCILLATORY AND PERIODIC SOLUTIONS

It is known from Theorem 1.3 that Eq. (2.19') with the initial condition x(0) = CQ has a unique solution (1.12). It is nonoscillatory if and only if PROOF.

b0 = e~a + a-lp(e-a

- 1) > 0,

and this condition is equivalent to (2.26).

■

If (2.25) is satisfied then, according to (1.74), the solution of (2.19') with x(0) = CQ ^ 0 has precisely one zero

*n = n + - l n ( l + - ) a

\

pj V>

u

in each interval (n, n + 1). If p = a/(e — 1), then x(t) = (e~at + (e~at - l)a _1 p)co,

for 0 < / < 1,

and x(t) = 0 on [l,oo). Theorem 2.9. Consider the advanced differential inequality x'(t) + a(t)x(t) + q(t)x([t + 1]) > 0

(2.27)

where a(t) and q(t) are continuous on [0,oo). Assume that l\m>'mijn+1

q(t)exp (- Jn+1 a(s)ds\ dt < - 1 .

(2.28)

Then (2.27) has no eventually positive solution. PROOF. If the assertion of the theorem is not true, then for suffi­ ciently large n x(t) > 0,

t > n.

From (2.27), we obtain y'{t) + q(t) exp ^

a(s) dsj y([t + 1]) > 0,

where y(t) = x(t) exp IJ a(s) ds ]

t > n,

2.2. OSCILLATORY PROPERTIES

95

or y'(t) + q(t) exp (- J"+1 a(s) ds) y(n + 1) > 0,

te[n,n

+ 1).

Taking the integral from n to n + 1, we have y(n + 1) M + £+1 q(t) exp (- J"+1 a(s) ds) dt) > y(n). Since y(t) > 0 for t > n, then 1 +

/n

l(t)exP[~Jt

a(s)dsjdt>0,

or

feinf /„" '(') ex P (~ It

a s d s di

( ) ) - _1'

which contradicts (2.28). Hence, (2.27) has no eventually positive so­ lution. ■ Theorem 2.10. / / condition (2.28) is satisfied, then the advanced differential inequality x'(t) + a(t)x(t) + q(t)x([t + 1]) < 0 has no eventually negative solution. PROOF is the same as in Theorem 2.9.

■

COROLLARY 2.2. Subject to condition (2.28), the advanced differen­ tial equation (2.20) has oscillatory solutions only. When a(t) and q(t) are constants, i.e. x'(t) + ax{t) + qx([t + 1]) = 0

(2.20')

the condition (2.28) reduces to 9< - ^ T -

(2-29)

Furthermore, (2.29) is a sharp condition. The following result is true.

96

2. OSCILLATORY AND PERIODIC SOLUTIONS

T h e o r e m 2.11. Every solution of Eq. (2.20') is nonoscillatory if and only if a> - ^ .

(2.30)

P R O O F . The conclusion immediately follows from Theorem 1.29. In fact, (2.30) is a particular case of the inequality

(«o + ^

T

) ( « i - ^

T

) < 0 ,

where a^ = 0, a\ = —
■

If (2.29) is satisfied then, according to Theorem 1.29, the solution of (2.20') with a;(0) = CQ 7^ 0 has precisely one zero

tn = n + l + i.n(l +

°)

Q' a \ q) in each interval (n, n + 1). If q = —aea/(e" — 1), then the only solution of (2.20') is x(t) = 0. The above method can also be applied to the linear differentialdifference equations

x'(i) + a(t)x(t) + £ Pi(t)x(t -

Ti)

= 0

(2.31)

x'(t) + a(t)x(t) - Y, Pi(t)x(t + Ti) = 0

(2.32)

»=i

and

where a(t) is continuous on R + , 0 < T\ < T 0 are continuous on R + for * = 1,2,... ,n. We give sufficient conditions under which (2.31) and (2.32) have oscillatory solutions only. The above results are caused by deviating arguments and are not valid when r,- = 0. We like to emphasize that our results for (2.31) and (2.32) generalize the results obtained by Ladas [154], and Ladas and Stavroulakis [160]. T h e o r e m 2.12. Consider the delay differential inequality x'(t) + a(t)x{t) + £ pi(t)x{t -Ti)<0 »=i

(2.33)

2.2. OSCILLATORY PROPERTIES

97

where a(t) is continuous on R + ; pi(t) > 0, i = 1,2,... ,n, and contin­ uous on M.+, and 0 < T\ < T
lim inf /

a(s) ds = Ai > —oo, i = 1,2,... ,n

(2.34)

lim inf/

pi(s)ds>

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

(2.35)

lim inf/

, J>,(s) ds > 0,

e = 1,2,... ,n.

(2.36)

t —► OO

J t — T"|

t—*oo

Ji—Ti/2

Then (2.33) has no eventually positive solution. PROOF. If the assertion of the theorem is not true, then for suffi­ ciently large c, x(t) > 0,

t > c.

From (2.33) we have, for t > c + T\ + r„ (x(t)exp(

j a(s)ds 11 + J2pi(t)x(t — r,)exp I j a(s)ds) < 0.

Let y(t) = x(t) exp I / a(s) ds j . Then (2.37) becomes y'(t) + E pi(t)y(t - Ti) exp [ft_Ti a{s) ds) < 0.

(2.38)

Since y(t - r,) > 0 for t > c + T\ + r„, and pi(t) > 0, i = 1,2,..., n, then y'(t) < 0. Let W t

()

=

z/(*-n) U\ >

t>

C+Ti

then lim inf wit) = A > 1.

+ Tn

2. OSCILLATORY AND PERIODIC SOLUTIONS

98

Dividing (2.38) by y(t), and integrating from t — T\ to t, we have -]nw(t)

+ gtl* " " " ' W w f \Js-n r o(r)<*r)
Since 2/(* - Ti) > y(t - Tl),

tOTt>C+Ti+T„

then - In «/(*) + E /* Pi(s)w(s) exp ( /* ■ , Jt—T%

a(r) dr) ds < 0. (2.39)

\JS—Ti

/

Now, we consider two possible cases: CASE 1. A isfinite,i.e. lim inf w(t) = A = finite > 1. Using (2.34) and (2.39), In A > A E eAi lim inf I* pAs) ds.

(2.40)

Using the fact that max(lnx — ax) = —Ina — 1,

x > 1,

from (2.40) one has n

rt

E eAi lim inf /

f^

<-><»

Jt-Ti

pAs) ds < e~l. y '

~

This contradicts (2.35). CASE 2. A is infinite, i.e. lim w(<) = A = +oo, i—►oo

Integrating (2.38) from t — T\/2 to t, we have

y(t) - y (* - y ) + E Jt[n/2Pi(s)y(s - r 0 «qp ( £ n a(r) dr) <** < o, or »(0-»f*-?-)+y('-7-i)E/ \

*/

/9P.(*)exp(r

:=1 ■ /< - T i/ 2

a(r)dr)
V"-*

'

/ 2 41)

99

2.2. OSCILLATORY PROPERTIES

using the fact that y(s - r.) > y(s - TI). Dividing (2.41) by y(t), and then by y(t - n/2), we obtain

'-^^SO'M-'GC,*)*)*^ and

l+

W%> ~ $^D S O From (2.42) and lim^oo y(t - n)/y(t)

v

lim *-oo

wexp

= +oo, we have, using (2.36),

yit-n/2) .. = y(t)

(C, *> *) * * £

+oo

and therefore

y

y(t-n)

l i m —r^ t-+coy(t-n/2)

which contradicts (2.43).

TTT- = + 0 0 ,

■

Theorem 2.13. Consider the delay differential inequality x'(t) + a(t)x(t) + J2Pi(t)x(t - Ti) > 0,

(2.44)

subject to the hypotheses of Theorem 2.12. Then (2.44) has no eventu­ ally negative solution. P R O O F . The result follows immediately from the observation that if x(t) is a solution of (2.44), then — x(t) is a solution of (2.33). From Theorems 2.12 and 2.13, it follows that Eq. (2.31) has no eventually positive or eventually negative solutions and therefore we are led to the following conclusion.

COROLLARY 2.3. Consider the delay differential equation (2.31), sub­ ject to the hypotheses of Theorem 2.12. Then every solution of (2.31) oscillates.

100

2. OSCILLATORY AND PERIODIC SOLUTIONS

Note that when a(t) and pi(t), i = 1,2,..., n, are constants, that is, in the case of the delay differential inequalities x'(t) + ax(t) + £

Pix(t

- n) < 0

(2.33')

x\t) + ax(t) + J2 Pi*(t ~ n) > 0

(2-44')

t=i

and »=i

and the delay differential equation x'(t) + ax(t) + £

Pix(t

-

Ti)

=0

(2.31')

t = l,2,...,n.

(2.45)

i=i

the conditions (2.34)-(2.36) reduce to g-l-ar,-

ptTi> , n We have the following results: We have the following results:

T h e o r e m 2.14. Assume that PiTn< , J = l,2,...,n. (2.46) n Then (2.33') has eventually positive solutions; (2.44') has eventually Then (2.33') has eventually solutions;solutions. (2.44') has eventually negative solutions; (2.31') haspositive nonoscillatory negative solutions; (2.31') has nonoscillatory solutions. P R O O F . First we show that (2.33') has eventually positive solutions. Looking for a solution of (2.33') of the form x(t) = ext it follows that F(A) = A + a + £>,e- A T i < 0. i=l

In view of (2.46),

1

\Tn

—1 " , — 1 1 - . ) = — + £/>.-e aTi+ri/T " < — + - = 0. T

n

i=l

T

n

Tn

101

2.2. OSCILLATORY PROPERTIES

Hence, there exists a A, namely, A = — a — l/r„ such that eXt is a positive solution of (2.33'). Following a similar procedure it is clear that — ext with A = — a — l/r„ is a negative solution of (2.44'). Finally, since F(-a)

> 0 and

F (-a - — ) < 0

there exists a A in each interval [—a — l/r n ,—a) such that ext is a nonoscillatory solution of (2.31'). ■ T h e o r e m 2.15. Consider the advanced differential inequality x'(t) + a(t)x(t) - £>,-(*)*(* + n) > 0,

(2.47)

»=i

where a{t) is continuous on M.+, pt(t) > 0, and 0 < T\ < T — oo, i = 1,2, ...,n (2.48) I+T\

/

Pi(s)ds>

e

liminf/'+n/2pi(s)ds>0, t—»-oo

,

4 = 1,2, . . . , n

i-l,2,...,n.

(2.49) (2.50)

Jt

Then (2.47) has no eventually positive solution. PROOF. Suppose that x(t) is a solution of (2.47) such that for suffi­ ciently large c, x(t) > 0,

t > c.

From (2.47) we obtain fexp (fc a(s) ds) x(t))

- f^Piit) exp (t a{s) ds) x(t + r.) > 0.

Let y(t) = x(t) exp I / a(s)ds),

t > c,

102

2. OSCILLATORY AND PERIODIC SOLUTIONS

then V'(t) - t Pi(t) exp (jf+Ti -a(s) ds) y(t + r.) > 0.

(2.51)

Since y{t + r,) > 0 for t > c, and p,(t) > 0 for i = 1,2,..., n, then y'(t) > 0

for

i > c,

and therefore y(t)
i=

l,2,...,n.

Set w{t)

= yJl+iil

for t

> c.

Then lim inf w(t) = A > 1. Dividing (2.51) by y(t) and integrating the result from t to t + ri we have In w(t) -Y.J

Pi(s) exp Qf

* - a ( r ) drj w(s) ds > 0 (2.52)

Now, we consider the following two cases: C A S E 1. A is finite. From (2.52) and (2.48) we have In A - XeAi J2 lim Urnml infIJ

^ p{(s) ds > 0.

Since max(lnx — ax) = — In a — 1, then rt+n Y\ lim int inf j/ Pi(s) ds < e~l~Ai i=l '

which contradicts (2.49).

x >1

2.2. OSCILLATORY PROPERTIES

CASE

103

2. A is infinite, that is, lim

..

t-OO

y(t)

= +oo.

Integrating (2.51) from t to t + n/2 we have

y(t +

j)-y(t) - V(t + n) £ Jtt+n/2Pi(s)

exp (/s5+T' - a ( r ) efr) ds > 0. (2.53)

Dividing (2.53) by y(t), and then by y(t + Ti/2), we obtain yjt + Ti/2)

- ^ ± ^ t p/2Pi(s)

exp ( f * -« ( r) dr) ds > 0, (2.54)

y(t)

y(t + n/2)

/!

- J r ^ j s r «w-p o r -w *) * > o (2.55) From (2.54), lim^oo y(t + T\)/y{t) = +oo, and (2.50), we have *-oc

y(t)

and therefore

i-

vtt + n)

j_

l i m —~r T-r = + 0 0 , t^oo y(t + T l / 2 )

which contradicts (2.55).

■

Theorem 2.16. Consider the advanced differential inequality x'(t) + a(t)x(t) - £ «(<)*(* + TJ) < 0, i=l

(2.56)

104

2. OSCILLATORY AND PERIODIC SOLUTIONS

subject to the hypotheses of Theorem 2.15. Then (2.56) has no eventu­ ally negative solution. PROOF. The result follows from the fact that if x(t) is a solution of (2.56), then -x(t) is a solution of (2.47). ■ Theorems 2.15 and 2.16 lead us to the following conclusion: COROLLARY 2.4. Consider the advanced differential equation (2.32), subject to the hypotheses of Theorem 2.15. Then every solution of equa­ tion (2.32) oscillates. If a(t) and p,-(i) are constants, that is, in the case of the advanced differential inequalities x'(t) + ax(t) - £ pix(t + Ti) > 0

(2.47')

x'(t) + ax(t) - £ Pix(t + Ti) < 0

(2.56')

and

and the advanced differential equation x'(t) + ax(t) - J2 PA* + rt) = 0

(2.32')

the conditions (2.48)-(2.50) reduce to gar.-l

Pin >

, n Theorem 2.17. Assume that

i = l,2,...,n.

(2.57)

gUTy-l

PiTn <

, i = l,2,...,n, (2.58) n ' then (2.47') has eventually positive solutions; (2.56') has eventually neg­ ative solutions; (2.32') has nonoscillatory solutions.

2.2. OSCILLATORY PROPERTIES

105

Looking for a solution of (2.47') of the form x{t) = eXt, it follows that PROOF.

F(A) = A + a - X > , e A r i > 0. »=i

In view of (2.58), F (— - a) = — - J2PieTi/Tn~aTi > — - — = 0. Hence, there exists a A, namely A = l/r„ — a, such that ext is a positive solution of (2.47'). Following a similar procedure it shows that — eM with A = l / r n — a is a negative solution of (2.56'). Finally, since F(-a)

< 0,

F(

a) > 0,

there exists a A in the interval (—a, —a+l/r„] such that eXt is a nonoscillatory solution of (2.32'). ■ Finally, we apply the results obtained for Eqs. (2.31) and (2.32) to discuss the oscillatory properties of solutions of x'(t) + a(t)x(t) + f^ Pi(t)x(Xit) = 0

(2.59)

t=i

and x'(t) + a(t)x(t) - £ Pi(t)x(Xit) = 0

(2.60)

i=l

where a(t) is continuous^,-(£)_ > 0 are continuous on E + , 1 > Ai > A2 • ■ • > A„ > 0 and 1 < Aj < A2 < . . . < A„ < +oo. T h e o r e m 2.18. Suppose that tint

lim inf i/ , lim inf

a(s) ds = A{ > - o o , i = 1,2,... ,n

lim inf [^

pAs) ds >

t—»oo

(2.61)

Jln(Xit) —I—A-

t—>oo

J\n{\it)

lim inf /'"' t—>oo

J\n(\/Xitj

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

(2.62)

n

vds) ds > 0, i = 1,2,... ,n.

(2.63)

106

2. OSCILLATORY AND PERIODIC SOLUTIONS

Then (2.59) has oscillatory solutions only. PROOF. Let t = es and A; = e _Ti , i = 1,2,... ,n. Define x{t) = x(es) = y(s), then y'(s) = e°x\t). Using the above transformations, (2.59) becomes y\s) + esa(es)y(s) + £ esPi(es)y(s - r,-) = 0. «=1

The result follows immediately by Corollary 2.3.

■

T h e o r e m 2.19. Suppose that /■ln(Ait)

lim inf / (—oo

lim inf / t—»oo

— a(s) ds = At > —oo, i = 1,2,... ,n (2

J\ni /•ln(A,<) Jlat

v

'

'

e~X~Ai

pAs)ds>

> )

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

V

(2

n

An(y/f[t) lim inf / pi(s)ds > 0, i = 1,2,... ,n. Then (2.60) has oscillatory solutions only. PROOF.

)

Let t = es, A,- = eTi,i = 1,2,... ,n, and *(<) = z(e*) = t/(s),

then (2.60) becomes y'(s) + e s a(e s )y(s) - £ c'p,-(e')y(« + TJ) = 0. Now, applying Corollary 2.4, we obtain the result.

■

(2

2.3. SOLUTIONS OF DELAY EPCA

107

3. Oscillatory and Periodic Solutions of Delay E P C A In this section, we study oscillatory properties of the linear delay EPCA x'{t) + a(t)x(t) + b(t)x([t - 1]) = 0,

(2.67)

where a(t) and b(t) axe continuous functions on [0, oo). A sufficient condition under which Eq. (2.67) has oscillatory solutions is given, and it is the "best possible" in the sense that when a and b are constants the condition reduces to b > ae _ a /4(e a — 1), which is necessary and sufficient. In the case of constant coefficients we find conditions under which oscillatory solutions are periodic. The following results were obtained by Aftabizadeh, Wiener, and Xu [8]. Theorem 2.20. Suppose that b(t) > 0, and for Jin^ sup £+1 b(t) exp (£

t>0

a(s) ds) dt > 1.

(2.68)

Then the delay differential equation (2.67) has oscillatory solutions only. We prove that the existence of an eventually positive (or negative) solution leads to a contradiction. For any k = 0 , 1 , 2 , . . . , Eq. (2.67) becomes PROOF.

x'(t) + a(t)x(t) + b(t)x(k - 1) = 0,

t e [k, k + 1),

(2.69)

or x(A; + l)exp ( /

a(s)ds) = x(k) -x(k-l)

J*+l b(t) exp (fk a(s) ds) dt. (2.70)

Now, assume that Eq. (2.67) has an eventually positive solution, i.e., x(t) > 0 for t > T, where T is sufficiently large. From (2.70) for any

108

2. OSCILLATORY AND PERIODIC SOLUTIONS

integer n> [T] + 3, we have x(n)exp(f

a(s)ds) = x(n — 1) - x(n - 2) j n _ b(t) exp (£

a(s) ds) dt,

(2.71)

t € [«, n + 1). Since a;(n — 2) > 0 and 6(£) > 0, we have x(n) exp r y ^ a(s) ds) < x(n - 1).

(2.72)

Also from (2.70) we have a;(n + l ) e x p l /

a(s)ds)

+ x{n - 1) £+1 b(t) exp (j* a(s) ds) dt = x(n),

(2.73)

t e[n,n + 1). From (2.72) and (2.73) we obtain x(n) il — exp ( j

a(s)ds) j

b(i) exp I j a(s) ds) dtj > x(n + 1) exp I /

a(s) ds I .

Since x(n) and x(n + 1) are positive, 1 > fn

b(t)exp(J

a(s)ds)dt

or 1 > Jim^ sup I

b(t) exp I /

a(s) ds I dt,

which is a contradiction to (2.68). The proof is the same in case of existence of an eventually negative solution. Hence, Eq. (2.67) has oscillatory solutions only. ■ Theorem 2.21. Assume that nliminf(exp(/n"

+1

a(S)^))

x lim^'mij"

b(t) exp f / a(s) ds) dt > - . (2.74)

109

2.3. SOLUTIONS OF DELAY EPCA

Then Eq. (2.67) has oscillatory solutions only. PROOF. Suppose that x(t) > 0 for t > T. Let N = [T] + 1. Then for any integer n > N, we have from (2.67), for t G [n, n + 1), x(n + l ) e x p l y

a(s)ds) +1 b(t) exp (j* a(s) ds) dt = x(n) + x(i(n - 1) £

(2.75)

or n+ + l) 1) x(n) x(n x(n) —exp / /yn+i a(s)ds)\ —— x(n) x(n — 1) V-Zn J + f"+1 b(t) exp ( f a(s) ds) dt = *^ .. (2.76) •/« V" J x(n — 1) Let wn = x(n)/x{n — 1). Then wn > 0. We consider two cases: C A S E 1. lim„_0O inf wn < oo. From (2.76), lim inf wn+\ lim inf wn lim inf I exp I / n—>oo

" T 1 n—too

n—»oo

+ lim inf I

V

r

w„

a(s) ds)) //

b(t)exp I I a(s)ds) dt < lim inf wn

which is impossible, because, using (2.74) and the fact that, if 4AC > 1,

A > 0,

then the inequality AX2 + C < X has no real solution. C A S E 2. lim^ooinf wn = oo Let r„ = l/ui„ = x(n — l)/x(n). Then linin^ooinf r„ = 0. From (2.75) we have

+

- ( r • w * ) ^ ■ ^ i ) r w -p {s: ■«*) x(n) ~ x(n + l ) ' This leads us to 0 < Jirn^inf exp f /

a(s) ds) < 0,

110

2. OSCILLATORY AND PERIODIC SOLUTIONS

which is a contradiction. So Eq. (2.67) cannot have an eventually pos­ itive solution. Existence of an eventually positive solution leads to a contradiction. Hence Eq. (2.67) has oscillatory solutions only. ■ LEMMA 2.3. Suppose that x(t) is a solution of Eq. (2.67) such that x(n) and x(n + 1) ^ 0 forn = 0 , 1 , 2 , . . . , b(t) > 0 (or b(t) < 0) for t > 0. Then there exists a T € (n,n + 1) such that x(T) = 0 if and only if x(n) ■ x(n + 1) < 0. PROOF. If x(n) ■ x(n + 1) < 0, it is clear that there exists T 6 (n,n + 1) such that x(T) = 0. We shall show that x(T) = 0 implies that x(n) ■ x(n + 1) < 0. For t £ [n,n + 1), Eq. (2.67) becomes x'(t) + a(t)x(t) + b(t)x(n - 1) = 0, or x(i) = exp I — / a(s) ds) x (x(n) - x(n - 1) J* b(s) exp (J* a(r) dr) ds] . (2.77) Since the solution x(t) of Eq. (2.67) is continuous on [0, oo), from (2.77) we get x(n + 1) = exp I — /

a(s) ds j

x (x(n) - x(n - 1) Jn+

b(s) exp (j* a(r) dr) ds) . (2.78)

Also from (2.77) and the fact that x(T) = 0, we have x(n) = x(n - 1) /

b(s) exp (J* a(r) dr) ds.

(2.79)

Let us assume that b(t) > 0. Then from (2.79), x(n) ■ x(n — 1) > 0. It is easy to see that, from (2.78) and (2.79), x(n + 1) • x{n — 1) < 0, and therefore x(n) • x(n + 1) < 0. If b(t) < 0, then we obtain the same result. ■ If a(t) and b(t) are constants, i.e., x'(t) + ax(t) + bx([t-l})=0,

(2.80)

2.3. SOLUTIONS OF DELAY EPCA

111

then condition (2.74) becomes b

>

ae

~\^ (2-81) K 4(e° - 1 ) ' We show that (2.81) is a sharp condition. We know that (2.80) with x(0) = co and x(—1) = c_i has a unique solution given by Af

x„(t) = c„e-a«-"> + £(e-"<*-"> - l)cn-i

(2.82)

for t E [n, n + 1), n = 0 , 1 , 2 , . . . , and c„ = 1 - ^ - r ( A ? + 1 ( c o - A2c_0 - \p\co

- A lC _0),

(2.83)

Al — A 2

where Ai, A2 are the roots of A2 - e~a\ + - ( 1 - c—) = 0. (2.84) a If these roots are real, assume Ai > A2. T h e o r e m 2.22. Eq. (2.80) has no oscillatory solution if either of the following hypotheses holds true: (i) b < 0 and CQ — A2c_i ^ 0; or (ii) 0 < 6 < ae-°/4(e Q - 1). PROOF. Either (i) or (ii) implies that (2.84) has real roots. If condi­ tion (i) holds true, then from (2.83) we have _fn Ai(c0 - A2c_t) - (eg - Aic_x)A2(A2/Ai)" c„-i " (co - A2c_i) - (c0 - Aic_1)(A2/Ai)n Since |A2/Ai | < 1 and c0 - A2c_i ^ 0, then lim„_oo c n /c„_i = Ai > 0. Hence, c„/c„_i > 0 for n > N. This shows that c„cn_i > 0 for n > N. Then Lemma 2.3 shows that there is no t € (n, n+1) such that x(t) = 0. So Eq. (2.80) has no oscillatory solution. If condition (ii) holds, then Ai > A2 > 0, provided b < ae _ a /4(e" — 1), and the same argument shows that c„c„+i > 0. If b = ae-a/i{ea - 1), then Aj = A2 = e~a/2, and (2.83) becomes cn = ((n + l)c 0 - nc-iA^A".

112

2. OSCILLATORY AND PERIODIC SOLUTIONS

If Co = c_iAi, then c„ = c_iA"+1, which implies c„c„+i > 0. If c0 ^ c_iAi, then c„ _ n(cp - c-iAi) + CQ c„_i n(c 0 - c_iAi) + c_i or lim n ^ 00 c„/c„_i = Ai > 0, so cnc„_i > 0 for n > N. Therefore in either case c„c„_i > 0, which shows that Eq. (2.80) has no oscillatory solution. ■ 1. If b < 0 and c0 = A2c_i, then c„ = c_iA2+1. Since A2 < 0, this shows that c„c„+i < 0. Therefore in this case (2.80) has oscillatory solutions. REMARK

COROLLARY 2.5. A necessary and sufficient condition for the solu­ tions of Eq. (2.80) to be oscillatory is either (2.81) or b<0

and

CQ = A2C_i.

T h e o r e m 2.23. The solution x = 0 of Eq. (2.80) is asymptotically stable as t —» +00 if and only if the moduli of the roots of (2.84) satisfy the inequalities |Ai| < 1 and |A21 < 1. Theorem 2.24. / / ae~a 4(e" - 1)

<

<

aea ea-l'

then every oscillatory solution of (2.80) tends to zero as t —> 00. Fur­ thermore, if —a < b < 0

and

0$ = c_iA2,

then every oscillatory solution of (2.80) tends to zero as t —+ 00. Eq. (2.80) exhibits unusually interesting properties concerning the existence of periodic solutions. LEMMA 2.4. The solution of Eq. (2.80) is periodic of period k if and only if Ck-\ = c_i and c* = CQ.

2.3. SOLUTIONS OF DELAY EPCA

PROOF.

113

Assume that x(t) is periodic of period k. Then x(t + k) = x(t)

for t G (—oo,+oo). This implies that ck_\ = c_i and ck = CQ. NOW, assume that ck^\ = c_i and ck = CQ. From (2.82) we get x0(t) = cQe-at + - (e~at - l) c_i,

0 < t < 1,

and ark(t) = c ,e- a ('-*) + - (e-<'-*> - l) c ^ ,

* < * < * + 1.

We see that zo(2) = £*(£), and, because of continuity, x0(l) = a = xk(k + 1) = Cjt+i. Also *i(<) = de-'C- 1 ) + - ( e - ^ - 1 ) - 1) co, xk+1(t) = ck+le-
+ - ( e -(*-*-l> - 1) c t )

l
This shows that xi(t) = x<;+i(i). Continuing this way, we can show that xk_i(t) = X2k-i(t). So x(t) is periodic of period k. ■ T h e o r e m 2.25. Assume that b > 0. TTien every oscillatory solution of Eq. (2.80) is periodic of period k if and only if o=

aea e

x

, and

, / 27rm\ a = — In I 2 cos —— I , \ K /

._ „_. (2.85)

where m and k are relatively prime and m = 1,2,..., \{k — l)/4]. PROOF. Let us assume that Eq. (2.80) has a periodic solution of period k. Then ck = CQ and ck^\ = c-\. Hence, from (2.83) we have (A*+1 - A*+1 - A, + A2)c0 - AXA2(A* - A*)c_: = 0, (A* -

A*)CQ

- (A{A2 - AiA* + Ax - A2)c_! = 0,

114

2. OSCILLATORY AND PERIODIC SOLUTIONS

or (Af - 1)(A£ - l)(Ai - A2)2 = 0. Since 6 > 0 and the solution is oscillatory, by Corollary 2.5, Ai and A2 are complex. So A* = 1 and \\ = 1 imply that Ax = e2™"'/*, A2 = e-^milk. But from (2.84), b.

_„.

,

2-KTU

1

_a

-(1 — e ) = 1 and cos—;— = - e , a k I which is (2.85). Now assume that (2.85) is satisfied; we show that every solution is periodic of period k. Using (2.84) and (2.85) we get . 27rm . 27rm A = cos —-— ± 1 sin —:—. k k A simple calculation shows that c/. = CQ and Ck-i — c_i. ■ T h e o r e m 2.26. Let 6 < 0 and CQ = c_iA2- Then every oscillatory solution is periodic of period 2 if and only if b = -<e<1 + 1 \ (2.86) K J ea - 1 COROLLARY 2.6. If b > 0, then for given CQ and c_i, the set of all equations of type (2.80) having periodic solutions is countable. REMARK 2. The above results were obtained with the implicit as­ sumption a ^ 0. If a = 0, then condition (2.85) becomes aea 6 = lim = 1. a—0 ea - 1 In this case, Eq. (2.80) with a = 0 and 6 = 1 has periodic solutions of period 6. 4. Differential Equations Alternately of Retarded and Advanced Type Differential equations of the form

•(«) = /(*(*),* ([* + £])) have stimulated considerable interest and have been studied by Afta­ bizadeh and Wiener [5, 7], Cooke and Wiener [53], Huang [126, 127], Jayasree and Deo [131], Ladas, Partheniadis, and Schinas [157], Papaschinopoulos [211], Wiener and Aftabizadeh [292], and Wiener and

115

2.4. EQUATIONS OF ALTERNATING TYPES

Cooke [293]. In these equations, the argument deviation T(t) = t—[t+^\ changes its sign in each interval n — \ 0 for n < t < n + \, which means that the equation is of alternately advanced and retarded type. It is of advanced type on [n — ^, n) and of retarded type on (n, n + | ) . Cooke and Wiener have studied in [53] the equation x'(t) = ax(t) + a0x (2 *-j^ ) ,

x(0) = CQ,

(2.87)

where [•] is the greatest-integer function. The argument deviation T(t) =

t-2

r* + i

(2.88)

is negative, for 2n — 1 < t < 2n, and positive, for 2n < t < 2n + 1. Therefore, Eq. (2.87) is of advanced type on [2n — 1,2n) and of retarded type on (2n, 2n + 1). Definition 2.2. A solution of Eq. (2.87) 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 6 [0,oo), with the possible exception of the points t = 2n — 1 (n = 1,2,...), where one-sided derivatives exist, (iii) Eq. (2.87) is satisfied on each interval 2n — 1 < t < 2n + 1. Let \(t) = eat + (eat - 1)0-^0,

(2.89)

Ai = A(l), A_1=A(-1). Theorem 2.27. Problem (2.87) has on [0,oo) a unique solution Xl

\ [('+i)/ 2 ]

x(t) = X(T(t)) (j±j if A_i ^ 0, where T(t) is given by (2.88).

Co,

(2.90)

116

2. OSCILLATORY AND PERIODIC SOLUTIONS

Assuming that xn(t) is a solution of Eq. (2.87) on the interval 2 n - l
x'n(t) = axn(t) + a0C2„. The general solution of this equation on the given interval is *„(*) = e ^ - ^ c - o - V i n , with an arbitrary constant c. Putting t = 2n gives C 2 n = C - a~ a,QC2n

and xn{t) = X(t - 2n)c2„.

(2.91)

For t = 2n — 1, we have xn(2n - 1) = c2n_i = A_ic2n,

c2„ = Al}c2n_i

and for t = 2n + 1, xn(2n + 1) = c 2n+ i = Aic2n. Hence, Ai

/ Aj \ "

c 2n+ i = ^ c 2 n _ i = ^ — j c2„ = A:Jc2n_i = A:} ( T - M From (2.91) it follows that for 0 < t < 1, a;0(<) = X(t)c0 and *o(l) = ci = Aic0. Therefore, c2„ = ( ^ )

co

Cl ,

CI.

2.4. EQUATIONS OF ALTERNATING TYPES

117

and *„(*) = A ( t - 2 n ) ( ^ - ) " c 0 ,

(2.92)

where A(i), Ai, and A_i are given by (2.89). Formula (2.92) is equivalent to (2.90). It was obtained with the implicit assumption a ^ 0, b u t the limiting case of (2.90) as a —► 0 is the solution of problem (2.87) with

a = 0.

■

T h e o r e m 2 . 2 8 . The solution of problem (2.87) has a unique back­ ward continuation on (—oo,0] given by formula (2.90) if Ai 7^ 0. PROOF. If x_n(t) is the solution of equation (2.87) on the interval —In — 1 < t < — 2n+ 1 satisfying the condition x_„(—2n) = c_2„, then from the equation x'_n(t) = ax-n(t)

+ a 0 c_ 2 „

it follows that x_n(t)

= e
At t = —In we get c = (1 + a _ 1 a 0 )c_2„ and x-„(t) = \(t + 2n)c_ 2 „. For t = - 2 n + l, z _ n ( - 2 n + 1) = c_ 2n+ i = Aic_ 2 „, C-2n = ^7 c -2n+lFor t =

-2n-l, z _ „ ( - 2 n - 1) = c_2„_i = A_ic_ 2 „.

Hence, A_i

(X-i\n

C_2„-l = - ^ C _ 2 n + 1 = [—)

C_J

118

2. OSCILLATORY AND PERIODIC SOLUTIONS

and c.-2n := A r

Ui

f

l C_i.

Finally, on — 1 < t < 0 we have x0(t) = X(t)co,

x 0 ( - l ) = c_i = A_ic0.

Thus,

c2n =

- iif)co

and

( A f\ "

= A(t + 2n) f y

£ _ H.W =

For £ < 0, this formula coincides with (2.90).

CQ.

■

T h e o r e m 2.29. The solution x = 0 of Eq. (2.87) is asymptotically stable as t —> +oo if and only if |Ai/A_i| < 1. PROOF. Since \T(t)\ < 1 and A(T) is continuous, the function A(T(<)) is bounded for all t. The proof then follows easily from (2.90) ■ T h e o r e m 2.30. The solution x = 0 of Eq. (2.87) is asymptotically stable as t —» +oo iff any one of the following hypotheses is satisfied: a e2

(l)

a < 0,

(n)

a > 0,

( ° + !) a0 > —j~—-rz[ea — iy 2a a(e + l) - ^a _ xy
(iii)

a = 0,

ao < 0.

PROOF.

For the function X(t) we have X'(t) = (a + a0)eat.

or -a;

a0 < - a ;

2.4. EQUATIONS OF ALTERNATING TYPES

119

If a + ao > 0, then X(t) is increasing, and assuming A(—1) > 0 leads to A(l) > A(—1), that is, Ai/A_i > 1. The conditions a + ao > 0 and A(—1) > 0 can be written as a e - 1 In this case, the solution x = 0 is unstable. The case -a < a0 <

a + a0<0,

a

A(-1)<0

is impossible. Indeed, the inequalities ao < —a,

ao >

a a

e- 1

are inconsistent because —a < a/(e° — 1). Prom a + a0 > 0

and

A(-l) < 0

it follows that ao > - / - r . ea — 1 The inequality Ai/A_i < 1 implies

e" + " V _ i) > e-« + 5»(e- - 1),

(2-93)

a a which is equivalent to a + ao > 0. On the other hand, Ai/A_i > —1 gives e<. +

a

l(e«._l)<_e-<._a«(e-a_l), a a

whence e2fl + l (ea - l ) 2

<

_ao a'

If a > 0, then a(e 2a + 1 ) o < " (e a _ 1)2 •

ffl

120

2. OSCILLATORY AND PERIODIC SOLUTIONS

This contradicts (2.93). For a < 0, we have g(e2a + 1 ) ao > — (e« - l ) 2 and since <

a(e2° + 1) (e» - l ) 2 '

ea - 1 hypothesis (i) ensures asymptotic stability of x = 0. Finally, the con­ ditions a + ao < 0 and A(—1) > 0 simply reduce to ao < —a. The same result follows from the inequality A] < A_i. Furthermore, from Ai > — A_i we obtain „2a

+1

ao

(ca _ 1)2 <

a

"

For a > 0, this confirms hypothesis (ii). The case a < 0 again leads to ao < —a. If a = 0, then Ai l + a0 < 1 = 1 - a0 A-i

holds for ao < 0. T h e o r e m 2.31. All nontrivial solutions of Eq. (2.87) have no zeros in (—00,00) if and only if ae" (2.94) 0. The case A_i < 0, Ai < 0 is impossible since it leads to inconsistent inequalities ae" ao > a ea-V e - 1 On the contrary, the case A_i > 0, \\ > 0 yields (2.94). If a + ao = 0, the only solution of problem (2.87) is x(t) = CQ. ■ a0<-

2.4. EQUATIONS OF ALTERNATING TYPES

121

Theorem 2.32. The problem x'(t) = a(t)x(t) + a0(t)x (2 ^ t l )

t

3.(0) = ^

(2.95)

has a unique solution on [0,00) if a(t) and ao(t) are continuous for t > 0, and C-i

^-1(*)«o(<) dt ± U~\2n),

n = 1,2,...

where U~l is the reciprocal of U and U(t) =

exv(j\(s)ds))

Theorem 2.33. The functional differential inequality

rt+11)

z'(r) +p(t)s(t) + g(*)* (2 ^

< 0,

(2.96)

with p(t) and q(t) continuous on [0,00), has no eventually positive so­ lution if Hm sup /22n"+1 q(t) exp ( £ p(s) d«) dt > 1.

(2.97)

PROOF. Following [4], we prove that the existence of an eventually positive solution leads to a contradiction. To this end suppose that x(t) is a solution of (2.96) such that x(t) > 0, for t > 2n, where n is a sufficiently large integer. For 2n — 1 < t < 2n + 1, inequality (2.96) becomes x'(t) +p(t)x(t) + q(t)x(2n) < 0, or y'(t) + q{t) exp (j^p(») d«) y(2n) < 0 where y(t) = x(t) exp Up(s)

ds\ .

(2.98)

122

2. OSCILLATORY AND PERIODIC SOLUTIONS

Integrating (2.98) from In to In + 1, we have y(2n + 1) < y(2n) (l - £ " + 1 q(t) exp (£/>(«) <*s) 0 for t > 2n, then

1

- llTq^exp (X p(s) ds ) d * > °'

or nlirn

sup^ n " + q(t) exp \J2nP(s) dsj

dt
This contradicts (2.97). So, (2.96) has no eventually positive solu­ tion. ■ T h e o r e m 2.34. If condition (2.97) is satisfied, the functional differ­ ential inequality x'{t) + p(t)x(t) + q(t)x ^2 t-~- \ > 0

(2.99)

has no eventually negative solution. From Theorems 2.33 and 2.34 it follows that subject to hypothe­ sis (2.97), the equation

x'{t) + p(t)x(t) + q(t)x \2 t + 1

= 0

(2.100)

has no eventually positive or eventually negative solutions and therefore we are lead to the following conclusion. T h e o r e m 2.35. Subject to condition (2.97), Eq. (2.100) has oscilla­ tory solutions only. COROLLARY 2.7. Eq. (2.95) has only oscillatory solutions on [0,oo) if Im^iafJ"

aQ(t) exp (- j

a(s) ds) dt < - 1 .

(2.101)

2.4. EQUATIONS OF ALTERNATING TYPES

123

REMARK 3. Condition (2.101) is sharp. For Eq. (2.87) with constant coefficients, (2.101) becomes

ae" a0 < -- a e -1 which is, according to (2.94), one of the two "best possible" conditions for oscillation. T h e o r e m 2.36. Inequality (2.96) has no eventually negative solu­ tion if

n% inf £"-1 *(') CXP ( £ ^ S ) dS) dt < ~l-

(2-102)

P R O O F . Suppose that x(t) is a solution of (2.96) such that x(t) < 0, for t > In — 1, where n is a sufficiently large integer. Integrating from In — 1 to 2n gives

2/(2") (l + £"_j q(t) exp [Jlnp{s) In — 1, then

1+

£-1 g(i) CXP (£ p ( s ) ds) d< > °'

or

feinf

£_i ?(*) exP ( £ K«) ds ) d* ^ _1 '

which contradicts (2.102).

■

T h e o r e m 2.37. If condition (2.102) is satisfied, (2.99) has no even­ tually positive solution. T h e o r e m 2.38. Subject to condition (2.102), Eq. (2.100) has oscil­ latory solutions only. COROLLARY 2.8. Eq. (2.95) has only oscillatory solutions on [0, oo) if Urn sup £ ^ a0(t) exp ( - £ a(s) 1.

(2.103)

124

2. OSCILLATORY AND PERIODIC SOLUTIONS

REMARK 4. Condition (2.103) is sharp. For Eq. (2.87) with constant coefficients, (2.103) becomes a ea — 1 which is, according to (2.94), one of the two "best possible" conditions for oscillation. Theorem 2.39. Ifa0 > a/(e"-l), solution (2.90) with the condition x(0) = co has precisely one zero in each interval In — 1 < t < 2n with integral endpoints. If a^ < —aea/(ea — 1), then (2.90) has precisely one zero in each interval 2n < t < In + 1. Theorem 2.40. The bounded nonvanishing solutions of Eq. (2.87) are periodic. They exist only for a^ = —a or a^ = —a(e2a + l ) / ( e a —l) 2 . In the first case, the solutions are constant; and in the second case, they are of period 4. PROOF. The conclusions follow from (2.90) and from the condition |Ai/A_i| = 1 which is necessary and sufficient for x(t) to be bounded and not vanish. ■ Wiener and Aftabizadeh [292] generalized the above results to equa­ tions of the form x'(t) = f(x(t),x(m

t

~)),

x(0) = co,

(2.104)

where [■] is the greatest-integer function, and k and m are positive integers such that k < m. The argument deviation

T(t) =

t-m

\t + k] m

(2.105)

is negative for mn — k < t < mn and positive for mn < t < m(ra+l) — k (n is integer). Eq. (2.104) is of advanced type on [mn — k,mn) and of retarded type on (mn, m(n + 1) — k). Definition 2.3. A solution of Eq. (2.104) on [0, oo) is a function x(t) that satisfies the conditions: (i) x(t) is continuous on [0,oo).

2.4. EQUATIONS OF ALTERNATING TYPES

125

(ii) The derivative x'(t) exists at each point t £ [0, oo), with the possible exception of the points t = mn — k(n = 1,2,...), where one-sided derivatives exist. (iii) Eq. (2.104) is satisfied on each interval I„ = [mn — k, m(n + 1) — k). Consider along with (2.104) the ordinary differential equation with a parameter ^

= f(x,ii).

(2.106)

If f(x, fi) is continuous and different from zero on a set fi, then on Cl there exists a general integral F(x,[i) = t + g(n), with an arbitrary function g(fi). Assume that the solutions of (2.106) can be extended over all t > 0, and let xn(t) be a solution of Eq. (2.104) on I„ satisfying the condition xn(mn) = cmn. Then we have the equation dx -£ = f(xn,cmn).

(2.107)

If \i = cmn, then F(xn,

Cmn) = t +

g(cmn).

For t = mn this gives F(cmn, cmn) = mn + g(cmn). Therefore, F(xn,cmn)

- F(cmn,Cmn) —*-

mn.

This can be written as fx"

dx

JCmn j[X,

=t-mn. Cmn)

(2.108)

126

2. OSCILLATORY AND PERIODIC SOLUTIONS

At t = mn — k we have xn = cmn-k, hence rcmn

dx

Jcm„-t

( 2 1 0 g )

f{X,Cmn)

Furthermore, at t = m(n + 1) - k we have xn = cm(n+i)-k and , M . +1) _» JCmn

dx / (#>

_k

(2.110)

C

mn)

Eqs. (2.109) and (2.110) also follow directly by integrating (2.107) from t = mn—k to t = mn and from t = mn to t = m ( n + l ) — &, respectively. If Eq. (2.109) has a unique solution with respect to cm„, C mn = / l l ( c m „ _ t ) ,

71=1,2,...,

(2-Hl)

and if Eq. (2.110) has a unique solution with repsect to c m („ + i)_t, Cm(n+1)-Jfc = hziCmn),

71 = 1 , 2 , . . . ,

(2.112)

then = Kcmn-k),

Cm{n+l)-k

71 = 1 , 2 , . . . ,

(2.113)

n = l,2,...,

(2.114)

where h = h^ o h\. From (2.113) cm(n+i)-k = hn(cm-k),

where hn is the n-th iteration of h. Substituting (2.114) in (2.111) gives cmn =Hn-i(cm-k), where i?„_i = hi o /i from (2.108) that

n_1

n = l,2,...,

(2.115)

. Finally, at £ = m — A; and n = 0, we have fCm-t

/

•/co

^

dx

r=m-fc.

/(x,C 0 )

2.116

From (2.116) we find cm_jb = ^(co) and substitute this value in (2.115). Then we substitute (2.115) in (2.108) and find x„(t, CQ), the solution of (2.104) on the interval I„. Note that if (2.106) satisfies uniqueness conditions, then (2.108) has a unique solution with respect to xn. In this case, Eqs. (2.110) and (2.116) have unique solutions with respect to cm(n+yyk and cm-k-

127

2.4. EQUATIONS OF ALTERNATING TYPES

We may always suppose f(x, c m n ) ^ 0, for x 6 (c m n _t, c m n ) , since the assumption f(c, cmn) = 0 for some c G (c m „_i,c m n ) leads to the conclu­ sion t h a t the constant function xn(t) = c is a solution of Eq. (2.104) on the interval mn — k < t < mn. However, this solution does not satisfy the condition xn(mn) = cmn because c ^ cmn. On the other hand, if /(cmnjCmn) = 0 for a particular n, then xn(t) = cmn is a solution of Eq. (2.104) on (mn — k, mn). But in this case, x(t) = cmn is the unique solution on [0,oo) of Eq. (2.104) with the initial condition x(0) = c m „. Hence, it satisfies problem (2.104) only if c m „ = CQ. This concludes the proof of the following: T h e o r e m 2 . 4 1 . Assume that (2.106) satisfies existence and unique­ ness conditions everywhere, its solutions can be extended over [0,oo), and f(co,C(j) ^ 0. If Eq. (2.109) has a unique solution (2.111) with re­ spect to cmn, then on [0,oo) there exists a unique solution of (2.104). / / /(co,co) = 0, then x(t) = CQ is the unique solution of (2.104) on [0,oo). COROLLARY 2.9. The linear equation with constant

(m

t + k~\\ ) ,

coefficients

x(0) = c 0

(2.117)

has a unique solution on [0, oo) if ao + ^

.

(2.118)

P R O O F . For (2.117), Eq. (2.109) becomes dx fCmn = k.

J

c mn — l OX + Cmn-i

a()Cmn

t h a t is, (a0(eka - 1) - a)cmn =

aekacmn-k.

T h e o r e m 2 . 4 2 . Problem (2.117) has on [0, oo) a unique x{t) = X(T(t)) (j±)

co,

solution (2.119)

128

2. OSCILLATORY AND PERIODIC SOLUTIONS

if X-i 7^ 0, where T(t) is given by (2.105) and A(<) = eat + —(eat - 1),

A! = X(m - k),

a

A_i = A(-fc). (2.120)

PROOF. Assuming that xn(t) is a solution of Eq. (2.117) on the in­ terval mn — k < t < m{n + 1) — k, with the condition xjjnn) = c mn , we have x'n(t) = ax„(t) + a0cmn. The general solution of this equation on the given interval is =

%n\J')

e

C

Cmn,

a with an arbitrary constant c. Putting t = mn here gives Oo Cmn — C

Cmn

a and *„(*) = A(t - mn)c m „.

(2.121)

For t = mn — k, we have Xn\mn

— K) = Cmn—i = A _ i C m n ,

Cmn

=

7

Cmn—t)

A_i

and for tf = m(n + 1) — k, xn(m(n + 1) - k) = Cm(„+i)_* = A,cm„. Hence, / Ai \ / Ax \n Cm(n+l)-t = I T I Cmn-k = I T 1 Cm-*, B_1

_J_

Cmn

"A_1

__WM

Cmn t

- -A_1lA_J

From (2.121) it follows that x0{t) = X(t)c0

Cm

-*'

2.4. EQUATIONS OF ALTERNATING TYPES

129

and x0(m - k) = cm_t = AiCo. Therefore, Ai\" ^mn —

xTj °°

and xn(t) = \{t-mn)[~J

co,

(2.122)

where X(t), Ai, and A_x are given by (2.120). Formula (2.122) is equiva­ lent to (2.119). It was obtained with the implicit assumption a ^ 0, but the limiting case of (2.119) as a —► 0 is the solution of problem (2.117) with a — 0. We also note that the inequalities (2.118) and A_i ^ 0 are equivalent. The following assertion is analogous to Theorem 2.28. Theorem 2.43. The solution of problem (2.117) has a unique back­ ward continuation on (—oo,0] given by formula (2.119) if \\ ^ 0 . Theorem 2.44. The solution x = 0 of Eq. (2.117) is asymptotically stable as t —► +00 if and only if |Ai/A_i| < 1. P R O O F . Since — k < T(t) < m — k and A(T) is continuous, the function X(T(t)) is bounded for all t. The proof then follows easily from (2.119). ■

LEMMA 2.5. For m ^ 2k, the equation
(2.123)

has a unique nonzero solution with respect to a. This solution is nega­ tive if m > 2k and positive if m < 2k. PROOF. For m > 2k, the function ipi(r)

m = r

-2rk + \

is decreasing on (0,ri) and increasing on (ri,l), where n

=

f2k\l/{m~k) (m)

130

2. OSCILLATORY AND PERIODIC SOLUTIONS

Since <£>i(l) = 0, then y>i(ri) < 0, and f\(r) is negative on ( r i , l ) . By virtue of y?i(0) > 0, we conclude that i(r) has a unique zero in (0, n ) . This proves the assertion for m > 2k, and the case m < 2k is treated similarly. T h e o r e m 2.45. Let a be the nonzero solution of (2.123) if m ^ 2k, and a = 0 if m = 2k. The solution x = 0 of Eg. (2.117) is asymptoti­ cally stable ast —> +oo if and only if any one of the following hypotheses is satisfied: (i)

a > a,

-a(ema + 1) ' < a0 <

a;

ma

-a(e

(")

a
a0>

(iii)

a — a,

ao < —a.

+1)
or a0 <

a;

PROOF. According to Theorem 2.44, we should verify the inequalities —1 < Ai/A_i < 1, where Ai and A_i are given by (2.120). Assuming, first, A_i > 0 implies ao <

a efca

_

1'

and Ai/A_i < 1 is equivalent to a0 < -a.

(2.124)

Since -a < a/(eka - 1), we retain only (2.124). Further, Ai/A_! > - 1 leads to (a0/a) —ema - 1. For a > a, we have (l/a) 0 and -a(ema + 1) a0 > —^-y-r '-.

, (2.125)

This inequality, together with (2.124), proves hypothesis (i). For a
ma a(ema ++1) 1)


'

(2.126)

2.4. EQUATIONS OF ALTERNATING TYPES

131

but in this case, a(ema + 1) -a < — ,
(2-127)

°>^rrp

and Ai/A_i < 1 is equivalent to QQ > —a. In this case, we retain (2.127). On the other hand, Ai/A_i > —1 gives % ( a ) < - ( e ™ + l), a and (2.126) follows from here for a > a. However, (2.126) and (2.127) are inconsistent. For a < a, we get (2.125), which together with (2.124) proves hypothesis (ii), since a eka

Condition (iii) is obvious.

_

1

<

a(ema +1)
■

Theorem 2.46. The problem x'(t) = a(t)x(t) + a0(t)x (m

\t + k

1) ,

x(0) = Co (2.128)

has a unique solution on [0, oo) if a(t) and ao(t) are continuous for t > 0, and fmn

U-\t)a0(t)dt^U-\mn),

n = l,2,...,

where U~l is the reciprocal of U and

U(t) = exp(j\(s)dsy PROOF. For the solution xn(t) of Eq. (2.128) on the interval I n = [mn — k, m(n + 1) — k)

132

2. OSCILLATORY AND PERIODIC SOLUTIONS

satisfying the condition xn(mn) = c mn , we obtain xn(t) = X(t,mn)cmn,

(2.129)

where X(t, 0) = v(t, 9) + fg v(t, s)a0(s) ds and

v(t,s) = U(t)U-\s). At t = mn — k and t = m{n + 1) - k, relation (2.129) gives Cmn-k = A(mn - k, mn)cmn and cm(n+i)-k = A(m(n + 1) -

k,mn)cmn.

Hence, cmn = A _1 (mn - k, mn)cmn-k

(2.130)

and cm(n+i)_jt = A(m(n + 1) - k,mn)\~l(mn

- k,

mn)cmn-k-(2.131)

From (2.131), the values cmn-k can be determined uniquely in terms of cm-k, where cm-k - A ( m - k,0)co. The proof is concluded by substituting cmn from (2.130) into (2.129), where cm„ is given by " X(mi - k,m(i - 1)) Cm

C

"- °n

X(mi-k,mi)

■

B

(2132)

In this section we shall discuss the oscillatory and periodic properties of solutions of the linear equation x'(t) = a(t)x(t) + a0{t)x (m -^-

) ,

where a(t) and ao(t) are continuous on [0, oo).

x(0) = CQ, (2.133)

2.4. EQUATIONS OF ALTERNATING TYPES

133

T h e o r e m 2.47. / / either of the conditions firm

Jwo^supl ^

m f

/ rmn

a0(t)expU

rm(n+l)—k

/

V

a(s)ds) dt > 1,

, ,

/ rmn

«o(0 exp (jj

m n

.

(2.134)

\

v

o(s) dsj
(2.135)

holds true, then every solution of Eq. (2.133) is oscillatory. PROOF. If Eq. (2.133) has no oscillatory solution, then x{t) > 0 (or x(t) < 0) for sufficiently large t. Let us assume that x(t) > 0 for t > mn — k, where n is a large integer. Then we have dt

(x(t) exp If™

a(s)dsj) = ao(t)x m

j exp I I

a(s)ds).

(2.136)

Taking the integral from mn — k to mn we get /

/ rmn—k

x(mn)lexpl/

\

rmn-k

a(s)ds]+J

I rmn—k

a 0 (f)expl/

\

\

a(s)ds\dt\ = x{mn — k).

Since x(mn) and x(mn — k) > 0, then rmn

/

( rmn

ao{t) exp I /

Jmn—k

or

\

a(s) ds)dt < 1,

\Jt rmn

lim sup /

/ / rmn

ao(t)exp ( /

\

a(s)ds) dt < 1,

which contradicts (2.134). Now taking the integral from mn to m(n + 1) — k of (2.136) we get, after some simplifications and using the fact that x(mn) > 0 and that x(m(n + l) - k) > 0, /

ao(2)exp(/

a(s)ds) dt > — 1,

or lim n—>+oo

rm(n+l)-k

mfJ r ( n + 1 ) Jmn

f rmn

\

a0(t) ex P (I"" a( s ) ds) dt > - 1 , aomexpl/ a{s)ds) dt > — 1,

134

2. OSCILLATORY AND PERIODIC SOLUTIONS

which contradicts (2.135). Proof is the same in case x(t) < 0 for large t. Thus, Eq. (2.133) has oscillatory solutions only. ■ When a(t) and ao(t) are constants, i.e., x'(t) = ax(t) + a0x (m - ^ - J ,

a0 ^ 0,

(2.133')

then the conditions (2.134) and (2.135) reduce to a0 > -

^

(2.134')

and «o < "

, % ea{m-k)

B m ne < -*>.

_ l

(2.135') V

/

The following theorem shows that (2.134') and (2.135') are "best pos­ sible" (sharp) conditions. Theorem 2.48. / /

- ^^-^{m-k)

< «o < ^rzi

( 2 - 137 )

then Eq. (2.133') has no oscillatory solution. P R O O F . Condition (2.137) implies that Ai/A_i > 0. So from (2.119) we deduce that the solution x(t) of (2.133') is always of one sign. ■

In view of Theorems 2.47 and 2.48 we conclude that: COROLLARY 2.10. Eq. (2.133') has no oscillatory solution if and only if condition (2.137) is satisfied. We note that A_i = 0 if a0 = a/(eak - 1) and Xx = 0 if n a

—

o=

a

-a(m-k)

7—n—re ea(m-k)

•

_ I

LEMMA 2.6. The solution x(t) of (2.133') is periodic of period p if and only if cp = CQ, where p is a positive integer. Proof follows from the uniqueness of solution and the autonomous type of Eq. (2.133').

2.4. EQUATIONS OF ALTERNATING TYPES

135

Theorem 2.49. Any solution of Eq. (2.133') is periodic of period 2m if and only if a =

° -^)(1

+ e<,m

)'

< 2 - 138 )

provided a is not a solution of (2.123). PROOF. If Eq. (2.133') has a periodic solution of period 2m, then by Lemma 2.6, Cim = CQ. Hence, / Ai \ 2 m C2m =

T

C0,

or 2m

h -■

(&)' This implies (2.138) or a = —ao. In case of a = —ao, the solution of Eq. (2.133') is constant. If (2.138) holds true, then Ai = -A_i and from Cmn = ( — 1) Co,

we have Cim = Co- Thus by Lemma 2.6, x(t) is periodic of period 2m.B In view of the above results we conclude that: Theorem 2.50. Let a be the nonzero solution of (2.123), ifm^ 2k, and a — 0 if m = 2k. Then every oscillatory solution of (2.133') tends to zero if and only if any one of the following conditions holds true: (i) (ii)

a
a0> - r ^ ( e a m + 1);

a > 5

—a , ^ a m

'

. —a + l)<°o<ea(m_t)_1

0a(rn-k)

Theorem 2.51. The only bounded solution of Eq. (2.133') that does not tend to zero as t —> ±oo is periodic.

136

2. OSCILLATORY AND PERIODIC SOLUTIONS

T h e o r e m 2.52. / / either of the conditions (2.134'), (2.135') holds true, then on each interval (mn — k,m(n + 1) — k) the solution of Eq. (2.133') has a zero tn, given by tn = mn +

iln(- +

aJ

0 a \a Let us again consider Eq. (2.133), where a(t) and ao(t) are continuous on (—00, +00) and are periodic of period m. Then from (2.132) one obtains

Cmn

= Co

(X(m - * , 0 ) \ "

U--M) / '

T h e o r e m 2.53. Assume that a(t), ao(t) are periodic ofperiod m and X(m - k, 0) 7^ 0,

X(-k, 0) ^ 0.

(2.140)

If I ao(r)exp(j

a(s)ds) dr = 1 — exp IJ

a(s)ds) , (2.141)

f/ien i?g. (2.133) /ias periodic solutions of period m, and if j ao(r)exp(l

a(s)ds)dr

= 2j_k a0(r) exp U a(s) ds) dr - exp ( £ "

a(s) ds j - 1, (2.142)

then Eq. (2.133) has periodic solutions of period 2m. PROOF. Conditions (2.140) ensure that Eq. (2.133) has a unique so­ lution on (—00,+co), provided a(t) and ao(t) are periodic of period m. If (2.141) holds true, then from (2.139) we have c m = CQ, which shows that the solution is periodic of period m. If (2.142) holds, then c2m = CQ, and the proof is complete. ■

2.5. OSCILLATIONS IN SYSTEMS OF EPCA

137

5. Oscillations in Systems of Differential Equations with Piecewise Continuous Arguments Linear systems of differential equations with the argument [t + |] have been explored by Wiener and Cooke [293]. Of special interest are oscillatory and periodic properties of the solutions, which depend on the eigenvalues of a certain matrix associated with the system. Therefore, the results can be extended to more general dynamical systems. Consider the initial-value problem

x'(t) = Ax(t) + Bx

(N)'

x(0) = Co, (—oo < t < oo) (2.143)

where A and B are real r x r-matrices and x is an r-vector. The argument deviation (2.144)

T(t) = t - t +

changes its sign in each interval [n — ^,n + ^) with integer n. Indeed, T(t) < 0 for n - \ < t < n and T(t) > 0 for n < t < n + ±, which means that Eq. (2.143) is alternately of advanced and retarded type. The function T(t) is of period 1 and equals t, for —\
M1/2 = M(1), V2y

M 1= M(1),

M_ 1/2 = M ( - ^j ,

M0 = Mzl/2M1/2. (2.145)

Theorem 2.54. / / the matrices A and M_i/2 are nonsingular, then problem (2.143) has on [0, oo) a unique solution

x(t) = M(T(t))Mlt+*]co

(2.146)

138

2. OSCILLATORY AND PERIODIC SOLUTIONS

where T(t) is given by (2.144). The solution of problem (2.143) has a unique backward continuation on (—oo,0] given by (2.146) provided the matrices A and Mi/% are nonsingular. Now we investigate the nonhomogeneous equation x'(t)

= Ax(t) + Bx(t

+ ^)+f(t),

x(0) = co

(2.147)

with constant matrices A and B and a locally integrable vector-function f(t), that is, absolutely integrable on every finite interval of (—oo,oo). The solution xn(t) of Eq. (2.147) on n — \
xn(t)

ds,

Jn

where M(i) is defined in (2.145). The relation xn+i(n + \) = x„(n +1) implies

M

c +i+ e (n+H)/(s)ds

(-^) " r^ "

= M(i)cn +

£+keA^-«>f(s)ds,

or M_1/2cn+l

rn+l

= M1/2cn + jT

eA(n+^f{s)

ds.

From here, c n+1 = M0cn + fn+u

(2.148)

fn+l = MZl1/2 £+1 e ^ " + H ) / ( s ) ds

(2.149)

where

Since the solution of (2.148) is sought in the form cn = Mg kn, then kn+i = K + M 0 " n_1 /n+i

2.5. OSCILLATIONS IN SYSTEMS OF EPCA

139

and

kn = k0+t

MoJfj.

i=l

By virtue of &o = Co, we obtain

x(t) = M(r(*))Mf *] L + CE] M^fA

+d]eA{t~s)f^ds'

*^°- (2-15°)

The solution #_„(£) of Eq. (2.147) on —n — | < t < —n + | satisfying the condition £_„(—n) = c_„ is x_„(i) = M(t + n)c_„ + /* eA^-s)f(s)

ds,

J—n

and its continuation x_„_i(<) on —n — | < £ < — n — | is given by x_„_i(*) = M(* + n + l)c_ n _! + /_
M g ) C_„_t + /;;_7 e^-*-H)/(,) ds = M (_I) c_„ + / 7 " 1 e^-»-H)/(s) *. Prom here, C-n-l = -^o" C_„ + /„ + l, where

fn+l = Mf/2 / 7 " 1 e^-»- M/(,) Hence, c-„ = M0-n

/

-\

[co + ZMlfA \

i=i

/

rf»-

(2-149')

140

2. OSCILLATORY AND PERIODIC SOLUTIONS

and +i]

3

+!1 M(T(t))M$ EMj / ijA x(t) ==M(T(«))il4' UL++ E M

+4/"-"/w«s,

t<0.

(2.150')

This proves the following result. Theorem 2.55. Problem (2.147) has on (—00,00) a unique solu­ tion (2.150), (2.150'), where M(t) and M0 are defined in (2.145) and fj,fj are given by (2.149), (2.149'), if the matrices A and Mo are nonsingular and f(t) is locally integrable. Similar results hold true for the problem x'(t) = Ax(t) + Bx([t\),

x(0) = co.

(2.151)

Theorem 2.56. If the matrix A is nonsingular, then problem (2.151) has on [0,00) a unique solution x(t) = M(t - [t])M[t]co.

(2.152)

The solution of problem (2.151) has a unique backward continuation on (—oo,0] given by (2.152) provided the matrices A and Mi are nonsin­ gular. Theorem 2.57. The problem x'{t) = Ax{t) + Bx{[t]) + f(t),

x(0) = co

(2.153)

has on [0,00) a unique solution [t]

x(t) = M(t - [t])M?] c0 + £ M? j'._x eA^f(s)

ds

+

jy^f(s)ds m

2.5. OSCILLATIONS IN SYSTEMS OF EPCA

141

if the matrices A and M\ are nonsingular and f(t) is locally integrable. This solution has a unique backward continuation on (—oo,0] given by x(t) = M(t - [t])M® L

+ £ M{'1 p+i

e ^ - ' + 1 - * ) / ( s ) ds J

+

f[t]eA^f{s)ds.

T h e o r e m 2 . 5 8 . The solution x = 0 of Eq. (2.143) is globally asymp­ totically stable as t —► + o o if and only if the eigenvalues Xj of the matrix MQ satisfy the inequalities \Xj\
j = l,...,r.

(2.154)

There exists a nonsingular matrix S such t h a t M0 =

SJS-\

where J is a diagonal or Jordan matrix with the diagonal elements Xj. Hence, cn = ( 5 J 5 - 1 ) " c 0 = SrS^co

(2.155)

or cn = EPj(n)y,

k
(2.156)

where the components of the vectors Pj(n) are polynomials of degree not exceeding k — 1. This implies that c„ —► 0 as n —► + o o if and only if (2.154) holds, and the conclusion of the theorem follows from the formula xn(t) = M(t - n)cn.

■

(2.157)

T h e o r e m 2 . 5 9 . The solution x = 0 of Eq. (2.143) is globally asymp­ totically stable as t —> + o o if and only if MQ —► 0 as n —* oo. The same is true for Eq. (2.151) with Mi instead of MQ.

142

2. OSCILLATORY AND PERIODIC SOLUTIONS

REMARK 5. The condition MQ —> 0 as n —► oo holds true if for some number N it appears that the moduli of all elements mfj of the matrix M0N do no exceed q/r, where 0 < q < 1 and r is the order of Mo, i.e.,

Kl<~, J

l
(2.158)

r

Indeed, in this case each element of the matrix M$N does not exceed r(q/r) ■ (q/r) = (? 2 / r )> each element of M03N does not exceed q3/r, and so on, which leads to the conclusion that (2.158) implies Mfi —> 0 as n —» +oo. Theorem 2.60. All solutions of Eq. (2.147) tend to zero as t —► +oo if the eigenvalues of MQ satisfy (2.154) and lim/(i) = 0 as t —* +oo. Theorem 2.61. If all eigenvalues of the matrix Mo are negative, then each component xl(t) of every solution of system (2.143) is oscillatory and, more precisely, has a zero in each interval n < t < n + 1, for sufficiently large n. PROOF. Let x'n(t) denote the i-th component of the solution xn(t) of problem (2.143) on the interval n — ^
'n=

EP<;(«)\".

i=

l,...,r.

For the continuation xn+i(t) of the solution o n n + | < < < n + | w e have

4 + i = X>.;(" + l)A"+1.7=1

If c'n = 0, for infinitely many n, then x'(t) is oscillatory. Assuming c*n ^ 0, for all sufficiently large n, we conclude that lim ^ ± i = A,- < 0 n—»oo

rt

(since the coefficients Pij(n) are polynomials of n), where A,- is one of the eigenvalues of MQ. Hence cj,+1/c}, < 0, starting with some n,

143

2.5. OSCILLATIONS IN SYSTEMS OF EPCA

which implies that x*(t) has a zero in each interval n < t < n + 1, for sufficiently large t. ■ REMARK 6. Under the conditions of Theorem 2.61 all components of every solution of system (2.143) have, for sufficiently large n, zeros in each interval n — \ a/(ea'2 - 1)

or

b < -aea/2/(ea'2

- 1)

holds true, then every solution of the scalar equation x'(t) = ax{t) + bx ( t + - J

(2.159)

is oscillatory [5]. Indeed, let a/2

+

/ a / 2 _ jx

-lfc

then either of the above inequalities implies A < 0. Furthermore, in each interval (n — | , n + | ) , the solution of (2.159) has a unique zero tn given by

=n

*" -Mnr)-

Theorem 2.62. / / the matrix MQ has a negative eigenvalue, then there exists an initial vector CQ such that the corresponding solution x(t) of problem (2.143) has an oscillatory component with a zero in each interval n < t < n + 1, for sufficiently large n. PROOF. Denote c„ = x(n) and let d0 = 5 _ 1 c 0 in (2.155), then c„ = SJndo. Assume, for simplicity, that J = diag(Ai, A2,..., Ar) and Ai < 0, and put d0 = {1,0,0,...,0}. This gives c„ = { s n A ^ i A " , . . . ,s r iA"} where 5 = («,_,), i,j = l , . . . , r . Since S is nonsingular, there exists an element Sn ^ 0, hence hm ^ ± i = Ai < 0,

144

2. OSCILLATORY AND PERIODIC SOLUTIONS

which proves that the component x'(t) of the solution x(t) has a zero in each interval (n, n + 1), for sufficiently large n. ■ Theorem 2.63. / / the matrix M_i/2 = M{—\) is negative definite, then every solution of Eq. (2.143) has a component with a zero in each interval [n — h,n]. If the matrix Mt/2 = M(^) is negative definate, the same property holds true for each interval [n, n + 5]. PROOF. By virtue of (2.157), the scalar product of vectors cn = x(n) and c„_ i/2 = x(n - | ) is (c n ,c„_ 1/2 ) = (cn,M_i/2cn) < 0, with the equality sign only if c„ = 0. However, if c„ = 0 for a particular integer n then x(t) E 0 on [n, oo). The second part of the theorem is proved similarly. ■ For Eq. (2.159), the condition M_ 1/2 < 0 reduces to b > a/(ea/2 - 1), and in this case every solution has a single zero in each interval [n—|, n]. The inequality M\ji < 0 becomes 6 < — ae a / 2 /(e a / 2 — 1) in which case the same conclusion holds true for each interval [n,n + t]. Theorem 2.64. // all eigenvalues of the matrix MQ are positive and A is a diagonal matrix, then each component x'(t) of every solution x(t) of system (2.143) is either nonoscillatory or identically equals zero, starting with some t. P R O O F . Let x(n) = cn, x(n - | ) = c„_i/ 2 , x(n + i) = c n+1 / 2 , then from (2.157),

c n -i/2 = M_i/2c„,

c„ +1/2 = Mi /2 c„

and c n + 1 / 2 = Mi/2Mzl/2cn_i/2.

(2.161)

Denote M2 = Mll2MZ[/2 and compare the eigenvalues of the matrices M2 and MQ = Since M2 = M_1/2M0Mzl/2,

(2.162) MZ\nM\n. (2.163)

2.5. OSCILLATIONS IN SYSTEMS OF EPCA

145

then M 2 - XI = M_1/2M0Mzl/2

- M_l/2XIMZll/2

= M_ 1/2 (M 0 -

XI)Mz\/2

and det(M 2 - A/) = det(M 0 - A/). Hence, the matrices M2 and Mo have the same eigenvalues, and simi­ larly to (2.156), we can write Cn-i/2 = £ ©(")*">

* < r,

(2.164)

where Aj are the eigenvalues of Mo and the components of the vectors qj{n) are polynomials of degree not exceeding k — 1. With the notation x'(n — | ) = cJ,_1/2> assume first that c'n_1,2 ^ 0 for all sufficiently large n. Then from (2.164), lim °-^!1 = Xi > 0, C

(2.165)

n-l/2

where A,- is an eigenvalue of MD (all A;- are positive). Therefore, the graph of xt(t) either does not cross the interval n — \
rfV(t) dt* ~N

dx\t) dt '

and ^

= /?„e^-"), fin^ \

n-\ n-\

146

2. OSCILLATORY AND PERIODIC SOLUTIONS

with /?„ constant in the above interval. If /?# = 0, for a particular N, then x\t) is constant in (iV - \,N + \). And if (3n ^ 0, then dxl/dt ^ 0 in (n — 4,n + | ) , which implies that x'(t) cannot cross this interval more than once. Hence, x'(t) has no zeros for sufficiently large t and is nonoscillatory. If c^—1/2 = 0 ^ or a ^ sufficiently large n, then x'(i) identically equals zero, starting with some t. It remains to verify whether the sequence {ctn_i/2} can contain a proper infinite subset of zero terms. If this is so, consider an infinite sequence of pairs (4-i/2> 4+i/2> w i t h 4-1/2 7^ 0 and 4 + 1 / 2 = 0. By virtue of (2.165), for a sufficiently small e > 0 such that e < \j(j = 1 , . . . , r), we have 0 = 4±i/2 C

>Xi-e,

n>

N(e).

n-l/2

This contradiction concludes the proof. T h e o r e m 2.65. // the matrix MQ is nonsingular and has a positive eigenvalue and A is a diagonal matrix, then there exists an initial vec­ tor Co such that the corresponding solution of problem (2.143) has a nonoscillatory component. PROOF. Let x(t) be the solution of problem (2.143), then from (2.161) and (2.162) it follows that X

\

+

2)

= Cn+1/2 =

M2 Cl 2

" / '

or c„+i/2 = M2nM1/2co. By virtue of (2.163), this can be written as c„+i/2 = RJnR'lMl/2c0,

R = M_ 1 / 2 5 = (Uj).

Since Mo is nonsingular, the same is true for Mi/ 2 , hence Mi/2co ^ 0 if Co 7^ 0. Assume, for simplicity, that J = diag(Ai,..., Ar) and Ai > 0, and put R-1M1/2CQ = {1,0,... ,0}. Then cn+i/2 = {'n-^i > *2iA",..., *riA"}

147

2.5. OSCILLATIONS IN SYSTEMS OF EPCA

and there exists an element tu ^ 0, hence

i im ^±iZi C

= Al>0.

n-l/2

The completion of the proof follows now from the last part of the pre­ vious theorem. T h e o r e m 2.66. Assume that the eigenvalues of Mo are positive dis­ tinct, the eigenvalues of A are real distinct, and A + B is nonsingular. Then there exists an initial vector CQ such that the corresponding solu­ tion of system (2.143) has a component which is either nonoscillatory or identically equals zero, for suffciently large t. P R O O F . Suppose that x'(t) is an oscillatory solution component; then by virtue of (2.165), there exists an infinite set of intervals [n—|, n + | ] in each of which x'(t) has an even number of zeros. Hence, the derivative dxl(i)/dt has a zero in each of these intervals, and our purpose is to show that this is impossibe for some i. From (2.145) and (2.157) we obtain

^

= e^-")(A + 5)M0"co,

( n - ± < i < n + i).

There exist real matrices Q and S such that Q~lAQ = 2) = diag(/zi, /x 2 ,..., fir), S~lM0S = J = diag(A!, A 2 ,..., AP) and therefore, *^L = Qe^-^Q-\A dt

+

B)SJnS-lco.

Denote P = Q-\A

+ B)S = (pij),

Q = {qij),

S-lc0 =

then

^f = (^-'V(A),

{d1,d2,...,dr},

148

2. OSCILLATORY AND PERIODIC SOLUTIONS

where PnW

= { p i l A J d i + • • • +pirK*r,

■ ■ • ,PrlA?dl + • • • +

PrAM-

Hence, dx' dt;f

r

=h_Ex M* - n)\nk

and hit(t - n) = E dkqmke^-n\

(2.167)

j=i

Choosing 5-^0 = { 1 , 0 , . . . , 0} gives -^L = hil(t-n)\n1.

(2.168)

Assuming there is an infinite sequence of integers n m and values £m such that n m — I < i m < nm + | and dx'nm/dt = 0 at t = tm, we conclude from (2.167) and (2.168) that M*m - nm) = E qiiPii^i{tm~nm) = 0. i=J Since —| < tm — nm < | , the analytic function /i,-i(z) has an infi­ nite number of zeros with a limit point in the interval [— ^, A]. Hence, hn (z) = 0 identically and QijPji=Q,

j =

l,...,r

In this equation the index i is fixed but we observe that, although the numbers nm and tm depend on i, the inequalities —\
2.5. OSCILLATIONS IN SYSTEMS OF EPCA

149

are oscillatory leads to the conclusion that kn(z) = 0 identically, for all i, that is, qijPji=0,

*,j = l , . . . , r .

From here, Pil=0,

j = l,...,r,

which is impossible since P is nonsingular.

■

Theorem 2.67. If the matrix Mo has no positive eigenvalues and no eigenvalues with equal moduli, then each component of every solution of system (2.143) is oscillatory. PROOF. Let the solution component x'(t) satisfy x'(n) = c'n and as­ sume that c'n ^ 0 for all sufficiently large n (otherwise, x'(t) is oscilla­ tory). If lim^oo c'n+l/c'n = A,-, where A,- is a negative eigenvalue of Mo, then x'(t) is oscillatory. Since the eigenvalues of MQ are simple, then r

i=i

with constant coefficients fc,j depending on the initial vector x(0) = Co, and if limcj, +1 /cj, ^ A, < 0, it remains to consider the possibility when the asymptotic behavior of c*n+i/c'n is determined by A:An+1 + kXn+l k\n+k\" ' where A, A are complex conjugate eigenvalues of MQ and k, k are com­ plex conjugate constants. But in this case, the behavior of ctn+m/cln as n —► oo is described by fcA"+m + kXn+m kXn+kXn ' for any natural m > 1. Denote 9 = arg A,

<j> = arg k,

150

2. OSCILLATORY AND PERIODIC SOLUTIONS

then kXn+m + kXn+m kXn + k"Xn

\X\m cos{n6 + <j> + m6) cos(nd + <j))

We may assume cos(n9 + <j>) ^ 0 for an infinite sequence of values of n, otherwise kXn + kX" = 0 for all sufficiently large n, and in this case a different eigenvalue would substitute for A in the preceding argument. We write cos(n# +
and choose m so that cos m6 is close to —1 (hence, sinm# is close to 0), which shows that cj, +m /cj, < 0 for infinitely many values of n and proves that z l (i) is oscillatory. ■ REMARK 7. Theorems 2.61, 2.62, 2.64, 2.65, 2.66 hold true for sys­ tem (2.151), under appropriate conditions on the matrix Mi defined in (2.145). EXAMPLE

2. Consider the system [7] x\t) = ax(t) + by([t}), y'(t) = cy(t) + dx([t]),

(2.169)

with the initial data x(0) = co,y(0) = bo. Let Ai and A2 be the roots of the characteristic equation A2 - (e" + ec)A + ea+c - — (ea - l)(e c - 1) = 0,

(2.170)

corresponding to Mi. If these roots are real, assume Ai > A2. A solution (x, y) of (2.169) is said to be oscillatory if x and y both have arbitrarily large zeros on [0,00). Otherwise (x,y) is said to be nonoscillatory. Denote F(s) = o ( e u - s) + ^(e° - 1)60,

G(s) = b0(ec -s) + -y - 1)0, x(n) = c„,

y(n) = 6„.

2.5. OSCILLATIONS IN SYSTEMS OF EPCA

151

The system (2.169) has no oscillatory solutions if ac(ea - ecf , , acea+c 4(e° - l)(e c - 1) 0 and the matrix A with entries an = a, 012 = 0,021 = 0,022 = c is diagonal. Also, (2.169) has no oscillatory solution if

for in this case Ai and A2 are real and c n + 1 ^F(A 2 )A 1 -F(A 1 )(A 2 /A 1 )"A 2 c„ F(A2) - F(Aj) (A2A)" ' b n + 1 ^G(A 2 )A 1 -G(A 1 )(A 2 /A 1 )"A 2 bn G(A 2 )-G(A 1 )(A 2 /A 1 )" • Hence, lim„_ooC„+i/c„ = Ai > 0 and limn_»006„+i/6„ = Ai > 0. The system (2.169) has oscillatory solutions if < -/c(e°-eC)2lV (2.171) v 4(e" - l)(e c - 1) ' because inequality (2.171) implies that Eq. (2.170) has complex roots. Also, (2.169) has oscillatory solutions if M

U

ace a+c > {ea _ 1 ) ( e c _ I)

^

d

F

^2)

= 0,

(2.172)

since inequality (2.172) means that (2.170) has real roots and A, >0,A 2 < 0 . From F(A2) = 0 we have c„ + i/c„ = A2 < 0 and bn+i/bn = A2 < 0 (clearly, F(A2) = 0 implies G(A2) = 0). Hence, every solution of (2.169) is oscillatory. Thus we conclude that a necessary and sufficient con­ dition for the system (2.169) to have oscillatory solutions only is ei­ ther (2.171) or (2.172).

152

2. OSCILLATORY AND PERIODIC SOLUTIONS

T h e o r e m 2.68. The solutions of problems (2.143) and (2.151) are periodic of period p if and only if cp = CQ, where p is a positive integer. PROOF. We prove the theorem for (2.151). If the solution x{t) is p-periodic, then x(t + p) = x(t) which implies cp = x(j>) = x(0) = c$. Conversely, since (2.151) is an autonomous system, it follows from the condition cp = CQ and from the uniqueness theorem that the solution coincides on the intervals 0 < t < 1 and p < t < p + 1. Continuity of the solution implies cp+\ = x(p +1) = x(l) = c\, and the above process can be repeated for the intervals [1,2) and [p + 1, p + 2) and for the following pairs, until we reach [p — 1,p) and [2p — 1,2p) and prove that the solution coincides on these intervals. Then by continuity c
T h e o r e m 2.69. The solution of problem (2.143) is periodic of pe­ riod p if and only if either of the following conditions is satisfied: (i)Mp0=I, (ii) Co is an eigenvector of Mfi corresponding to the eigenvalue A = 1

of Ml PROOF. The conclusion follows from the formula cp = MQCQ and from the criterion cp = c$ for the existence of periodic solutions, which yield

(M 0 p -/)c 0 = 0.

■

T h e o r e m 2.70. The equality MQ = I takes place if and only if the matrix MQ is diagonalizable and all its eigenvalues A,- satisfy A? = 1. PROOF. Assume MQ = / and write MQ = SJS-1, onal or Jordan matrix, then SJPS-1 = i.

where J is a diag­

2.5. OSCILLATIONS IN SYSTEMS OF EPCA

153

Hence, Jp — I which means that J is diagonal and its eigenvalues are p-order roots of unity. On the other hand, if Mo is diagonalizable, then M 0 = S1)S'1, where 2) is a diagonal matrix and Ml = 533 p 5 _ 1 . Since Mo and 2) are similar, they have the same eigenvalues. By virtue of \Vj = 1, for all eigenvalues of Mo, we conclude that 2) p = I and Ml = I. U COROLLARY 2.11. All solutions of system (2.143) are p-periodic if and only if MQ is similar to a diagonal matrix and all its eigenvalues are p-order roots of unity. EXAMPLE 3. Let bd < 0 in (2.169), then every oscillatory solution of system (2.169) is periodic of period p if and only if bd

=,aC{C~\ c (e°-l)(e -l)

ande° + e< = 2 c o s ( ^ ) , V p I

y(2.173) !

where m and p are relatively prime and m = 1,2,..., [(p—1)/4]. Indeed, since bd < 0 and the solution is oscillatory, the roots Ai and A2 of (2.170) are complex. Therefore, the conditions Af = 1 and A<> = 1 are equivalent to (2.173) If bd > 0, the roots Xu A2 of (2.170) are real. Since Ai + A2 = e° + ec, the cases Ai = 1,A2 = —1, and Ai = A2 = —1 are impossible. The system (2.169) cannot have solutions of period p = 1, except constant solutions. Indeed, the case when all solutions of (2.169) have period 1 is dismissed because the equality M\ = / implies a — b — c = d = 0 (the entries of Mi are mn = e", mi 2 = b(ea — l)/a, m2i = d(ec — l)/c, m22 = ec). If some solutions of (2.169) are 1-periodic, the initial vector x(0) = Co, y(0) = 60 is an eigenvector of Mi corresponding to the eigenvalue Ai = 1. Hence, F(Ai) = c0(ea - 1) + h-{ea - 1)60 = 0, G(A1) = 6 0 ( e c - l ) + - ( e c - l ) C o = 0, and c

=

F(A2) Ai — A 2

^

=

G(X2) Ai — A 2

154

2. OSCILLATORY AND PERIODIC SOLUTIONS

which shows that the above solutions are constant. It remains to in­ vestigate the possibility Ai = ea + ec + 1,

A2 = - 1 .

(2.174)

If A2 = - 1 is an eigenvalue of Mj, then A^ = 1 is an eigenvalue of M?. Therefore, (2.169) has solutions of period 2 if and only if the initial vector (co, &o) 1S a n eigenvector of Mi corresponding to its eigenvalue A2 = —1. Eqs. (2.174) for the roots of (2.170) are equivalent to the condition v (ea - l)(e c - 1) ' We conclude that system (2.169) has periodic solutions of period p — 2 if and only if the conditions

F(A1) = ^ ( e a - l ) 6 0 - ( e c + l)c 0 = 0 and (2.175) hold true. Theorem 2.71. In addition to the conditions of Theorem 2.55, as­ sume that f(t) is periodic of period 1 and let fo = Mzl/2Jo1eA^-^f(s)ds.

(2.176)

Then problem (2.147) has a solution of integer period p if and only if either of the following conditions is satisfied: (i) Sp = 0, f(t) and Co are arbitrary, where

sP-

= t vr,

(ii) Sp^0 and go = (I — MQ)CQ — /o is either zero or an eigenvector of Mo corresponding to an eigenvalue A ^ 1, which is a p-order root of unity. PROOF. Since f(t) is of period 1, the substitution s —► s + n changes formula (2.149) to

fn =

M-_\l2jlQeAW-°)f{s)ds,

2.5. OSCILLATIONS IN SYSTEMS OF EPCA

155

that is, /„ = /o, for all integers n. Following the proof of Theorem 2.55, we can write cp = Mfikp and p

kp = co + E M ) " 7 r Hence, cp = M0pc0 + £ AiJ"7o, and it remains to use the condition cp = CQ for the existence of p-periodic solutions to (2.147), to obtain (/ - M$)co - Spf0 = 0.

(2.177)

Since / - M0P = SP(I - M 0 ), then SP((I - M0)c0 - /„) = 0. Furthermore, the substitution s —► — n — 1 + s changes (2.149') to /» = /o, where /o =

-MJ%j\AW-*f{s)ds,

that is, /o = -Mo'VoThe condition c_p = Co leads to c0 = MoPc0 +

J2MoP+jfo

or M f o = c0 + E M^/o. This can be written as c0 = M f o + E M ^ V o , which is equivalent to (2.177) and proves the theorem. Clearly, the equation Sp = 0 implies M$ = I, for p > 2. ■

156

2. OSCILLATORY AND PERIODIC SOLUTIONS

COROLLARY 2.12. In addition to the conditions of Theorem 2.55, as­ sume that f(t) is of period 1. Then every solution of system (2.147) is of period p > 2 if and only if Sp = 0.

T h e o r e m 2.72. Every solution of system (2.147) in which f(t) is of period 1, is of period 1 if and only if Mg = I and /o = 0, where /o is given by (2.176). In this case x(t) = co + /oT(<) eA™-*f(s) T(t)

=

ds,

(2.178)

t-[t+\\,

if none of the eigenvalues of A is of the form 2nin, i = y/—l and n = 0,±l,±2,... . P R O O F . The criterion c\ = c0, c_i = to (2.147) is equivalent to

CQ

for solutions of period p = 1

(I - M0)c0 = /oThis equation is satisfied by arbitrary values of CQ if and only if MQ = I and /o = 0. Hence, all /_,- = 0, and from (2.150),(2.150') we get x(t) = M(T(t)),co + I* eA{t-^f{s)

ds,

(2.179)

Jn

n = [t + k). Furthermore, MQ = / means Mi/ 2 = M_!/2, that is, e(i/2M + (e d/2M

_

I)A-iB

= e -d/2M +

( e -d/2M _

I)A~lB,

whence e(i/2M

_ e-(i/2M _ (e-(i/2M _

l/2)A

e(

)A~lB.

Since A has no eigenvalues of the form 2nin, the matrix eA/2 — e~A^ is nonsingular and B = —A. By virtue of (2.145), M(t) = I, and the substitution s —> s + n changes (2.179) to (2.178). Simultaneously we have shown that if all solutions of system (2.143) are of period p = 1, they are constant, provided none of the eigenvalues of A is 2win. ■ Oscillatory properties of n-dimensional systems (1.49) have been stud­ ied by Ladas [155] and Gyori, Ladas, and Pakula [111], who have shown that every solution of (1.49) oscillates (componentwise) if and only if its

2.6. A PIECEWISE CONSTANT ANALOGUE

157

characteristic equation has no positive roots. Stability and oscillation of neutral EPCA with both constant and piecewise constant delays have been investigated by Partheniadis [213]. These properties for a second order EPCA alternately of retarded and advanced type were explored by Ladas, Partheniadis, and Schinas [157]. Huang has discussed os­ cillatory and periodic solutions of a system of two first order linear equations alternately of advanced and delay type in [126], and oscilla­ tions and asymptotic stability for first order neutral equations of the same type in [127]. Papaschinopoulos [211] has obtained results con­ cerning asymptotic stability and oscillatory behavior for a class of third order linear neutral EPCA. Gopalsamy, Gyori, and Ladas [93] have studied oscillations of delay equations with continuous and piecewise constant arguments. Oscillatory properties of some classes of nonlinear EPCA may be found in [109], where it is shown that under appropri­ ate hypotheses a nonlinear EPCA oscillates if and only if an associ­ ated linear equation oscillates. Research on nonlinear EPCA, includ­ ing their stability and oscillatory properties, is still developed insuffi­ ciently. However, certain progress in this direction has been made by Carvalho and Cooke [38], Furumochi and Hayashi [86], Gopalsamy, Kulenovic, and Ladas [95], Huang [125], Ladas [155], Seifert [241], and Vlahos [269]. The characteristic equation for linear EPCA with both con­ stant and piecewise constant delays was discussed by Grove, Gyori, and Ladas [103]. Finally, Chapter 8 of the book by Gyori and Ladas [110] is devoted to EPCA. 6. A Piecewise Constant Analogue of a Famous FDE The functional differential equation x'{t) = ax(t) + bx(Xt),

0
(2.180)

was the subject of many profound studies, starting with the celebrated work of T. Kato and J. B. McLeod [138]. We consider the equation x'(t) = ax(t) + bx(h

— J,

x(0) = CQ

(2.181)

with real constant coefficients, where [■] is the greatest-integer function, 0 < A < 1, and h is a positive constant.

158

2. OSCILLATORY AND PERIODIC SOLUTIONS

Definition 2.4. A solution of Eq. (2.181) 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 * £ [0, oo), with the possible exception of the points t — nh/\ (n = 0 , 1 , . . . ) , where one-sided derivatives exist, (iii) Eq. (2.181) is satisfied on each interval nh

(n + l)h

Theorem 2.73. Problem (2.181), with constant coefficients and pa­ rameters 0 < A < 1, h > 0, has a unique solution on [0, oo). PROOF. Eq. (2.181) on the interval nh/X < t < (n + \)h/X takes the form x\t) = ax(t) + bsn,

sn = x(nh).

The solution of this equation, satisfying the condition x{nh/X) = c„ is x(t) = c„e0('-n/,/A> + a-1bsn{e
- 1).

(2.182)

Continuity of the solution at t = (n + l)h/X implies c n+1 = eah'xcn + a~lb{eahlx - \)sn

(2.183) on A

The solution of this equation is sought in the form c„ = e ''/ Jfcn, whence kn+1 = kn + a-xb{eahlx - l)e-«"+WSn,

k0 = Q,.

From here, kn = c0 + a-lb(eah'x

- 1) £

e-ahi'xSi^

and c„ = coe'WA + a-lb{eah'x

- 1) f, c"*
(2.184)

Substituting (2.184) in (2.182) determines the solution of (2.181) suc­ cessively on each interval I„ = [nh/X, (n + l)h/X], since s 0 = co and Si = x(ih), Hi e I m , where m = [iX] < i. The delayed action of the

2.6. A PIECEWISE CONSTANT ANALOGUE

159

second term on the right of (2.181) is clearly visible in (2.184), where the value c„ of the unknown solution x(t) at t = nft/A depends on its values S{ at lagging times ti = ih (i = 0 , . . . , n — 1). ■ EXAMPLE 4. For the equation x'(t) = ax(t) + bx(h ^j- V

x(0) = co

we have I„ = [2nh, (2n + 2)ft], and Eq. (2.183) takes the form s2n+2 = e2ahs2n + a~lb{e2ah - l)s„.

(2.185)

This difference equation is not of finite order. At t — (2n + l)ft, for­ mula (2.182) gives s 2n+1 = eahs2n + a-lb{eah - l)s„.

(2.186)

From (2.185) we obtain *2„ = Coe2anh + a-lb(e2ah - 1) £

e ^ ^ s ^ .

Finally, (2.185) and (2.186) yield the relation S2n+2 - (e°

+ l)s2n+l + e" S2n = 0.

T h e o r e m 2.74. If a + b > 0, all nontrivial solutions of Eq. (2.181) are unbounded, monotonic, and none of them has a zero in (0, oo). PROOF. Assume, for instance, CQ > 0 and consider the case a > 0. By virtue of the inequality a + b > 0, the derivative x'(t) = (a + b)coeat is positive on 0 < t < ft/A. Hence, x(t) is positive and increasing on this interval, which implies C\ > si > 0. For ft/A < t < 2ft/A, the derivative x (t) = (ac\ + &s1)ea('~/l/A) retains the sign of ac\ + bsi, which can be written as (a + b)si + a{c\ — s\) > 0. Assuming, by induction, cn > sn > 0 implies x'{t) > 0 on the interval I„, because it retains the sign of acn + bsn = (a+b)sn + a(cn — sn) > 0. Furthermore, from (2.183) we have c„+i - c„ = a-\eahlx

- l ) ( o c + bsn),

(2.187)

160

2. OSCILLATORY AND PERIODIC SOLUTIONS

and since acn + bs„ > (a + b)sn > (a + 6)co, then c„ +1 - c > a-\eah/x

- l)(o + 6)co

and c„ > c0 + na - 1 (e o h / A - l)(a + 6)c0, which proves that the solution x(t) is unbounded and increasing. Consider the case b > 0. The solution of (2.181) is positive and increasing onO < t < h/X. Hence, CQ < «i < C\ and from the equations c2 =

ah/x e

Cl

+ a-lb(eah'x

ci = eah/xc0 +

- l)s:,

a-1b{eah/x-l)c0

we see that c2 - d = eahl\Cl - c0) + a-lb{eahlx - 1)(», - q,) > 0. (2.188) Then, (2.187) gives ac\ + bs\ > 0, which means that x'{t) > 0 and x(t) is increasing on h/t < t < 2h/\. The assumption Co < c\ < ■ ■ ■ < c„ implies Co < s\ < ■ • ■ < sn and c n+1 - c = eaA/A(cn - c„_!) + a-16(ea/'/A - l)(s„ - s n _i) > 0. This proves that the solution of (2.181) is positive and increasing on [0,oo). Prom (2.183) we have c n+1 > {eah'x + {eah'x - l)a-xb)sn,

(2.189)

and from (2.182) at t = nh we obtain sn = cmeah{-nX-m^x + a-lbsm{eah^x-m^x

- 1),

where m = [n\]. Indeed, s„ = x(nh) and m/X < n < (m + 1)/A. If m = nX, then s„ = c m , and subtituting in the right of (2.189) cm = eah'xcm^

+ a-lb(eah'x

-

\)sm.u

we obtain the estimate c n + 1 >(e"''/ A + (e a ''/ A -l)a- 1 6) 2 S m _ 1 . If m ^ nA, then we write Sn > (^("A-mJ/A

+

(ea/,(„A-m)/A _

i^-lfc)^

2.6. A PIECEWISE CONSTANT ANALOGUE

161

and substitute this expression in (2.189). The inequality a + b > 0 implies eat + a~lb(eat — 1) > 1, for t > 0, and it follows by induction that c„ —» oo as n —► oo. ■ Theorem 2.75. If 0 < b < —a, then all nontrivial solutions of Eq. (2.181) monotonically tend to zero ast —► +oo and, hence, none of them has a zero in (0,oo). PROOF. Assuming CQ > 0 shows that x(t) is decreasing and positive on [0, h/X]. Hence, 0 < c\ < s\ < CQ, and from (2.188) it follows now that ci — c\ < 0. Thus, x(t) is decreasing and positive on [h/X, 2h/X]. We conclude by induction that x(t) is decreasing on (0, oo), and (2.182) implies x(t) > 0. ■ COROLLARY 2.13. If Eq. (2.181) has a nontrivial oscillatory solution, then a + 6 < 0,

b < 0.

(2.190)

Theorem 2.76. If a < 0 and b < 0, then all nontrivial solutions of Eq. (2.181) oscillate. PROOF. Assuming the existence of a positive solution (for large t) implies c„ > 0 and sn > 0, for all sufficiently large n. Hence, 0 < c n+1 < eaA/Ac„ < c n , which means the solution is decreasing to zero, at least as eanhlx. There­ fore, c m+ i < sn < cm where m = [nX], which indicates that s„ decreases as eamhlx. The inequality c„ +1 < eah'xcn + a-lb(eah>x - l)c m + 1 that follows from (2.183) is impossible, since for large n the right-hand side becomes negative. ■ Theorem 2.77. For a + b < 0 and b < 0, every nontrivial solution of Eq. (2.181) has a zero in (0, oo). Let Co > 0 and assume that x(t) > 0, for t > 0. Since a + b < 0, the solution x(t) is decreasing on 0 < t < h/X, and c\ < s\. On h/X < t < 2h/X, the derivative x'(t) retains the sign of ac\ + bs\, which is written as (a + b)c\ + b{s\ — c\) < 0, since b < 0. Hence, x(t) is PROOF.

162

2. OSCILLATORY AND PERIODIC SOLUTIONS

decreasing on h/\ c„ > 0 implies x'(t) < 0 on I n , since acn + bsn = (a + b)cn + b(sn - c„) < 0. Therefore, 0 < c n+1 < {eah'x + {eah'x -

\)a-lb)cn,

which shows that c„ and s„ decrease as (eah^ + (eahlx — l)a _ 1 6) n and (e"*/* + (C«*/A _ i) a -i6)m > respectively. This violates (2.183). ■ T h e o r e m 2.78. / / |6| < —a, then the solution of (2.181) tends to zero as t —* +oo, for any CQ. PROOF.

Denote Mn = sup|x(i)|,

t G [(n — l)h,oo),

ah x

q= e / + a-l\b\(eah'x-l). Then |c n _i| < Mn, |s„_i| < M„, and from (2.183) we get \cn\ < qM„, while the condition |6| < —a implies q < 1. By induction, we conclude from (2.183) that |c„+;| < qMn, i > 1. Furthermore, on each inter­ val [nh/X,(n + 1)A/A] the function \x(t)\ attains its maximum at an endpoint of this interval. Hence, the inequality |c[„/^]| < qM„ leads to ^[n/A] < qM„. Therefore, Min/x] < and the proof is completed by lowering the subindex [1/A] times successively. ■ This is, perhaps, the first attempt to explore equations of type (2.181), and many questions remain open.

CHAPTER 3

Partial Differential Equations with Piecewise Continuous Delay The first fundamental paper [290] in this direction appeared in 1991. It has been shown in [290] that partial differential equations (PDE) with piecewise constant time naturally arise in the process of approximating PDE by using piecewise constant arguments. Thus, if in the equation ut = a2uxx — bu, which describes heat flow in a rod with both diffusion a2uxx along the rod and heat loss (or gain) across the lateral sides of the rod, the lateral heat change is measured at discrete times, then we get an equation with piecewise constant argument (EPCA) ut(x,t) — a2uxx(x,t) t G [nh,(n + l)h],

— bu(x,nh), n = 0,1,...

where h > 0 is some constant. This equation can be written in the form ut(x, t) = a2uxx(x, t) - bu(x, [t/h]h),

(1)

where [■] designates the greatest-integer function. The diffusion-convection equation ut = a?uxx - rux describes, for instance, the concentration u(x,t) of a pollutant car­ ried along in a stream moving with velocity r. The term a uxx is the diffusion contribution and — rux is the convection component. If the 163

164

3. PARTIAL DIFFERENTIAL EQUATIONS

convection part is measured at discrete times nh, the process results in the equation ut(x, t) = a2uxx(x, t) - rux(x, [t/h]h).

(2)

These examples indicate at the considerable potential of EPCA as an analytical and computational tool in solving some complicated prob­ lems of mathematical physics. Therefore, it is important to investigate boundary-value problems (BVP) and initial-value problems (rVP) for EPCA in partial derivatives, and explore the influence of certain dis­ continuous delays on the behavior of solutions to some typical problems of mathematical physics. 1. Boundary-Value Problems for Partial Differential Equations with Piecewise Constant Delay We consider the boundary-value problem (BVP) consisting of the equation

du(x,t) dt

+

'5> u(x .0 p( KBXJ

=Q

=

u

(£) (*'

ft),

where P and Q are polynomials of the highest degree m with coefficients that may depend only on x, the boundary conditions LjU = £ (Mjk4k-X\Q)

+ Njku(h-l\l))

= 0,

(3.2)

*=i

[Mjk and Njk are constants, j = 1 , . . . , m) and the initial condition u(x,Q) = u0(x).

(3.3)

Here [•] designates the greatest-integer function, (x,t) £ [0,1] x [0,oo), and h = const. > 0. Eqs. (3.2) will be written briefly as Lu - 0.

(3.2')

Definition 3.1. A function u(x,t) is called a solution of the above BVP if it satisfies the conditions: (i) u(x, t) is continuous in Q = [0,1] x [0, oo).

3.1. BOUNDARY-VALUE PROBLEMS

165

(ii) du/dt and dku/dxk (k = 1 , . . . , m) exist and are continuous in ft, with the possible exception of the points (x, nh), where one-sided derivatives exist (n = 0 , 1 , 2 , . . . ) . (iii) u(x, t) satisfies Eq. (3.1) in il, with the possible exception of the points (x,nh), and conditions (3.2), (3.3). Let u„(x, t) be the solution of the given problem on nh < t < then

^ £ ^ 1 + pUn{x,t) = Qun(x),

(n+l)h,

(3.4)

where un(x) =

un(x,nh).

Write un(x, t) = wn(x, t) + vn(x), which gives the equation -^P- + Pwn + Pvn(x) = Qun(x), and require that ^

+ Pwn = 0,

(3.5)

Pvn(x) = Qun(x).

(3.6)

Assuming both wn and vn satisfy (3.2') leads to an ordinary BVP (3.6), (3.2'), whose solution is denoted by vn(x) =

P~1Qun(x),

and to BVP (3.5), (3.2'), whose solution is sought in the form wn(x,t) = X(x)Tn(t). Separation of variables produces the ODE T'n + \Tn = 0 with a solution Tn{t) =

nh

e-W-

\

(3.7)

166

3. PARTIAL DIFFERENTIAL EQUATIONS

and the BVP P (■—) X-\X

= 0,

LX = 0

(3.8)

where L is defined in (3.2) and (3.2'). If BVP (3.8) has an infi­ nite countable set of eigenvalues Xj and corresponding eigenfunctions Xj(x) £ C m [0,1], then the series oo oo

wn(x,t) = ZCnie-W-nVXj(x), w„(x, t) = £ Cnje-x*-nVXj(x),

Cnj = const. Cnj = const,

represents a formal solution of problem (3.5), (3.2') and oo oo un(x ,t) = E C e -\j(t- -nh)^ :(X) + P~l lQU; .(*) nj «„(*, t) = £ Cnje-W-^Xfa) + P- Qun(x)

(3.9)

is a formal solution of (3.1), (3.2). At t = nh we have oo oo

l Qu u„(x) n(x). un(x) == £ CnjfXj{x) A>(r) + pP-lQu n(x). = Ecn.

(3.10)

Therefore, assuming the sequence {Xj} is complete and orthonormal in Cm[0,1] yields for the coefficients Cnj the formula Cnj = / o ' Xj{x)(I - p-lQ)un(x)

dx,

(3.11)

(n = 0 , l , 2 , . . . ) . Substituting the initial function UQ(X) G C m [0,1] in (3.11) produces the coefficients CQJ, and putting them together with UQ(X) in (3.9) as n = 0 gives the solution u0(x,t) of BVP (3.1), (3.2), (3.3) on the interval 0 < t < h. Since U(,(x,h) = u\(x,h), we can find from (3.11) the numbers C\j and then substitute them along with u\{x) in (3.9), to obtain the solution ui(x,t) on h < t < 2h. This method of steps allows us to extend the solution to any interval nh < t < (n + l)h. Furthermore, continuity of the solution u(x, t) implies un(x (x, ,(n (n + l)h) l)/i) ==- uUn+l(x (n + l)h) un+i(x), l)h) = «n+i(a:), n+i(x, ,(n

3.1. BOUNDARY-VALUE PROBLEMS

167

hence, at t = (n + l)h we get from (3.9) the recursion relations oo

«n+i(*) = £ C V - ^ X ^ x ) + P ^ Q u ^ x ) .

(3.12)

Therefore, oo

un+1(x) = «„(*) - D C n/ (1 - e ^ ' ^ X ^ x ) y=i

and 1 l (I P-l1Q)u {x) {I-- P-pQ)un(x) (/ - pg)«n+l (x) = (/ Q)«„(X) n+1 oo

oo

l X Z-- P -P'Q ^x). - E CW(1 -- e e--A^'*)(J Q)Xi{x).

-Ecnj(i-

Multiplying by -Xfc(x) and integrating between 0 and 1 yields the re­ cursion formulas Cnn+l,k +l,k = =C CV ,fc -— C nk

oo

X h

J3 Cnj(l -e— e* y)X)Xj*, ik,

-Ec»;(ii=i

where

-i:

l

1 X Xk(x)(I(x)(I --PP-lQ)Xj(x) Q)Xj(x) dx. dx. Xjkifc'■= jf X

T h e o r e m 3.1. Formula (3.9), with coefficients Cnj and functions u„(x) defined by recursion relations (3.11) and (3.12), represents a for­ mal solution of BVP (3.1), (3.2), (3.3) in [0,1] x [nh,(n + l)h], for n = 0 , 1 , . . . , if BVP (3.8) has a countable number of eigenvalues Xj and a complete orthonormal set of eigenfunctions Xj(x) £ Cm[0,1] and the initial function UQ(X) G Cm[0,1] satisfies (3.2). A different method can be used if we look for a solution with con­ tinuous derivatives d2u/dt2 and dk+1u/dtdxk (k = 1 , . . . ,m) for t E (nh, (n + l)/i). In this case we differentiate (3.4) with respect to t and obtain the equation d

y»^i>(d\

&

+

* (

:£)-»• n

du du «n "~ dt'

y

168

3. PARTIAL DIFFERENTIAL EQUATIONS

whose solution is sought in form (3.7). Again, separation of variables produces T„(t) and BVP (3.8). Integrating the solution OO oo

nA Vn(x ,*)* = E Bnje -*/(*-- )X»(a;)

yn(x, t) = £ Bnje-W-^Xjix) between n/i and t gives between nh and t gives - e"-A,-(<- ■ » " ) ) ^ ( x ) u n (x ,*) == un(x) f. 2^ ^ £»;(! A,u n (x, i) = «„(*) + 53 - H i ^-^A

(3.13)

(3.14)

A

3=1

J

Continuity of the solution at t = (n + l)h implies ^X^x) .. »oo Bnni(l(l --- e-ex-s^h)XAx) «„+i(x) 53 ——— «„+i(ar) = uw„(x) -Hi n(x) + £ Ai >—J±±. i=i j=i

/0,rN (3.15)

*j

Prom (3.4) and (3.13) at t = nh we have nh) = yn(x,nh) == (QVn(x, t l •»(*), n(x), (Q - P ) P)u oo

n/i) := yn(x,nh) Vn(x,

oo

53 BB njX i (*). njXj(x),

=j=\E

and consequently,

i=l

B n i = /oJ X,(x)(Q - P)tc(x)
(3.16)

Theorem 3.2. Series (3.14), with the functions un(x) and coeffi­ cients Bnj defined by (3.15) and (3.16), formally represents a solu­ tion of BVP (3.1), (3.2), (3.3) whose derivatives dun/dt, dkun/dxk (k = 1 , . . . , m) are continuous in [0,1] x [nh, (n + l)h] and d2un/dt2, dk+1un/dtdxk are continuous in [0,1] x (nh, (n + l)h) if, in addition to the other conditions of Theorem 3.1, the initial function «o(x) and (Q-P)uo(x) satisfy (3.2). The solution un(x,t) of the nonhomogeneous equation

*ti)+p(|)a(M, = «(£)«(I,[i],)+/(,.)(,17)

3.1. BOUNDARY-VALUE PROBLEMS

169

on nh < t < (n + l)h is also sought in the form oo

un(x,t) = Y,Xj{x)Tnj(t),

(3.18)

.7=1

where Xj(x) are the eigenfunctions of the operator P. Upon multiply­ ing (3.17) by Xf.{x), then integrating between 0 and 1 and changing k to j , we obtain Kj(t) + XjTnj(t) = qnj + fj(t), d

Qnj =

Jo Xi(x}Q t ) u"(x) dx>

fj(t) = Jo1Xj(x)f(x,t)dx, whence Tnj(t) = (Tnj(nh) +

Xfq^e-W-***

\fqnj+J*he-W-*)fj(s)ds, (nh
+ l)h)

Tnj(nh) = / un(x)Xj(x)

dx,

that is, Tnj(t) = (£ X;{x){I - XjlQ)un(x) + Xj1 _£ Xj(x)Qun(x)

dx) e-W~nV dx + fnh e-x*-^fj(s)

ds. (3.19)

The principal role of the operator P emerges from the above three methods of constructing the solution. Let

Py=tpjy(m~j\ j=0

where pj are real-valued functions of classes C"1--' on 0 < x < 1 and p0(x) ^ 0 on [0,1]. Assuming C m [0,1] is embedded in £ 2 [0,1] with the

3. PARTIAL DIFFERENTIAL EQUATIONS

170

inner product (»•*) = I

y(x)z(x)dx,

BVP (3.8) is called self-adjoint if (Py,z)

=

(y,Pz),

m

for all y, z € C [0,1] that satisfy the boundary conditions Ly = Lz = 0. If BVP (3.8) is self-adjoint, then all its eigenvalues are real and form at most a countable set without finite limit points. The eigenfunctions corresponding to different eigenvalues are orthogonal. T h e o r e m 3 . 3 . Boundary-value problem (3.1), (3.2), (3.3) has a solu­ tion in [0, l]x[n/i, ( n + l ) / i ] , for eachn = 0 , 1 , . . . , given by formula (3.9) if the following hypotheses hold true: (i) BVP (3.8) is self-adjoint, all its eigenvalues Xj are positive. (ii) For each Xj, the roots of the equation P{z) — Xj = 0 have nonpositive real parts. (iii) The initial function UQ(X) 6 C m [0,1] satisfies (3.2). PROOF. According to (3.6), we find the solution vo(x) = P~1QUQ(X) of the equation PVQ(X) = QUQ(X) satisfying the boundary conditions Lv0 = 0. Then the difference u0(x) - P-lQua(x) G C m [0,1] satis­ fies (3.2'), and therefore we conclude from (3.10) t h a t the Fourier series ^ZCQJXJ(X) converges to it absolutely and uniformly on [0,1], where {Xj(x)} is the set of the orthonormal eigenfunctions of (3.8). Since Xj > 0, the series in (3.9) also converges absolutely and uniformly on [0,1] x [0, h]. Furthermore, the same is true on [0,1] for the se­ ries in (3.12) at n = 0, and ui(z) satisfies (3.2). Hence, u\(x) should be used now to find the solution v\(x) = P~lQu\(x) of the equation Pvi(x) = Qui(x) satisfying Lv\ = 0, then to calculate the coefficients C\j by (3.11) and the solution ux(x, t) of the given B V P on [0,1] x [h, 2h], according to (3.9). This procedure can be continued successively to construct the solution u„(x,t) for any n > 0. From (3.10) we con­ clude that all un(x) satisfy (3.2'). Differentiating (3.9) term by term with respect to t produces a series which converges to du„/dt uniformly

3.1. BOUNDARY-VALUE PROBLEMS

171

on [0,1] x [nh + 6,(n + l)h], for sufficiently small 6 > 0, since Xj > 0. Furthermore, it follows from (3.11) that C

"> = V 1 £(PXi)(J

~ P~lQ)un{x) dx

and since Xj(x), un(x), and P~lQun(x)

satisfy (3.2'), then

Cnj = A71 jf1 Xj(x)(P - Q)un(x) dx. Hence, ICnil < A71 / Xjdx) V

((Pun-Qun)2dx)

'

V

'


Let po(x) = 1 and X = pm, then in any domain G of the complex /9-plane the equation

{£)y-Xy=

0

has m linearly independent solutions j / i , . . . , ym which are regular with respect to p 6 G, for sufficiently large \p\, and satisfy the relations

ytX)(x)=Pr-1eput*H-1 (k,r =

+ 0(p-1)),

l,...,m)

where W\,... ,u)m are the different m-order roots of unity [204]. There­ fore, by virtue of condition (ii) and estimates (3.20), differentiating series (3.9) term by term r times (r = 1 , . . . , m) with respect to x pro­ duces series that converge uniformly on [0,1] x [nh + 6, (n + l)h], for sufficiently small 5 and large Aj. Letting t = (n + l)h in each of these series and taking into account (3.12) shows that un+i(x) € Cm[0,1] if un(x) 6 C m [0,1]. By virtue of (iii), the proof is complete. REMARK 1. We assumed in this theorem that po(x) = 1, where po(x) is the leading coefficient of the operator P(d/dx). If po = const. ^ 1, then dividing the equation Py — Xy = 0 by po produces an equation

3. PARTIAL DIFFERENTIAL EQUATIONS

172

whose leading coefficient is 1. If po(x) ^ const, on [0,1] and retains its sign, then we may assume po(x) > 0 and use the substitution [204] Xl =

JoP°1/m^ds/l

Po1/m(s)ds^

to reduce the above equation to a new one in the interval 0 < Xi < 1, with a constant leading coefficient. REMARK 2. The Fourier coefficients used in the above three methods of solving BVP (3.1), (3.2), (3.3) are closely interrelated. Indeed, differ­ entiating (3.9) with respect to t and comparing with (3.13) shows that Bnj = —AjCnj. Furthermore, comparing (3.9) with (3.18) and (3.11) with (3.19), we have to prove that y 1 \-JxXj{x)Qun{x)

dx = f Xj{x)p-lQun(x)

dx.

Since P~1Qun(x) satisfies (3.2), then P-lQun{x)

oo

r\

= £ Xk(x) I jb=i

-

Xk(x)P-1Qun{x)dx,

70

and applying the operator P to this equation yields oo

-1

Qun(x) = Y, \kXk(x)

Xk(x)P~1Qun(x)dx.

It remains to multiply this expansion by Xj(x) and to integrate between 0 and 1. EXAMPLE 1. In [0,1] x [nh, (n + l)h], the solution un(x, t) of Eq. (1) with boundary conditions un(x,nh) = un(x) is sought in form (3.18). Separation of variables produces

Xj(x) = V2sin(njx),

T'nj{t) + a^j2Tnj{t)

=

-bTnj(nh),

whence

Tnj(t) = Cnje-aV^-nh) -

~±^Tnj(nh). 0, 71 J

3.1. BOUNDARY-VALUE PROBLEMS

173

We put t = nh in this equation and get C

*i = i1 + -^p) **("*),

that is, Tnj(t) = Ej{t -

nh)Tnj(nh),

where b H =: ee-°a22^^< --- ((l! - -ec- -WV *^)) --JL^. Ej{t) Ej(t) =

(3.21)

At t = (n + l)/i we have Tnj((n + l)h) =

Ej(h)Tnj(nh)

and since Tnj((n + l)h)=Tn+1J((n

+ l)h),

then Tn+lJ((n

+ l)h) =

Eji^Tnjinh)

and

Tnj{nh) = EJ(h)T0i(0)Therefore, Tnj(t) = Ej(t -

nh)Ef(h)T0j(0)

and «n(x, t) = £ V2E]{h)T^)Ej{t

- nh) sm.{irjx).

i=i

Putting 2 = 0, n = 0 gives wo(z) = ]£ 7bj(0)v2sin(7r,;'x) dx and T0j(0) = v 2 /

uo(a;)sin(7rj'x)dx.

(3

174

3. PARTIAL DIFFERENTIAL EQUATIONS

If \Ej(h)\ < 1, then solution (3.22) decays exponentially as t —► oo, uniformly with respect to x. From (3.21) it follows that this is true if ea'*'h +1 ™2-2

2

2

2

2

-a 7r < b < a 7r

ea

2

7r 2 /> _

^"

Furthermore, from the equations Tnj(nh) = E?{h)T0j(0), we see that Tnj(nh)Tnj((n+l)h) holds true if

Tnj((n + l)h) =

EJ+1{h)T0j(0)

< 0 HEj(h) < 0. The latter inequality b>

a2n2

e°***h

_

(3.23)

r

Hence, under condition (3.23) each function Tnj(i)(j = 1,2,...) has a zero in the interval [nh, (n+l)/i], in sharp contrast to the functions Tj(t) in the Fourier expansion for the solution of the equation u< = a2uxx — bu without time delay. Moreover, the inequality Ej(h) < 0 takes place for sufficiently large j and any b > 0. Therefore, for b > 0 and sufficiently large j , the functions Tnj(t) are oscillatory. EXAMPLE 2. Eq. (2) on nh
whose solution is sought in form (3.7). Separation of variables leads to the equations X"{x) + XX(x) = 0,

rn(t) + a2XTn(t) = 0,

and the boundary conditions un(0,t) — un(l,t) yn(x,t) = £

= 0 give Xj = j2n2

V2Tnj(nh)e-aV^-nhhm(njx).

and

3.1. BOUNDARY-VALUE PROBLEMS

175

Since yn(x, nh) = a?u"n(x) - ru'n(x),

u„{x) = un(x, nh),

then oo

a 2 < ( x ) - ru'n(x) = £

V2Tnj(nh)sm(7rjx)

and T„j(nh) = —a2n2j2y/2j

un(x)

sin(7rjx)dx + rwjV2 / un(x) COS(TTJX) dx.

Finally - >/2 rBJ-(nfe) (1 - e -°W('-"ft)) sin(ffjx) u„(x,t) = un(x) + 2J — TTT^. • 2 =i a 7rV ; Given the initial function u(x,0) = uo(x), we can find the coefficients T(y(0) and the solution uo( x )0 on 0 < i < L Since uo(a:,/i) = "1(^)5 we can calculate the coefficients Tij(h) and the solution u\(x,t) on h < t < 2h. By the method of steps the solution can be extended to any interval [nh, (n + l)h]. EXAMPLE

3. The equation

__g2_gMM) 2m0 5x2

<MM) 9 at

+K

/ m \ W p[ftJ A J u

is a piecewise constant analogue of the one-dimensional Schrodinger equation -q2 iqi/jt(x, t) = -—ij)xx{x, t) + V(x)rp(x, t). ZTTIQ

If u(x, t) satisfies conditions (3.2) and (3.3), with m = 2, then separa­ tion of variables produces a formal solution 00

un(x,t) = £ Cnje-i^t-"h^Xj(x)

+

P~lQun{x),

176

3. PARTIAL DIFFERENTIAL EQUATIONS

for nh < t < (n + l)h. Here, Xj(x) are the eigenfunctions of the operator q2(d2/dx2)/2mo, and P~lQun(x) is the solution vn(x) of the equation g 2 <(x) =

2m0V(x)un(x)

that satisfies (3.2) and Cnj are given by (3.11). The Fourier method can be also used to find weak solutions of the boundary-value problem (3.1), (3.2), (3.3) and it is easily generalized to similar problems in Hilbert space. First, we remind a few well known definitions. Let H be a Hilbert space and let P be a linear operator in H (additive and homogeneous but, possibly, unbounded) whose domain T)(P) is dense in H, that is 1>(P) = H. The opera­ tor P is called symmetric if (Pu,v) = (u,Pv), for any u,v € 2)(-P)If P is symmetric, then (Pu,v) is a symmetric bilinear functional and (Pu,u) is a quadratic form. A symmetric operator P is called positive if (Pu, u) > 0 and (Pu, u) = 0 if and only if u = 0. A symmetric operator P is called positive definite if there exists a con­ stant 7 2 > 0 such that (Pu,u) > 7 2 ||u|| 2 . With every positive oper­ ator P a certain Hilbert space Hp can be associated, which is called the energy space of P. It is the completion of T>(P), with the inner product (u,v)p = (Pu,v); u,v £ ^>{P)- This product induces a new norm ||u||p = (Pu, u) 1 ' 2 , u € D(-P), and if P is positive definite, then IMI < 7 _ 1 \\U\\P- Since T)(P) is dense in H, it follows by using the latter inequality that the energy space Hp of a positive definite operator P is dense in the original space H. Assuming P is positive definite, we may consider the solution u(x, t) of (3.1), (3.2), (3.3) for a fixed t as an element of HP. If T>(Q) C H, then Qu(x, [t/h]h) may be treated as an abstract function Qu([t/h]h) with the values in H. Therefore, the given BVP is reduced to the abstract Cauchy problem du + Pu = Qu( h), t>0,u\t=0 = u0eH. (3.24) ~dt If (3.24) has a solution, we multiply each term by an arbitrary func­ tion g(t) £ Hp in the sense of inner product in H and get on the

3.1. BOUNDARY-VALUE PROBLEMS

177

interval nh < t < (n + l)h the equation (du fdu

\

Id*' $)+(« ,g)p

= (Qun,g),

where un = u(nh). Conversely, if u <= C1((nh,(n + l)h);V(P)) for all integers n > 0 and satisfies (3.25), then it also satisfies Eq. (3.24). Indeed, if u £ "D(P), then ( U , # ) P = (Pu,g), and (3.25) can be written (du I — + Pu-

\ Qun,g\ = 0,

n/i < £ < (n + l)/i.

Since Hp is dense in H, then u(i) is a solution of Eq. (3.24). Definition 3.2. An abstract function u(t): [0,oo) —► H is called a weak solution of problem (3.24) if it satisfies the conditions: (i) u(t) is continuous for t > 0 and strongly continuously diferentiable for t > 0, with the possible exception of the points t = nh where one-sided derivatives exist, (ii) u(t) is continuous for t > 0 as an abstract function with the values in Hp and satisfies Eq. (3.25) on each interval nh < t < (n + l)h, for any function g(t): [0, oo) —► Hp. (iii) u(t) satisfies the initial condition (3.24), that is, \im\\u(t)-uQ\\H

= 0.

A weak solution u(t) is also an ordinary solution if u(t) € 'D(P), for any t > 0, and u(x, t) —» ^0(2;) as £ —► 0 not only in the norm of H but uniformly as well. It is said [191] that a symmetric operator P has a discrete spectrum if it has an infinite sequence {Aj} of eigenvalues with a single limit point at infinity and a sequence {Xj} of eigenfunctions which is complete in H. Suppose the operator P in (3.25) is positive definite and has a discrete spectrum and assume the existence of a solution u(t) = u(x,t) to Eq. (3.25) with the condition u(0) = UQ. On the interval nh < t < (n + l)h this solution can be expanded into series (3.18), where Tj(t) = (u(t),Xj). To find the coefficients Tj(t), we

178

3. PARTIAL DIFFERENTIAL EQUATIONS

put g(t) = Xk in (3.25) and since Xk does not depend on t, then (du(t)

[ dt ,Xk) (u,X (u ,Xkk)P)P

= ft(u(t),Xk) = no,

= (u,PXkk)) = = X\k(u (u,X = (Pu,X (Pu,Xkk) = (u,PX kTk(t), ,xkk)) ==■-XhT

which again leads to the equation T;j(t) + XjTnj(t) =

(Qun,Xj)

and to a generalization of (3.19). By selecting a proper space H, a weak solution corresponding to conditions (3.2) can be constructed. The proof of the following theorem is omitted. Theorem 3.4. / / P and Q are linear operators in a Hilbert space and P is positive definite with a discrete spectrum, then there exists a unique weak solution of problem (3.24). 2. Initial-Value Problems for Partial Differential Equations with Piecewise Constant Delay This topic has been explored recently by Wiener and Debnath [296]. We consider the initial-value problem (IVP) du(x,t) dt

+

p(

s)"(l'()

t h. ")•

(3.26)

u(x,0) = UQ{X) where P and Q are polynomials of the highest degree m with constant coefficients, [•] denotes the greatest-integer function, h = const. > 0, and (x,t) G Cl = (-00,00) x [0,oo). Let un(x, t) be the solution of the given problem on nh < t < then Eq. (3.4) follows where un(x) = un(x,nh). Write un(x,t) = wn(x,t) + vn(x),

(n+l)h, (3.27)

3.2. INITIAL-VALUE PROBLEMS

179

whence dwn — + Pwn + Pvn(x) = Qun(x), which leads to Eqs. (3.5) and (3.6). If vn(x) is a solution of ODE (3.6), then at t = nh we have wn(x, nh) = un(x) - vn(x),

(3.28)

and it remains to consider Eq. (3.5) with initial condition (3.28). It is well known that the solution E(x, t) of the problem ~

+ Pw = 0,

w(x, 0) = tuo(ar),

(3.29)

with WQ(X) = 6(x), where 8(x) is the Dirac delta functional, is called its fundamental solution. The solution of IVP (3.29) is given by the convolution w(x,t) = E(x,t)*w0(x).

(3.30)

Hence, the solution of problem (3.5), (3.28) can be written as wn(x, t) — E(x, t — nh) * wn(x, nh),

(3.31)

and the solution of (3.4), (3.27) is un(x, t) = E(x, t - nh) * (un(x) - v„(x)) + v„(x), (nh
(3.32)

(n + l)h).

Continuity of the solution at t = (n + l)h implies un(x, (n + l)h) = un+i(x, (n + l)h) = un+1(x), that is, u„+i(x) = E(x, h) * (u„(x) - v„(x)) + vn(x).

(3.33)

Formulas (3.32), (3.33) successively determine the solution of IVP (3.26) on each interval nh < t < (n + l)h. Indeed, from Pv0(x) = QUQ(X) we find vo(x) and substitute both UQ(X) and VQ(X) in (3.32) and (3.33) to obtain uo(x,t) and u\(x). Then we use ui(x) in (3.6) to find vi(x) and substitute u\{x) and v\{x) in (3.32) and (3.33), which yields ui(x,t) and ui(x). Continuing this procedure leads to u„(x,t), the solution

180

3. PARTIAL DIFFERENTIAL EQUATIONS

of (3.26) on [nh,(n + l)h]. The solution vn(x) of (3.6) is defined to within an arbitrary polynomial q(x) of degree < m. Since q(x) is a solution of Eq. (3.29) with the initial condition w(x,0) = q(x), then q(x) — E(x,t) * q(x), and q(x) cancels in the formulas (3.32), (3.33). This concludes the proof of the following assertion. Theorem 3.5. If Eq. (3.29) with w(x,0) = UQ(X) has a unique solu­ tion ont (E (0,oo), then there exists a unique solution of IVP (3.26) on (0, oo) and it is given by (3.32), for each interval nh < t < (n + l)h. COROLLARY 3.1. There exist unique solutions of Eqs. (1) and (2), with u(x,0) = UQ(X), in the class of functions that grow to infinity slower than exp(x2) as \x\ —» oo. For Eqs. (1) and (2) we have vn(x) = a~2b I (x — s)un(s) ds

and

v„(x) = a~2r j u„(s) ds,

respectively, and E(x,t) = exp(—x2/4a2t)/2a^/Tti. Formula (3.32) for the solution of the problem ut(x,t) = a2uxx(x,t)

- buxx (x, - hj ,

u(x,0) = u0(x)

on nh < t < (n + \)h becomes un(x,t) = (l

j ) E(x,t-nh)*n„(x)

+

—un(x),

where E(x,t) is the same as in Eqs. (1) and (2). The above method may also be used to solve IVP for PDE of any order in t with piecewise constant delay or systems of such equations. In the latter case, P and Q in (3.26) are square matrices of linear differential operators and u(x, t) is a vector function. Thus, the solution un(x, t) of the problem utt(x,t) = a2uxx(x,t) - buxx(x,[i\), u(x, 0) = / 0 (x), «| (x, 0) = g0(x)

181

3.2. INITIAL-VALUE PROBLEMS

o n n < £ < n + l i s sought in the form un(x,t) = w„(x,t) + v„(x) whence a2v'^(x) - 6<(x,n) = 0 and d2wn/dt2 = a2d2wn/dx2. Setting u(x,n) = fn(x), ut(x,n) = g„(x) gives vn(x) = a-2bfn(x),

w(x, n) = (1 - a~2b)f„(x),

wt(x, n) = gn(x),

and b \ fn(x-a(t-n)) fn{x -- a(t - n)) ++ fnfn(x(x ++ a(ta(t-n)) - n)) =- ^\fn(*) u«■(*,*) n(x,t) / n ( * )++ ((ll -- ±) a2) 2 1

rx+a(t-n)

+ 7T 2a

,,„ , 9n{s) ds. Jx-a(t-n) Jz-a(t-n)

Putting t = n + 1 produces the recursion relations b \ fn{x ~- a) + fn(x +fla) fn+i(x) —fn{x)+\l-- ^) w * ) == £/.(«)+(i /■('-°)+/^+ ) a2) 2 1

f*+"

+ 5- /

. . .

M«) «s,

+ 2^"( X + a ) + 9n{x - a)). Loaded partial differential equations have properties similar to those of equations with piecewise constant delay. The IVP for the following class of loaded equations

^-'(s)^ + £*(s)-<"<<>-

u(x,0) = UQ(X) (3.34)

was considered in [28] and [296], where (x,t) G R" x [0,T], the *,- £ (0,T] are given, P(s) and Q(s) are polynomials in s = (s\,... ,s„), and £ |Qj(s)| ^ 0. Eq. (3.34) arises in solving certain inverse problems for systems with elements concentrated at specific moments of time. The Fourier transform U(s,t) of u(x,t) satisfies the equation Ut(s,t) = P(is)U(8,t)+i

Qj{is)U{s,tj),

182

3. PARTIAL DIFFERENTIAL EQUATIONS

whence, U(s, t) = £/0(s)eP(,'s)' + k(P(is), t) £ Qj{is)U(s, tj),

(3.35)

where UQ(S) is the Fourier transform of uo(x) and

k(P(is),t) =

j\p^dy.

Denote A i = Uo(s)epWi,

kj = k(P(is), tj),

B=£

Qj(is)U{s, i=i

tj), (3.36)

then multiply by Qj(is) each of the equations U(s, tj) = Aj + kjB, and add them. Hence,

j - 1,..., q

B = E AjQj(is) + BJ:

kjQj(is)

or 1 - £ kjQj(is)) B = Y, AjQj(is).

(3.37)

The equation A(s) = 1 - E Qj(is)k(P(is),

tj) = 0

(3.38)

is called the characteristic equation for (3.34) and its solution set Z is called the characteristic variety of (3.34). It is said [28] that (3.34) is absolutely nondegenerate if Z = 0 , nondegenerate of type a if a = inf | I m s | < oo,

s 6 Z ^ C",

and degenerate if Z = C". The case Z = 0 implies A(s) = const., since A(s) is meromorphic, and a meromorphic function that is not

3.2. INITIAL-VALUE PROBLEMS

183

constant assumes every complex value with at most two exceptions. The equation A(s) = C can be written as P(w) + £ Qi(w) - £ Q,(«)e p(, ' s) '' = CP(is) and is possible for q > 1 only if P(s) = const, otherwise exp(P(is)tj) would grow faster than any polynomial, which breaks the latter equal­ ity. For q = 1 we have _ P(i8) + Q1(i8)-Q1(is)ep^ A{S) ~~ P(iS) and in this case Z = 0 is equivalent to P(is) + Qi(is) = 0. On the other hand, A(s) = 0 is equivalent to P(is) + £ Qj(is) - £ Qj(is)ep(isK

= 0,

which implies P(s) = const. This establishes the following proposition which was stated in [28] without proof. LEMMA 3.1. Eq. (3.34) is absolutely nondegenerate if and only if ei­ ther of the following conditions holds true: (i)

P{s) = Cu

EQi(^(Ci,t/) = C2^l;

or (ii)

q = l,

P ( s ) + Q ! ( S ) = 0.

Eq. (3.34) is degenerate if and only if P(s) = Cu

ZQj(s)k(Cutj)

= l.

Substituting B from (3.37) in (3.35) leads to the proof of the following theorems which were formulated in [28]. T h e o r e m 3.6. The uniqueness classes for the solution of the Cauchy problem for an absolutely nondegenerate equation (3.34) are the same as those for the equation {without "loads") ut(x,t) = Pu(x,t).

184

3. PARTIAL DIFFERENTIAL EQUATIONS

T h e o r e m 3.7. The homogeneous degenerate IVP (3.34) has nontrivial solutions, with compact support.

{UQ{X)

= 0)

T h e o r e m 3.8. Suppose that (3.34) is of finite type a(0 < a < oo) and that u(x, t) is a solution of (3.34) with uo(x) = 0. If \u(x,t)\ < CeaM,

x£Rn,

te [0,2],

(3.39)

and a < a, then u(x,t) = 0. For any a > a there exists a solution u(x,t) ^ 0 o/(3.34) with UQ(X) = 0 satisfying (3.39). Boundary-value problems for loaded PDE and some of their applica­ tions were studied in [129] and [205]. The uniqueness classes for the solution of the Cauchy problem for the equation ut(x,t) = Pu(x,t) were explored in [43] and [87] and consist of the functions that grow no faster than exp(a|:r| a ) as |x| —► oo, where a > 1 depends on the degree of P(s). Integral transformations can be also used in the study ofEPCA. T h e o r e m 3.9. The solution of the problem ut(x, t) = a2uxx(x, t) - buxx(x, [t]),

u(x, 0) = u0(x)

(3.40)

is given by the formula

1

■fco-sC/K -

b \' / b y-'] (b „,

.,

- fA) JjB(x,* - [t])\ * uo(x),> (3.41) + (l (l-^ E(x, t + ji-[t])J*iio(x) where E(x,t) = exp(—x2/4a2t)/2a^/Tri and E(x,0) * uo(x) = uo(x). PROOF. For n < t < n + 1, Eq. (3.40) becomes ut(x,t) = a2uxx(x,t)

-

buxx(x,n),

and the Fourier transform U(s,t) = ^(^(a:,*)) satisfies the equation Ut(s, t) = -a2s2U(s, t) + bs2U(s, n), whence U(s,t) = Ce-a2^t-")

+

±U(s,n).

185

3.2. INITIAL-VALUE PROBLEMS

At t = n we have U(s,n) = C + \u(s,n),

C = (l -

\)u(s,n),

and U(s, t)=(±+(l-

A ) e - ^ 2 ( ' - « ) ) ff(., n).

(3.42)

At t = n + 1 this gives t/( S ,n + l ) = ( A + ( l - A ) e - ^ 2 ) c / ( S , n ) and

^ , n ) = (A + (i-A) e -^ 2 ) n c/( 5) o). Substituting the binomial expansion of U(s,n) in (3.42) yields for­ mula (3.41). ■ Now we again turn our attention to Eqs. (1) and (2) from the intro­ duction. T h e o r e m 3.10. The solution of Eq. (1) with the initial condition u(x,0) = UQ(X) is given by the formula u(x,t) = u0(x) * \Fxx{x,t)

- -^F(x,t)

+

-^F\x,h

where

i•

k F(x, t) = [ E ([t[h]) ^j k)h) i} £ {{) (-l) E(x, t-(j-

j=o \ 3 ) a *=o W

x2j+l

*WTTyHix}' H{x) = 1, for x>0, PROOF.

and H(x) = 0, for x < 0.

For nh < t < (n + l)h, we have ut(x,t) = a2uxx(x,t)

— bu(x,nh),

(3 43)

-

186

3. PARTIAL DIFFERENTIAL EQUATIONS

and the two-sided Laplace transform U(s,t) = L(u(x,t)) equation Ut(s, t) = a2s2U(s, t) - bUn(s),

satisfies the

Un(s) = U(s, nh)

whence

^.') = (^"-"',(1-^)+^)w ( S )gives = (e' 2 '" 2 ( l At t = (n + l)ht/ n+1 this "av)

Un(s) = {e^

and

+

av)

^(S)

+ (1 - e " 2 " * 2 ) ^ )

U0(s),

U0(s) = £( U o (x)). Hence,

Un(s) = U0(s) E f j ^ S

(j[)(-l)^<^

and

*'>=W5^|(!)(-I)'P'M-'W i _ «V(l-y-t)*) , j!_pa>hs*(n+k-j ')■ a2s2 a*sl which proves the result. Theorem 3.11. 77ie solution of Eq. (2) wttfi u(x,0) = uo(aO w given by the formula u(x, t) =

Uo (x)

* tar(x,*) - ^Fx(x,

where F(x,t) is defined in (3.43).

t) + ^Fx(x,h

t\\

3.2. INITIAL-VALUE PROBLEMS

187

T h e o r e m 3.12. The solution of problem (3.26) is given by the for­ mula u(x, t) = u0(x) * l £ ^

+Qj+l

h ]

j (& ( £ ) />■(*) * *)(*, t)

{k) Pj+l(X) * iFj (*' * [1]) " Fj{X>t]) )'

where

Fj(x,t) = £ M ( - l ) * ^ ( x , t - (j - *)A), E(x,t) is the fundamental solution of (3.29), and Pj(x) is the inverse Laplace transform of P~*(s). P R O O F . The solution u„(x,t) of (3.26) on nh < t < (n + l)h satis­ fies (3.4), (3.27), and for its two-sided Laplace transform in x we obtain the equation

Ut(s, t) + P(s)U(s, t) = Q(s)Un(s),

Un(s) = U(s, n),

whence U(s,t) = Ce-p^t-n^

+

P-\s)Q(s)Un(s).

At t = nh we have Un(s) = C +

P-\s)Q(s)Un(s)

and U(s,t) = (e-pW-nh)

+ (1 - e-'MC*-*)) p~*Q) Un(s).

At t = (n + l)/i this gives Un+l(s) = {e~pW + (1 - e - W ) P- J Q) % ( . ) , hence, Un(s) = {e-pW

+ (1 - e-^W") P " 1 ^ " C/0(S)

3. PARTIAL DIFFERENTIAL EQUATIONS

188

and

Un(s) = Uo(s) t {^jQjP-jpQ Q ( - 1 ) V W " - ^ . Therefore, J J U (.)fj: (f)Q 9 =^)(g(;)^>P- i

U(s U(>,t) ,*) = =

-(j-k)h) -i)*« ,-.?(«)(*(1)(-l)'e-^Xt-(i-*W

- ' ! #

k p-P(*)/.(n + £ (nW+1^J'_1 E (j)(-i)-i)V e- (sMn-j+k) -j+k)

*=o *=0 W w

i=o yy//

-1)V -P(«)(t- (j-k)h)\

^

# ) < which leads to (3.44)

■

Consider the initial-value problem ~ = A(D)u(x, t) + f(t, u(x, [t])), u(x,0)

(3.45)

= UQ(X),

where u(x,t) and u0(x) are m-vectors, x = (xi,£2>... ,#JV) € K^,

A{D) = £ 4*£>a, a = (ai,a 2 ,...,aAr), |a| = a t + a 2 H D = D?---D%», Dk = iO/dxk (k = a

1-%, l,2,...,N),

the coefficients yla are given constant matrices of order m X m, and the m-vector / G C 1 ([n,n + 1) x £ 2 (R JV ),i5 2 (R JV )), n = 0 , 1 , 2 , . . . . The number r is called the order of the system. It is assumed that u0 € £2(RN), and the solutions sought are such that u(x,t) E £2(RN), for every t > 0. Let fii(s), fi2(s),..., fim(s) be the eigenvalues of the matrix A(s). The system % = A(D)u

(3.46)

3.2. INITIAL-VALUE PROBLEMS

189

is said to be parabolic by Shilov if ReHj(s) <—c\s\h + b,

j =

l,...,m

where h > 0, c > 0, and b are constants. T h e o r e m 3.13. Problem (3.45) has a unique solution on RN x [0, oo) if system (3.46) is parabolic by Shilov, the index of parabolicity h co­ incides with its order r, and f £ Cl([n,n + 1) x £2(RN),£2(RN)), n = 0,1,2,.... P R O O F . For a fixed t we may consider the solution u(x,t) as an ele­ ment of £ (IR^), and f(t, u(x, [t])) may be treated as an abstract func­ tion f(t,u([t])) with the values in £2. Therefore, IVP (3.45) is reduced to the abstract Cauchy problem

-£ = Au + f(t, u([t])),

u\t=Q = u0e £2.

(3.47)

Applying to (3.46), with the initial condition u(x,0) = uo(x), the Fourier transformation "3f in x produces the system of ordinary dif­ ferential equations Ut(s,t) = A(s)U{s,t),

(3.48)

with the initial condition U(s,0) = UQ(S), where U(s,t) = 7(u(x,t)), Uo(s) = y(«o(a;)), and A(s) is a matrix with polynomial entries de­ pending on s = (si, S2i • • • j SN)- The solution of (3.48) is given by the formula U(s,t) =

etA^U0(s).

Parabolicity of (3.46) by Shilov implies that the semigroup T(t) of oper­ ators of multiplication by eiA^"\ for t > 0, is an infinitely smooth semi­ group of operators bounded in £ (M.N). Together with the requirement h = r, this ensures that the Cauchy problem for (3.46) is uniformly correct in £2(M.N) and all its solutions are infinitely smooth functions of t, for t > 0. Since / is continuously differentiable, problem (3.45) has on [0,1) a unique solution u(t) = T(t)u0 + I* T(t - s)f(s, u0) ds.

190

3. PARTIAL DIFFERENTIAL EQUATIONS

Denoting ui = u(l), we can find the solution u(t) = T(t - l ) U l + f T(t - s)f(s, ui) ds of (3.45) on [1,2) and continue this procedure successively. If f(t,u([t})) = Bu([t}), where B is a constant matrix, the solution of (3.45) for t G [0, oo) is given by

u(t)=(T(t-[t})

+ J^]T(t-S)Bds)j x n

(T(1)

k=[t] V

+ I* , T{k - s)B ds) u0, / 1 • *-

/

in accordance with (1.38). The theorem holds true if / includes also derivatives of u(x, [t]) in x of order less than r, provided the initial function UQ(X) is sufficiently smooth. ■ 3. A Wave Equation with Discontinuous Time Delay The purpose of this section is to investigate the influence of terms with piecewise constant time on the behavior of the solutions, especially their oscillatory properties, of the wave equation. Research in this direction was initiated in 1991 by Wiener and Debnath [297, 299]. First, we shall discuss separation of variables in systems of PDE. Consider the BVP consisting of the equation Ut(x, t) = AUxx{x, t) + BUxx(x, [<]),

(3.49)

the boundary conditions U(0,t) = U(l,t) = 0,

(3.50)

U(x,0) = U0(x).

(3.51)

and the initial condition

Here, U(x,t) and Uo(x) are real m x m matrices, A and B are real constant m X m matrices and [•] denotes the greatest-integer function.

3.3. WAVE EQUATION

191

Looking for a solution in the form U(x,t)=T(t)X(x)

(3.52)

gives T'(t)X(x)

= AT(t)X"(x)

+

BT([t])X"(x),

whence (AT(t) + BTW^T'it)

= X"(x)X-\x)

= -P2,

which generates the BVP X"(x) + P2X(x) = 0, X(0) = X(1) = 0

(3.53)

and the equation with piecewise constant argument T\t) = -AT(t)P2

- BT([t])P2.

(3.54)

The general solution of Eq. (3.53) is X(x) = cos(xP)Ci + sin(xP)C 2 , where oo / ' _ 1 , \ n T 2 n p 2 n (

oo f _ 1 "\n„2n+l p 2 n + l

cos(xP)=E * , sin(xP) = E ( * T; v „=o ( 2 ") ! (2n + l)! n=o and Ci,C2 are arbitrary constant matrices. From X(0) = 0 we con­ clude that C\ = 0, and the condition ^ ( 1 ) = 0 enables us to choose sin P = 0 (although this is not the necessary consequence of the equa­ tion (sin P)C2 = 0). This can be written eiP - e~iP = 0, e2iP = I. As­ suming that all eigenvaluespi,p2j • • • >Pm 2) = diag(pi,p2,...,pm), we have exp(2i52)5of P are1 )distinct = / , 5and e 2 , J ,S^PS 5-1 = = /, 1 1 2, 2) = e diag(pi,/> we= have exp(2i52)5/ , Se^S' and ° = I. Therefore, diag(7r;'i, 7172, ■ ■ )•, =i".?m)> where the=jk/ ,are 2 ,---,i>m), 2) and e 2, ° = integers, andI. PTherefore, = 52) 5 - 12) , = diag(7r;'i, 7172, ■ ■ •, njm), where the jk are integers, and P = 52) 5 - 1 , P 2 = S&S-' = 5 d i a g ( ^ 2 , TT2;2, . . . , n2j2JS~\

192

3. PARTIAL DIFFERENTIAL EQUATIONS

sin(xP) = SsinixtyS-1 more, we can put

- 5diag(sin7rjix,... ,simTJmx)S~l.

Further­

Pj = diag(7r(m(j - 1) + 1 ) , . . . , nmj),

(3.55)

(.7 = 1,2,...) in (3.53) and obtain the following result. Theorem 3.14. There exists an infinite sequence of matrix eigenfunctions for BVP (3.53), Xj(x) = y/2 diag(sin7r(m(j - 1) + l ) x , . . . ,sin7rmjx), 0 = 1,2,...) which is complete and orthonormal in the space £ [0,1] ofrnxm trices, that is,

(3.56) ma­

JQlXj(x)Xk(x)ds = l[^ j * * where I is the identity matrix. REMARK 3. The matrices SXj(x)S~l nonsingular 5.

satisfy Theorem 3.14 for any

Theorem 3.15. Let E(t) be the solution of the problem T(t) = -AT(t)P2,

r(0) = /

(3.57)

and let M(t) = E(t) + (E(t) - I)A~lB.

(3.58)

/ / the matrix A is nonsingular, then Eq. (3.54) with the initial condition T(0) = CQ has on [0, oo) a unique solution T(t) = M(t - [t])M®(l)C0.

(3.59)

P R O O F . On the interval n < t < n + 1, where n > 0 is an integer, Eq. (3.54) turns into

T'(t) = -AT(t)P2

- BCnP2,

C„ = T(n)

193

3.3. WAVE EQUATION

with the general solution T(t) = E(t - n)C -

A~lBCn.

At t = n we have C„ = C - A~lBCn, whence C = (I + A~lB)Cn

and

l

T(t) = (E(t -n) + (E(t - n) -

I)A~ B)Cn,

that is, T{t) = M{t - n)Cn.

(3.60)

At t = n + 1 we have Cn+l = M(1)C„ and C„ = M"(1)C 0 .

(3.61)

Hence, n)Mn(l)C0,

T(t) = M{t which is equivalent to (3.59)

■

Theorem 3.16. / / ||M(1)|| < 1, then \\T(t)\\ exponentially tends to zero as t —> +oo. EXAMPLE 4. For the scalar parabolic equation ut(x, t) = a2uxx(x, t) + buxx(x, [t]) we have m = 1 and Pj = nj, according to (3.55). For Eq. (3.57) with A = a2 and P = Pj, we have Ej(t) = e-a^^H and Mj(t)

= e-<W - A(i _ e -<W).

Hence, the inequality |Mf(l)| < 1 is equivalent to _ l

< e

- ^

2

_ ^

( 1

a*

_

e

- ^

2

) < l ,

whence -a

2

.
2l

2

2-2

+ e-°2^ 1 _ a w

194

3. PARTIAL DIFFERENTIAL EQUATIONS

Since the function (1 + e~')/(l — e - ') is decreasing, all functions Tj(t) exponentially tend to zero as t —► oo if and only if - a2 < b < a 2 .

(3.62)

lib < —a2, then all Tj(t) monotonically tend to infinity as t —► oo; and if b > ( l

! _

e

- a V

then all Tj(t) are unbounded and oscillatory. For any b > a2, there exists a positive integer jo such that the Tj{t) are unbounded and oscil­ latory, for j > jo- Indeed, letting b = a2 + e and solving the inequality ,,2

+ e>

,1 + e"- a M j a

2

11 — e.a»«»j»

gives -aMj2

<

€ 22

2 a + e* e' 2a

which holds for any positive e and sufficiently large j and implies Mj(\) < - 1 . If b = -a2, then Mj(t) = 1, Tj(t) = Tj(0), and u(x,t) = U(j(x), for all t. Therefore, the condition |6| < a2 is neces­ sary and sufficient for the series 00

u(x,t) = J2Tj(t)Xj(x)

(3.63)

to be a solution of the scalar BVP (3.49)-(3.51), with A = a2 and B = b, if UQ(X) is three times continuously differentiable. The coefficients 7)(0) are given by

Tj(0)=j\0(x)Xj(x)dx, where Xj(x) = ^sin(njx)

and u0(x) G C3[0,1] satisfies u0(0) = u 0 (l) = 0.

3.3. WAVE EQUATION

195

Theorem 3.17. The solution T = 0 of Eq. (3.54) is globally asymp­ totically stable as t —» +oo if and only if the eigenvalues Xr of the matrix M(l) satisfy the inequalities |Ar|
r = l,...,m.

(3.64)

LEMMA 3.2. All entries of every solution of the equation T\t) = A{T{t)Ai,

(3.65)

with constant m x m matrices A\ and Ai, are linear combinations of terms tk exp(a\ ctj t), where a\ and cq are eigenvalues of A\ and Ai and k is a non-negative integer. PROOF. Assume, for simplicity, that the eigenvalues of A\ and Ai are distinct, and let

SfUjSj =2)! =diag(a ( 1 1) ,4 1) ,---,«£ ) ), S? A2S2 = 2>2 = diag(a(12), a%\ ..., ag>). Then the substitution T — S\V changes (3.65) to V'(t) = DiV(<)A2>

(3.66)

and the substitution V = WSi1 transforms (3.66) into W'(t) = 2)iW(t)© 2 . The entries of W{t) are c,-j exp(a,- 'o^ '<), with arbitrary constants c^, and the completion of the proof follows from the formula T = SiWS^1If some of the eigenvalues are multiple, polynomial factors will replace the constants c,j ■ Theorem 3.18. If all eigenvalues of A have positive real parts and U0(x) e C 3 [0,1] ; \\A~lB\\ < 1, then BVP (3.49)-(3.51) has a solu­ tion (3.63). This series and all its term-by-term derivatives converge uniformly. P R O O F . The functions Tj(t) satisfy Eq. (3.54), with Pj from (3.55). Therefore, setting in the above lemma Ai = —A, Ai = Pf and noting that R e a | ^ < 0, ay > 0, we conclude that Ej(t) —► 0 as j —► 00 and

196

3. PARTIAL DIFFERENTIAL EQUATIONS

all t > 0, where Ej(t) is the solution of (3.57). Hence (3.58) implies Mj(t) —► -A~lB as j —> oo and ||My(t)|| < 1, for sufficiently large j , by virtue of the condition ||>1—2JB|| < 1. Prom the formula Tj(t) =

Mj(t-[t])Mf](l)C0j

we note that ||Tj(t)|| —► 0 exponentially as t -> oo, for sufficiently large j . Therefore, for any e > 0, there exist t = 0 a^d j — N such that ||i?7v(£)|| < e, for t > to, where Rp/(t) is the remainder of series (3.63) after the iV-th term. For 0 < t < to and large j , we have ||Tj(t)|| < ||COJ||J and in this case the uniform convergence of series (3.63), together with its respective derivatives in t and x, follows from the smoothness of the initial function UQ(X) and the formula Tj(0) = C0j - /oX U0(x)Xj(x) dx.

U

Separation of variables in the matrix equation with constant coeffi­ cients Ut(x, t) = A0U(x, t) + A2Uxx(x, t) + B0U(x, [t]) + B2Uxx(x, [t]) leads to (3.53) and to the EPCA T{t) = A0T(t) - A2T(t)P2 + B0T([t]) -

B2T([t])P2,

which can be also investigated by the above method. Separation of variables in the equation with constant coefficients uti(x,t)

= a2uxx(x,t)

- buxx(x, [t])

(3.67)

and boundary conditions (3.50) yields Xj(x) = v^sin(7rjx) and leads to the EPCA T'J(t) + a2-K2j%(i) = b^j%{[t\).

(3.68)

For brevity, omit the subindex j and use the substitution T'(t) = V(t), which changes (3.68) to a vector EPCA w\t) = Aw(t) + Bw([t]),

(3.69)

197

3.3. WAVE EQUATION

where w = col(T, V) and A =

( - a V j 2 o) '

B=

[bn2j2 Oj •

Eq. (3.69) on the interval n < t < n + 1 becomes w'(t) = Aw(t) + Bcn,

cn = w(n)

with the solution w(t) = M(t - n)c„, where M{t) = eAi + (eAt -

I)A~lB.

(3.70)

Therefore, Eq. (3.69) with the initial condition w(0) = CQ has on [0, oo) a unique solution given by the right-hand side of (3.59), where M(t) is defined in (3.70). T h e o r e m 3.19. For b < 0, the solution w = 0 of Eq. (3.69) is un­ stable. PROOF. Computations show that eM = cos(wf)/ + w _1 sm(ujt)A and At _ j _

cos wt — 1 a; v— (jjsmujt

sin ut

cos Ljt — \/

where u> = anj. Also (eAt - I)A-lB

=

6(1 - cos u)t)la2 0' , (bui sin uit) j a

Oj

Hence,

M(t) =

cos wt + ba~2(l — cosut) u 2

(ba~ — l)u sin uit

l

sinuit

coswt

)

198

3. PARTIAL DIFFERENTIAL EQUATIONS

and det Mil) = 1

r + — cos w. a a1 The condition b < 0 implies det M(l) > 1 and shows that at least one of the eigenvalues A of M(l) satisfies |A| > 1. Therefore, ||w(*)|| —► oo as t —► +oo, for some initial vector CQ ^ 0. ■ 1

T h e o r e m 3.20. For b > a2, the solution w = 0 of Eq. (3.69) is unstable. Calculations yield

PROOF.

det(M(l) - XI) = A2 - 2 (cos w + - r sin2 ^ ) A + 1 r + -^cosu \ a1 2/ a^ az and the expressions Ai = s + d, A2 = s — d for the eigenvalues Ai, A2 of M ( l ) , where s = cos u H—r2 sin 2 a-

W

2 n\ • 2 . °6 • 4 W 1 S,nfa; + 8,n

Z' 6

J2

2' ^a =U~ J 2

?

2-

The condition b > a2 shows that d > 0 and Ai > 1. The latter inequality implies ||w(£)|| —» 00 as t —► +00, for some initial vector

c 0 ^0.

■

Theorem 3.21. The solution w = 0 of Eq. (3.69) ts asymptotically stable as t —> +00 i/ and on/y if 0 < 6 < a2,

(3.71)

and w ^ 27rn, n = 0 , 1 , 2 , . . . . PROOF. The condition d2 < 0, which means t h a t the eigenvalues of M ( l ) are complex, leads to , u> b2 cos — 2 > (2a2-6)2' whence 6 < a 2 (1 - tan 2 j j

or

6 < a 2 f 1 - cot 2 ~\ .

3.3. WAVE EQUATION

199

Since |Ax| = |A2| and detM(l) = AiA2, the inequality |Ai| < 1 is equivalent to detM(l) < 1, that is, to b > 0. Therefore, in the case of complex eigenvalues, a criterion for asymptotic stability is 0 < b < max o2 fl - t a n 2 ^ ] ,a 2 ( l - c o t 2 - ] ) . The inequality d2 > 0 in the case of distinct real eigenvalues leads to b > max (a 2 (1 - tan 2 j ) , a2 (l - cot2 - ] ] , and the inequahties Ai < 1, A2 > — 1 yield b < a2 Hence, in this case a criterion of asymptotic stability is max (a 2 (l - t a n 2 - ) ,a2 fl - c o t 2 ^ ) ) < b < a2. Finally, if b = max (a2 (l - tan 2 j J , a2 [ 1 - cot2 j j j , then d = 0 and Ai = A2 = cosw + ba~2 sin2 u>/2, whence cos u) < Aj < cos2 UJ/2 and |Ai| < 1. According to (3.64), this implies asymptotic stability and completes the proof of criterion (3.71). 4. If b = a2, then Ai = 1, A2 = cosui, and the solutions of (3.69) are bounded but not asymptotically stable. REMARK

REMARK 5. In Theorems 3.19 and 3.20 it was implicitly assumed that to ^ 2nn. If u> = 27rn, then Ai = A2 = 1, which leads to the existence of unbounded solutions for (3.69).

COROLLARY 3.2. / / the coefficient a is irrational, then (3.71) is a criterion of asymptotic stability of the solutions to (3.68) for all j . P R O O F . Recalling that LU = UJJ — anj, we note that the equality anj = 2im is impossible for any irrational a. ■

200

3. PARTIAL DIFFERENTIAL EQUATIONS

COROLLARY 3.3. For any rational a, there exist infinitely many in­ tegers j such that the corresponding solutions Wj(i) of (3.69) are un­ bounded. PROOF. The equation w, = 2nn implies j = 2n/a, and j will be an integer for infinitely many integer n. ■ Theorem 3.22. Each component of every solution of Eq. (3.69) os­ cillates if and only if either of the following conditions holds true: (i)

b < max (a 2 (l - t a n 2 - ) ,a 2 ( l - c o t 2 T ) ) >

(ii)

max (a2 ( l - tan 2 - J ,a 2 f 1 - cot2 — j j < b

a2 2 sin2 |

and cosw < —|. PROOF. Hypothesis (i) means that the eigenvalues of M ( l ) are com­ plex, and in this case all nontrivial solutions of (3.69) oscillate com­ ponentwise. In the case of real eigenvalues, we require d2 > 0 and Ai = s + d < 0, which also implies \2 = s — d < 0, where s and d are defined in Theorem 3.20. The condition d2 > 0 leads to the left side inequality in (ii), and the condition Ai < 0 yields the right-side inequality in (ii). Finally, the restriction cosw < —1/2 arises from the comparison of the inequalities s < 0 and Ai < 0. ■ REMARK

a2

, (u>\

2a2

6. The restriction cos LO < —1/2 implies — sin I — ] < —-.

In conclusion, it is worth noting that the asymptotic properties of Eq. (3.68) depend on the algebraic nature of the coefficient a. For b < 0, all solutions of Eq. (3.68) are unstable and oscillatory; for b > a2 all solutions of Eq. (3.68) are unstable and nonoscillatory. These two cases hold true for both rational and irrational values of a. For 0 < b < max ( a 2 ( l - t a n 2 ^ ) , a 2 ( l - c o t 2 ^ ) ,

3.4. BOUNDED SOLUTIONS

201

all solutions of (3.68) are asymptotically stable and oscillatory, pro­ vided that u) ^ 27m. However, as indicated in Corollary 3.3, for any rational a, there exist infinitely many integers j such that Uj = 27m, which leads to the existence of unbounded solutions for (3.68). Fur­ thermore, since w = UJJ = anj the inequality cosui < —1/2 breaks down for infinitely many integers _;'. Therefore, under hypothesis (ii) of Theorem 3.22, there are infinitely many solutions of Eq. (3.68) which are asymptotically stable and oscillatory, as well as infinitely many so­ lutions which are asymptotically stable and nonoscillatory (u> ^ 27m). Also, for u =£ 2im and a 2 /2sin 2 (w/2) < b < a2, the solutions of (3.68) are asymptotically stable and nonoscillatory. Problems of this nature deserve further investigation. 4. Bounded Solutions of Retarded Nonlinear Hyperbolic Equations The study of nonlinear partial differential equations with piecewise constant arguments has been originated by Shah, Poorkarimi, and Wiener [243]. They inserted into a hyperbolic PDE different types of delay (piecewise constant, discrete, continuous) and observed its in­ fluence on the boundedness of the solutions. The existence problem for the corresponding equation without delay has been discussed in [260], after reducing it to a Volterra integral equation in two variables. Such problems appear in a mathematical model for the dynamics of gas ab­ sorption which leads to a nonlinear hyperbolic equation with charac­ teristic data [59]. We shall deal with the hyperbolic equation of the form uxt + a(x, t)ux + b(x, t)ut = C(x, t, u(x, t), u(x, [t])),

(3.72)

where a, b, and C are defined for (x,t) e Cl = {0 < x < l,t > 0}, and C:Ox]RxM—>R, and where [■] denotes the greatest-integer function. To get a unique solution of (3.72) in fl, we impose conditions on the characteristics: u(x, 0) =
u(0, t) = u0(t),

t > 0. (3.73)

Without loss of generality, we can assume that b(x,t) = 0 in (3.72). In other words, as we shall see in the following lemma, (3.72) assumes a

202

3. PARTIAL DIFFERENTIAL EQUATIONS

simplified form, namely, uxt + a(x, t)ux = C(x, t, u(x, t), u(x, [t])).

(3-74)

LEMMA 3.3. Suppose X(x,t) is twice continuously differentiahle in ft and satisfies u(x,t) = e*x'%(x,t).

(3.75)

Then Eq. (3.72) can be written in the form vxt + a(x, t)vx = C(x, t, v(x, t), v(x, [<])),

(3.76)

with no derivative Vt, where a and C can be expressed in terms of a, b, and C. PROOF.

From (3.75) one has ut = exvt + exXtv,

ux = exvx + exXxv,

and uxt = exvxt + ex\xvt + exXtvx + exXxtv + exXxXtv. Then Eq. (3.72) can be written as exvxt + {Xx + b)exvt + (Xt + a)exvx + (Xxt + XxXt + aXx + bXt)exv = C(x,t,exv,ex^%(x,

[<])).

We want that Xx + b = 0, which implies

X = -J*b(s,t)ds.

(3.77)

Hence, the substitution (3.75) with A given by (3.77), reduces (3.72) to (3.76) where a(x, t) = A,(x, t) + a(x, t) and C(x, t, v(x, t), «(*, [t])) = e~xC(x, t, exv, ex^%(x,

[*]))

- (Kt + K^t + aXx + bXt)v

3.4. BOUNDED SOLUTIONS

203

From (3.77), assuming that b(x, t) is differentiate with respect to t, we obtain Ax(x, t) = -b(x, t),

Xt = - j * bt(s, t) ds,

Xxt = -bt(x, t)

and X(x,[t]) =

-JQXb(s,[t])dS.

Therefore, a(x,t) = a(x,t) — J

bt(s,t)ds

and C(x, t, v, v(x, [t])) = e~xC(x, t, exv, ex^%(x,

[<]))

+ (bt(x, t) + a(x, t)b(x, t))v(x, t). The data on the characteristics for (3.76) is v(x,0) = e~X(p(x), 0<x
v(0,t) = u(0,t) = u0(t), t > 0

with A from (3.77). Thus, Eq. (3.76) is of the form (3.74).

■

Before discussing the existence problem for solutions to Eq. (3.74) under conditions (3.73), it is appropriate to transform this equation with data on charateristics into an integral equation of Volterra type in two variables. Denoting ux(x,t) = w(x,t) changes (3.74) to wt(x, t) + a(x, t)w(x, t) = C(x, t, u, u(x, [t])).

(3.78)

Let w„(x,t) be a solution of Eq. (3.78) on the interval n < t < n + 1, satisfying the condition wn(x,n) = c„(x), where n > 0 is integer. Then [t] = n, and integrating (3.78) with respect to t from n to t gives wn(x, t) = cn(x) exp ( - Jn a(x, s) ds) + j exp ( - J a(x, T) dr J C(x, s, u(x, s), u(x, n)) ds. (3.79)

204

3. PARTIAL DIFFERENTIAL EQUATIONS

LEMMA 3.4.

K(x) = I

Jn

Let

expl—/ \ J°

a{x,T)dT) C(x, s,u(x, I

s),u(x,n))ds, (3.80)

then cn(x) = CQ(X) exp (— I

PROOF.

a(x,s)ds)

+ E 6 t (x) exp v( - Jkra(x,s)ds), n>\. *=o ' Putting t = n + 1 in (3.79) and taking into account

(3.81)

wn(x, n + 1) = wn+i(x, n + 1) gives c„+1(x) = a„(x)c„(x) + bn(x),

(3.82)

where a„(x) = exp( — j

a(x,

s)ds\.

We look for the solution of (3.82) in the form Cn

= d„

n-l

na.

(3.83)

m

m=0

As a result, one obtains d.,, dn = dr.+1 "-- «L

b„ nt1

rc=o «m and 71-1

4 = 4+E^/n«m. t=0

/

(3.84)

m=0

Hence, (3.81) follows from (3.83) and (3.84) since n-l

n m=0

II am = exp ( - j

a(x, s) dsj ,

dQ =

CQ.

■

T h e o r e m 3.23. Assume for problem (3.73), (3.74) the conditions:

3.4. BOUNDED SOLUTIONS

205

(i) uo(t) is bounded and continuously differentiable on t > 0, with the possible exception of the points [t] G [0, oo), where one-sided derivatives exist; (ii) m > 0 in $7; (iv) C(x, t, u, v) is continuous o n f i x l x R , with C(x, t, 0,0) bounded on Cl, and satisfies the Lipschitz condition in u, v: \C(x,t,u,v)

-C(x,t,u,v)\

< L ( | u - « | + |w-w|).

(3.85)

Then there exists a unique continuous solution of problem (3.73), (3.74) defined in fl and bounded there. PROOF.

Integrating (3.79) with respect to x we get

u(x,t) = Uo(t) + +

c„(r)expl— /

dr I exp I — / a(r,T)dr\

a(r,s)ds\dr C(r,s,u(r,s),u(r,n))ds,

(3.86)

where cn{r) are given by (3.81). To apply to (3.86) the method of successive approximations, we construct a sequence {u,(x,t)} according to the recursion relations: ux{x,t) = 0, ui+i(x, t) = u0(t) + j * c„(r) exp f- J^ a(r, s) dsj dr + I dr I exp (— / a(r, r)dr ] C(r, s, u,(r, s), u,(r, n)) ds,

(3.87)

(i > l , n < t < n + 1,0 <x < I). The following estimates are satisfied in fi: \ui+l(x, t) -

Ui(x,

t)\ < J* dr fn e-m^-^\C(r,

s, Ui(r, s), u{(r, n))

- C(r, s, u,_i(r, s), u,_i(r, n))| ds.

206

3. PARTIAL DIFFERENTIAL EQUATIONS

Taking into account Lipschitz condition (3.85), we obtain \ui+l(x,t)

- Ui(x,t)\ < Lj'drfce-^-'Wu^s)

- u,-_i(r,«)| + \ui(r,n) -

Ui^i(r,n)\)ds,

which leads to the estimates \ui+l(x,t)-Ui(x,t)\
,

(0<x
(3.88)

(i - 1)!

where A = sup|u 2 0M)ln From (3.87) we have U2(x,t) = u0(t) + J c n ( r ) e x p f - y

a(r,s)ds)dr

+ J' dr J* exp (- f a(r, r) dr\ C(r, s, 0,0) ds. (3.89) Since «o(0 a n < l C(x,t, 0,0) are bounded, it remains to evaluate c„(x). Putting t = 0, n = 0 in (3.79), we conclude that c0(x) =
,

YJ bk(x)exp[k=0

V

.„

N

a(x,s)ds) Jk

n-1

< B "£, . '

fc=0 00


B

L —km

= ~

1

-e~m'

k=0

This proves that the coefficients cn(x) are uniformly bounded for all n > 0 and x G [0,/]. Therefore (3.89) implies that U2(x,t) is bounded in ft and A < oo. Inequality (3.88) shows that the sequence {u,(x,*)} uniformly converges in fi, which in turn implies the existence of a

3.4. BOUNDED SOLUTIONS

207

continuous solution u(x,t) in fl for Eq. (3.86). Also, from (3.88) one can see the boundedness of u(x, t) in fi. Since oo

u(x,t) = £ ( u l + i ( M ) -

Ui(x,t)),

i=l

then

Hx,t)\
(* -

1)!

To prove the uniqueness of solution u(x,t), we assume that there exist two bounded solutions u(x,t) and v(x,t) to Eq. (3.86). Hence, M M ) - v(x,t)\ <

JoXdr^e-m^C(r,s,u{r,s),u(r,n)) -C(r,s,v(r,s),v(r,n))\ds

(3.90)

and LHx M M ) - v(x,t)\ <

,

(3.91)

m where

H = sup(|u(a:, t) - v(x, t)\ + \u(x, It]) - v(x, [t])\). si

Applying estimate (3.91) to (3.90), we find . „L2x2 m2 JO Jn m Continuing this iterative process leads to the conclusion that 2LHr .

\u(x,t) - v(x,t)\
\u(x,t) - v(x,t)\ < —

,

<
n>l.

Since ,. H (2Lx\n lim — "^°°n! V m /

n

=0,

then sup|u(a;, t) — v(x, t)\ = 0, n and this proves the uniqueness of solution to Eq. (3.86).

■

208

3. PARTIAL DIFFERENTIAL EQUATIONS

Consider the differential-difference equation uxt{x,i) + a(x,t)ux(x,t) = C(x,t, u(x,t),u(x,t (h = const. > 0)

— h))

(3.92)

and pose for (3.92) the following initial and boundary conditions: u(x, t) = ip(x, t); u(0,t) = u0(t);

0 < x < I, -h < t < 0, t>0.

(3.93)

Here 0; (ii) 0} and satisfies a(x,t) > m > 0 in fl; (iv) C(x, t, u, v) is continuous and bounded on £1 x M. x K and satisfies the Lipschitz condition \C(x,t,u,v)

-C(x,t,u,v)\

< L\u-u\.

(3.94)

Then there exists a unique continuous solution of problem (3.92), (3.93), defined in Cl and bounded there. PROOF. Denote ux(x,t) = w(x,t), then wt(x, t) + a(x, t)w(x, t) = C(x, t, u(x, t), u(x, t — h)).

(3.95)

To solve this equation, we use the method of steps. On the interval 0 < t < h, Eq. (3.95) becomes wt(x, t) + a(x, t)w(x, t) = C(x, t, u(x, t), ip(x, t — h)).

3.4. BOUNDED SOLUTIONS

209

Integrating from 0 to t we obtain w(x, t) =
(3.96)

and integrating (3.96) from 0 to a: gives u(x, t) = u0(t) + £ (ptf, 0) exp (- jo a(£, r) dr) d£ + JoZ /o* exp ( - Jl a(f, 9) d0} C(f, r, u(£, r), v(£, r - h)) dr d£. (3.97) To prove existence-uniqueness, we apply to (3.97) the method of suc­ cessive approximations. Put m(x, t) = u0(t) + j * (p((£, 0) exp f- JQ a((, r) dr\ d£, then since \u0(t)\
\
and a(x, t) > m > 0, we have \ui(x,t)\


with some constants A, K, N. Furthermore, t) = Jl a((, e xexp p ( (-/% t ( £ ,r)r )dr) d r )d£ # = «„(<) u0(t)-+f /j ;* ^^,(0£ ), 0) u2(a:,t) + \l H exp ex( - / a(£, 0) ^)c(e,i dfl) C(f, r,■^lC^T-).^r --h))drd(, fc)) dr d£, U l (£, r), ytf, 7 "-

/o7o' p(-/>^^

and

-/>))R

|u2(x,

M M < —x, m where \C(x,t,u,v)\

<M.

210

3. PARTIAL DIFFERENTIAL EQUATIONS

Also, by virtue of (3.94), \u3(x, t) - u2(x, t)\ < jT fQ exp ( - fT a(f, 9) dfl) x\C(Z,TMt,r),
< m- Jf» M£,r)-«i(£,r)K M_ (Lx\2 ±

~ L \m)

2!'

and similarly,

. ,

.

,

v

M /-Lx\° 1

\u4(x,t)-U3(x,t)<J-{—)

y-

Continuing this procedure gives the estimate \un+i{x,t)-un{x,t)\<

— [— L \m)

—, n!

n = l,2,...

and since oo

u(x,t) = ui(x,t) + ^2(ui+i(x,t)

-

u{(x,t)),

«=1

then M ILx\ \u(x,t)\ < A + — exp — , Lt

(0
0<x
\ TTl J

For the solution of (3.92), (3.93) on the interval h < t < 2h we have ti(x, t) = « 0 (0 + j P uj0)(e, h) exp ( - j£ o(C, r) rfr) d£ + /fl" fh exp ( - jf* a(£, 9) d6J Ctf, r, «(£, r ) , ««»(£, r - ft)) dr d£, where u! 0 '(s, *) is the solution of (3.92), (3.93) on 0 < t < ft. Repeating the above calculations yields M (Lx\ \u(x,t)\ < M + -r expf — 1 ,

(ft
2ft, 0 < x < I)

3.4. BOUNDED SOLUTIONS

211

where A, = sup (\u0(t)\ + JoX \uf\t,

h)\ exp ( - fh a(£, r) dr) df) .

To evaluate ?40)(a;,/i), we use Eq. (3.96) from which v£\x, h) = ux(x, 0) exp ( - J a(x, r) dr) + JQh exp (- £ a(x, 9) d6j C(x,

T, UW(X,

T),


l4 0 ) (*, h)\ < Ke~mh + —(1 - e~mh) < Ke~mh + —, m m and M < N + (l<e-mh + —) I. In the region (2h
{•£)

T

where

A2
-Imh

Finally, it is easy to see that M (Lx\ \u(x, t)\ < An + — exp — , L \m) with

M + —e -mh m

ml

(nh
+ l)h, 0 < x < I)

(Ke-nmh \ nmh + M_ " ^ e_imh A„ A < N + (Ke+ — " E e- imh) Il n
((V/<:K++arn— £g.—), e - ' '/ z


M °°

1=0

=N+

"2 ,=o

\

m/

/

m/l V m l - e- m/l / Thus all values An are uniformly bounded, and the proof is complete.

212

3. PARTIAL DIFFERENTIAL EQUATIONS

Now we state a theorem on the boundedness of solutions of nonlin­ ear hyperbolic equations with argument X(t) £ C[0, oo) satisfying the condition 0 < X(t) < t. T h e o r e m 3 . 2 5 . Assume for the functional

differential

uxt + a(x, t)ux = C(x, t, u(x, t), u(x, X(t))),

equation (3.98)

with the conditions (3.73) the hypotheses: (i) uo(t) is bounded and continuously differentiable on t > 0; (ii) ip(x) is continuously differentiable on [0,1]; (iii) a(x,t) is continuous in Cl = {0 < x < /; t > 0} and satisfies a(x,t) > m > 0 in ft; (iv) C(x, t, u, v) is continuous on fi x R x R, with C(x, t, 0,0) bounded on fl, and satisfies a Lipschitz condition in u and v. In addition, \{t) is continuous on [0, oo) and 0 < \{i) < t, limA(i) = oo as t —> o o .

Then there exists a unique continuous solution of problem (3.73), (3.98), defined in O and bounded there.

CHAPTER 4 R e d u c i b l e Functional Differential E q u a t i o n s In [273]-[278] a method has been discovered for the study of a spe­ cial class of functional differential equations — differential equations with involutions. This basically algebraic approach was developed also in a number of other works and culminated in the monograph [223] by Przeworska-Rolewicz. Although papers continue to appear in this field, some aspects of the theory still require further investigation. In connection with the purposes of this chapter we mention only such top­ ics as higher-order equations with rotation of the argument, equations in partial derivatives with involutions, influence of the method on the study of systems with deviations of more general nature, and solutions in spaces of generalized and entire functions. 1. Differential Equations w i t h Involutions In studying equations with a deviating argument, not only the gen­ eral properties are of interest, but also the selection and analysis of individual classes of such equations which admit simple methods of investigation. In this section we consider a special type of functional differential equations that can be transformed into ordinary differen­ tial equations and thus provide an abundant source of relations with analytic solutions, as well as heuristic ideas for equations with linear argument transformations. Silberstein [252] studied the equation x'(t) = x (-\

, 213

0
(4.1)

214

4. REDUCIBLE FDE

and assumed a solution of the form x(t) = tk + \tm, where k, m, and A are constants. Substituting in (4.1) gives

ktk~l + \mtm-1 = rk + \rm and k + m = l,

X = k,

km = 1.

Therefore, k2 - k + 1 = 0, and simple computations yield the solution. In [274] we proved that the solution is obtained very easily by a differentiation of (4.1). As a matter of fact, x"(t) = whence,

-X)- - ^ ( 0 .

t2x"(t) + x(t) = 0.

(4.2)

Consequently, x(t) = Vi Ci cos l — lnA+dsin

(^—\nt\

.

Substituting x(t) in (4.1), we obtain C\ = y/3C2, and finally, x(t) = CVicos f-^-lnt - ~) . For a more general equation tnx'(t) -x(-Jt we have x'(t) = t~nx(l/t) x"(t)

0 < * < oo

and

= - r - V (~) - nrn-lx Q .

4.1. DIFFERENTIAL EQUATIONS WITH INVOLUTIONS

215

Hence, x"(t) =

-rn~Hnx{t) - nrn-ltnx'{t),

and we obtain Euler's equation t2x"(t) + ntx'(t) + x(t) = 0. The roots Ai and A2 of the corresponding characteristic equation A2 + (n - 1)A + 1 = 0 are complex for — 1 < n < 3 and real elsewhere. By means of the condition x(l) = x'(l), we find the solution of the original FDE: x(t) = Ct, for n = - 1 ; x(t) = Ct(l - 2In t), for n = 3; x(t) = C(tXl + Ai**2), x{t) = Cta cos(b In t) -\

for n < - 1 or n > 3; — sin(6 In t)

1 -n where a = Re A = —-—,

for - 1 < n < 3 b = Im A

Obviously, the clue to the solution is the fact that f(t) = 1/t maps the interval (0,00) one-to-one onto itself and the relation /(/(*)) = *,

(4-3)

or, equivalently,

/(*) = rl® is satisfied for each t 6 (0,00). A function f(t) ^ t that maps a set fi onto itself and satisfies on Cl condition (4.3), is called an involution. In other words, an involution is a mapping which coincides with its own inverse. Let fl (t) = / ( * ) , fn+l(t) = f(fn(t)), n=l,2,... denote the iterations of a function / : Cl —* Ct. A function / : ft —* SI is said to be an involution of order m if there exists an integer m > 2 such that fm(t) = t for each < £ f i , and /„(<) ^ t for n = 1 , . . . , m — 1. It is easy to check that the following functions are involutions.

4. REDUCIBLE FDE

216 EXAMPLE

1. f(t) = c - t on R = (—00,00), where c is an arbitrary

real. EXAMPLE 2.

(-at

ioit>0,

W-U for«<0, I a on R, where a > 0 is arbitrary [223]. EXAMPLE 3.

\t~k /(')

"

| r i/*

for 0 < t < 1, for

t

>

1;

on (0,oo) where fc is an arbitrary positive integer [223]. EXAMPLE 4. The function f(z) = ez, were e = exp(27re'/m), is an involution of order m on the complex plane. EXAMPLE

5. The function [178]

t, t G (-oo,0)U(m,+oo), t + 1, iG ( 0 , l ) U ( l , 2 ) U - - - U ( m - 2 , m - l ) , /(*) = t — (m — 1), ( 6 ( m - l , m ) is an involution of order m on O = ( - 0 0 , 0 ) U (0,1) U ■ ■ • U (m - 1, m) U (m, + 0 0 ) .

Definition 4 . 1 . A real function /(*) ^ i of a real variable t, defined on the whole axis and satisfying relation (4.3) for all t, is called a strong involution [274]. We denote the set of all such functions by I. The graph of each / G I is symmetric about the line x = t in the (t, x) plane. Conversely, if T is the set of points of the (t, x) plane, symmetric about the line t — x and which contains for each t a single point with abscissa t, then T is a graph of a function from I. One of the methods for obtaining strong involutions has been suggested by Shisha and Mehr [251]. Assume that a real function g(t, x) is defined on the set of all ordered pairs of real numbers and is such that if g(t,x) = 0 then g(x,t) = 0 (in particular,

217

4.1. DIFFERENTIAL EQUATIONS WITH INVOLUTIONS

this is fulfilled if g is symmetric, i.e., g(t, x) = g(x, t)). If to each t there corresponds a single real x = f(t) such that g(t,x) = 0, then / £ I. For example, g(t,x) = t +

x-c,

then f(t) =

c-t.

If we take g(t,x) = t3 + x 3 - c , then f(t) = v/c~^73. Every continuous function / £ I is strictly decreasing [26]. Hence, lim f(t) = +oo, t—>—oo

lim f(t) = - o o .

(4.4)

t—*-|-oo

Theorem 4 . 1 . A continuous strong involution f(t) has a unique fixed point. PROOF. The continuous function ip(t) = f(t) — t satisfies relations of the form (4.4) and, therefore, has a zero which is unique by virtue of its strict monotonicity. ■ We also consider hyperbolic involutory mappings m

=

?L±li yt — a

(a 2 + / ? 7 > 0 )

(4.5)

which leave two points fixed. We introduce the following definition. Definition 4.2. A relation of the form F(t, x(fi(*)), • . . , * ( / * ( * ) ) , . . . , l W ( / i ( ' ) ) » • ■ • - x{n\fk{t)))

= 0,

in which fi(t), ■. ■, fk(t) are involutions, is called a differential equation with involutions [274].

218

4. REDUCIBLE FDE

Theorem 4.2. ([274]). Let the equation x'(t) = F(t,x(t),x(f(t))

(4.6)

satisfy the following hypotheses. (i) The function f(t) is a continuously differentiable strong involu­ tion with a fixed point *o(ii) The function F is defined and is continuously differentiable in the whole space of its arguments. (iii) The given equation is uniquely solvable with respect to x(f(t)): x(f(t)) = G(t,x(t),x'(t)).

(4.7)

Then the solution of the ordinary differential equation

*"«> - §+W)At)+W)mF('('>•*<*<»•*(% (where x(f(t))

is given by expression (4.7)) with the initial conditions x(t0) = x0,

x'(t0) = F(t(,,x0,x0)

(4.9)

is a solution of Eq. (4.6) with the initial condition x(t0) = x0.

(4.10)

PROOF. Eq. (4.8) is obtained by differentiating (4.6). Indeed, we have

*"< ( >=f + lr' <,)+ ra*' (/W)m But from (4.6) and the relation /(/(<)) = t it follows that

x'(f(t)) = F(f(t),x(f(t)),x(t)). The second of the initial conditions (4.9) is a compatibility condition and is found from Eq. (4.6), with regard to (4.10) and /(
= F(x(t)).

U

219

4.1. DIFFERENTIAL EQUATIONS WITH INVOLUTIONS

T h e o r e m 4.3. ([274]). Assume that in the equation x'(t) = F(x(f(t)))

(4.11)

the function f(t) is a continuously differentiable strong involution with a fixed point to, and the function F is defined, continuously differentiable, and strictly monotonic on (—00,00). Then the solution of the ordinary differential equation x"(t) =

F'(x(f(t)))F(x(t))f'(t),

x(/(0) =

F-\x'(t))

with the initial conditions x(t0) = x0,

x'(t0) = F(x0)

is a solution of Eq. (4.11) with the initial condition x(to) = XQ. COROLLARY 4.1. Theorems 4.2 and 4.3 remain valid if f(t) is an involution of the form (4.5), while the equations are considered on one of the intervals (—00, a/7) or (a/7,00). REMARK 1. Let to be the fixed point of an involution f(t). For t > to, (4.6) and (4.11) are retarded equations, whereas for t < to they are of advanced type.

EXAMPLE 6. By differentiating the equation [274]

(4 12)

W-3TH

'

and taking into account that x '(a

-t) =

1

x(tY

we obtain the ordinary differential equation d'2x _ J_(dx\2 dt* ~x(t)[dt)

. ■

.

(4 13)

-

220

4. REDUCIBLE FDE

The fixed point of the involution f(t) = a - t is t0 = a/2. The initial condition for (4.12) is '1^1 —o. the corresponding conditions for (4.13) are X

»f)=X°'

\2) 12/ x,{0

Eq. (4.13) is integrable in quadratures: (t-a/2\ 5— • V xo I This is the solution of the original Eq. (4.12). The topic of paper [179] is the equation F(t,x(t),x'(t),...,x^(t)) = x(f(t)), x[t) = x0exp

(4.14)

where x is an unknown function. T h e o r e m 4.4. ([179]). Let the following conditions be satisfied: (1) The function f maps the open set Cl into Cl, tt being a subset of the set R of real numbers. (2) The function f has derivatives up to, and inclusive of, the order mn — n for each t £ Q, f'(t) ^ 0 for each t € O. (3) The function F(i, ui, «2, • - • > w„+i) is mn — n times differentiable of its arguments for each t G Cl and ur £ R(r = 1 , . . . , n +1) and dF/dun+1^0. (4) The unknown function x has derivatives up to, and inclusive of, the order mn on f2. (5) The function f has iterations such that fl(t)

= f(t),...,fk(t)

= f(fk_l(t)),...,fm(t)

= t

for each t 6 ft, where m is the smallest natural number for which the last expression holds. In this case there exists an ordinary differential equation of order mn such that each solution of Eq. (4.14) is simultaneously a solution of this differential equaiton.

4.1. DIFFERENTIAL EQUATIONS WITH INVOLUTIONS

221

Let us consider the functional differential equation [180] n / l (t), *(/!(*)), • • • , * ( *°(/l(t)), ■ • ■ , *(/„(*)), • • •, * <M (/«(*))) = 0, (4.15) where x is an unknown function and where the following conditions are fulfilled: (1) The functions / i , . . -, /„ form a finite group of order n with respect to superposition of functions, fi(t) = t, and map the open set fi into ft, ft being the largest open set wherein all expressions appearing in [180] are defined. (2) The functions x and fT (r = 1 , . . . ,n) have derivatives up to the order p, where p = max(&i,..., kn), so that f'r(t) ^ 0 for every t £ ft and r = 1 , . . . , n. (3) For the function F at least one relation dF/(dx^s\fr)) ^ 0 is valid for s = 0 , . . . ,p; r = 2 , . . . , n and every t G fi. T h e o r e m 4.5. ([180]). / / conditions (l)-(3) are satisfied, then ev­ ery p-times differentiable solution of Eq. (4.15) is a component of the solution of a system of ordinary differential equations with argument t only. This system is obtained from Eq. (4.15). To investigate the equation x'(t) = f(x(t),x(—t)), denotes y(t) = x(—t) and obtains

y'(t) = -x'(-t)

= -f(x(-t),x(t))

=

Sharkovskii [248]

-f(y,x).

Hence, the solutions of the original equation correspond to the solutions of the system of ordinary differential equations

§ = /(M),

d

i = -f(y,*)

with the condition x(0) — 2/(0). From the qualitative analysis of the solutions of the associated system he derives qualitative information about the solutions of the equation with transformed argument. The linear case is discussed in some detail. Several examples of more general equations are also considered. Boundary-value problems for differential

222

4. REDUCIBLE FDE

equations with reflection of the argument have been studied in [104, 105, 106, 291]. 2. Linear Equations In this section we study equations of the form In this section we study equations of the form n

k

+m

Lx(t) = 1k(t)l •S \t) = *(/(*)) Lx(t) = i=0 £ <*(*)*«(*) =- *(/(<)) + m

(4.16)

with an involution f(t). with an involution f(t). Theorem 4.6. ([274]). Suppose that the initial conditions x^(t0)

= xk,

k = 0,...,n-l

(4.17)

are posed for Eq. (4.16) in which the coefficients ak(t), the function ip(t), and the strong (or hyperbolic (4.5)) involution f(t) with fixed point to belong to the class C"(—oo, oo) (or C"(a/7, oo)). Ifn > 1, then assume f'(t) ^ 0. We introduce the operator

M

<" 8 >

= W)7f

Then the solution of the linear ordinary differential equation £ ak(f(t))MkLx(t) k=o

- x(t) = £ ak(f(t))MkrP(t) k=o

+

^(f(t)) (419)

with the initial conditions x(k)(t0) = xk, M

k=

0,...,n-l,

*M0L o ==** + MV(0L«o, * = 0 , . . . , n - l

is a solution of problem (4.16), (4.17).

(4.20)

223

4.2. LINEAR EQUATIONS

PROOF.

By successively differentiating (4.16) n times, we obtain *(/(*)) = Lx{t) - it>(t) = M°Lx(t) - M V ( t ) ,

Am

- W)iLx(t) ~ w^'m=MLt(t)

~Mm-

M M!il W - MVW. * " < > « » - / M ^ " - - 7^)| *W =

««(«*))-^^i^^o--

/

w

> '

v

w

Mnntp(t), ip(t). = MnnLx(t) Lx(t) - M

These relations are multiplied by a0(/(*)), ai(/(<)). ■■-,«»(/(<)),

respectively, and the results are added together:

± ak(f(t))xW(f(t)) k=0

= ± ak(f(t))MkLx(t)

- £ a t (/(t))Mfy(t).

k=0

*=0

By virtue of /(/(*)) = t, it follows from (4.16) that

±ak(f(t)>ik\f(t)) = x(t) + Hf(t))k=0

Thus, we obtain Eq. (4.19). In order that the solution of this equation satisfies problem (4.16), (4.17), we need to pose the following initial conditions for (4.19): the values of the function x(t) and of its n — 1 derivatives at point
= «<*>(/(<)) + AfV(*)»

k= k

by substituting the values to and xk for t and x^ \t).

0,...,n-l ■

The equation

*«(*) = x ( i )

(4.21)

4. REDUCIBLE FDE

224

was studied in [235] by assuming a solution of the form x(t) = tk + \tm, substituting which in (4.21) yields k(k - 1) • - • (k - n + l)tk-n + \m(m - 1) • • • (m - n + l ) r ~ " - r * - \rm

= 0.

Equating powers of t and corresponding coefficients shows that k + m = n, A = k(k — 1) • • • (k — n + 1), Am(m — 1) • ■ • (m — n + 1) = 1. Eliminating A gives k(k - l) 2 • ■ • (k - n + l)\k

-n)

= (-1)".

Lettingp = k2 — kn and combining & and k — n, (k — l ) 2 and (k — n+1) 2 , etc., we obtain [p + (n - l)]2[p + 2(n - 2)]2 • - • [p + ^(n 2 - l)]2p = - 1 , \p + (n-

l)]2[p + 2(n - 2)]2 • • • [p + -n 2 ] 2 p = 1,

n odd

n even

so that A: depends on the solutions of a n-th degree equation (the indicial equation) in p. Let p\,... ,p„ be distinct solutions of this equation, then to each root p,- there correspond two values of k. Associated with each of these two values of k we have a value of A, and the product of these two values of A is unity, implying that the two solutions are identical. Thus, the general solution of (4.21) is given by

*(o = £ *(tki+wk% with arbitrary constants c;. The case of multiple roots pi requires obvi­ ous modifications. The application of Theorem 4.6 to Eq. (4.21) refines these results.

225

4.2. LINEAR EQUATIONS

Theorem 4.7. ([274]). Eq. (4.21) is integrable in quadratures and has a fundamental system of solutions of the form ta(\nt)j sm(blnt),

ta(lnt)j cos(blnt),

(4.22)

where a and b are real and j is a non-negative integer. PROOF. By and n-fold differentiation Eq. (4.21) is reduced to the Euler equation

£ b^x^Xt) = x(t).

(4.23)

k=n+l

For n = 1 this follows from (4.2). Let us assume that the assertion is true for n and prove its validity for n + 1. In accordance with for­ mula (4.18), we introduce for Eq. (4.21) the operator M =

-t^. dt On the basis of (4.19) and (4.23) we have Mnx^n\t)= n

n+i

M x( \t)=

2n

b{^tkx^(t),

£ k=n+l 2n

£ &i"W*+1>(*).

k=n+l

Then Mn+lx(n+l){t)

=

_t2l_

g

k

b^t

x(k+1\t)

dt k=n+l

= - E kb^]tk+lx^\t)

- £

k=n+l

b£hk+2x(k+2\t).

k=n+l

Consequently, the equation

x^n+1\t) = x (jj is reduced by and (n + l)-fold differentiation to the Euler equation M

n+Ix(n+l)^ _

x

^

4. REDUCIBLE FDE

226

At the same time we established the recursion relation 4" +1 ) = _(jfc - l)4 n _\ - 4-2 ,

&w = o,

n + 2 < f c < 2 n + 2,

°2n+l — U>

connecting the coefficients of the Euler equations £

b^x^t)

= x(t)

and

fc=n+l

' i f 6t ,+1) «*x«(*) = *(«), k=n+2

which correspond to the equations xW(t) = x(-)

and

x(n+1\t)

= x (jj .

It is well known [46] that the Euler equation has a fundamental system of solutions of the form (4.22), where a+bi is a root of the characteristic equation and j is a non-negative integer smaller than its multiplicity. The theorem is proved. EXAMPLE

7. The study of the nonhomogeneous equation [274]

x'(t) = x(-]+ip(t),

0
i(j(t)

eC\0,oo)

x(l) = x0 reduces to the problem

t2x"(t)+x(t)=t^(t)-xp(^y x(l)

= x0,

X'(1)

= X0 +

I/J(1).

The solution is x(t) = xQV~tcos ( ^ I n t ) + ~

^jyn^L

( y + V-(l)) v^sin f ^ l n t ]

[„vW-,(I)],„.

227

4.2. LINEAR EQUATIONS

Theorem 4.8.

([274]). The solution of the equation x'(t) = atpx (- j ,

0 < t < co,

(4.24)

x(l) = x0 is representable in closed form. PROOF. After differentiation, Eq. (4.24) is reduced to the form t2x"(t) - (3tx\t) + a2x(t) = 0 with the initial conditions x(l) = x0,

x'{\) = ax0.

Putting t = exp u, we obtain x"(u) - ( 1 + P)x'(u) + a2x(u) = 0, The roots of the characteristic equation A2 - (1 + f3)\ + a2 = 0 are =

At 2

1+/?

s

(1+/?) 2 — a2.

Consider various cases: 2 (1 ao = ((1) 1 ) 4/i s A >

(l±«!_ ^>o,o, x0

A *(*) = V A, - ^A K« " *»)**' + ( l - " ^ l

( 2 ) ^ - ^ x(t) = (3)

= 0,

xotWV2

! , { a _I±£]] n 7 ;

(1+/3) 2 - a 2 = - A 2 < 0,

x(0=^of (1+/?)/2 cos( A In t) + °

(1 +

. ^ ) / 2 sin( A In <)

228

4. REDUCIBLE FDE

For a = 1, P — 0 the latter formula yields the solution of Silberstein's equation (4.1). ■ EXAMPLE 8. The equation afx(t) + bf+lx'(t)

= cf+lx

(-) + dtsx' (-) ,

(t>0)

(4.25)

where a,b,c,d,r,s are real constants and x is an unknown function, was explored in [180]. Let b2 - d2 ^ 0. By putting x(l/t) = y(t), Eq. (4.25) becomes afx(t) + bf+1x'(t)

= cts+1y(t) - dts+2y'(t).

(4.26)

If 1/t is substituted for t in (4.25), we get arry(t)

- brr+1y'(t)

= crs-lx(t)

+ drsx'(t).

(4.27)

From (4.26) and (4.27), Euler's equation is obtained: t2x"(t) + (r- s)tx'(t) + (Bs -Br

+ A2-B2

+ B)x{t) = 0,

where A = (be - ad)/(b2 - d2), B = (cd - ab)/(b2 - d2). If b = d and a = c, Eq. (4.25) is equivalent to the system of equations ax(t) + btx'(t) = u(t),

u (- j = r _ s _ 1 u ( i ) .

If b = d and a ^ c, (4.25) reduces to the functional equation

X

{]) = f~S~lxW-

In the case of b = —d and a = —c, Eq. (4.25) reduces to the system ax(t) + btx'(t) = u(t),

u (-] =

-f-s-1u(t).

In the case of b = — d and a =fi —c, Eq. (4.25) reduces to the functional equation

x

(7) = -'""^w-

229

4.2. LINEAR EQUATIONS

The equation x'(t) = x(f(t)) with an involution f(t) has been stud­ ied by Kuller [150]. Consider the equation [178] with respect to the unknown function x(t): x'(t) = a(t)x(f(t))

+ b(t).

(4.28)

(1) The function / maps an open set fl onto fi. (2) The function / can be iterated in the following way

h(t) = f(t), ...,fk(t)

= /(A_i(<)),..., fm(t) = t,(te

fi)

where m is the least natural number for which the last relation holds. (3) The functions a(t), b(t), and f(t) are m — 1 times differentiable on f2, and x(t) is m times differentiable on the same set. Theorem 4.9. ([178]). Eq. (4.28), for which conditions (1) - (3) hold, can be reduced to a linear differential equation of order m. EXAMPLE 9. Consider the equation [178] **(*) = *(/(*)),

f{t) = {\-t)~\

(4.29)

and SI = (-oo, 0) U (0,1) U (1, +00). For / we have f3(t) = t onto. In this case (4.29) is reducible to the equation t\l

- tfx"'{t)

- 2*2(1 - t)x"{t) - x(t) = 0.

Theorem 4.10. In the system x'{i) = Ax(t) + Bx(c - t),

x ( | j = x0

(4.30)

let A and B be constant commutative r x r-matrices, x is an r-vector, and B is nonsingular. Then the solution of the system x"(t) = (A2 -

*(|)=*o, is the solution of problem (4.30).

B2)x(t),

x'(^j = (A + B)x0

4. REDUCIBLE FDE

230

In [152] it has been proved that the equation t2x"(t) - Vl2x ( j ) = 0,

0 < t < OO

has the general solution *(*) =

Cl (VsV

+ r 2 ) + c2 [sin(V31nf) -

^cos^ln*) 3 + y/l2

while the equation

t2x"(t) + s/Ux (jj = 0 has the general solution x(t) = c3(t2 - \ / 3 r 2 ) + c4 sin (V31nf) + ■ •*" J-

v3

cos^lnf)

It follows from here that, by appropriate choice of Ci,C2,C3, and C4, we can obtain both oscillating and nonoscillating solutions of the above equations. On the other hand, it is known that, for ordinary secondorder equations, all solutions are either simultaneously oscillating or simultaneously nonoscillating. It has also been proved in [152] that the system

x'(t) = A(t)x(t) +

f(t,x(j)y

1 < t < 00,

'K))l*'hG) where 6 > 0 and q > 1 are constants, is stable with respect to the first approximation. For the equation

t aktkxW(t) = « ( ! ) ,

k=o

we prove the following result.

w

0 < t < 00

(4.31)

4.2. LINEAR EQUATIONS

231

T h e o r e m 4.11. Eq. (4.31) is reducible by the substitution t = e* to a linear ordinary differential equaiton with constant coefficients and has a fundamental system of solutions of the form (4.22). PROOF. Put t = es and x{es) = y(s), then tx'(t) = y'(s). that

Assume

k t«**W(<) xW(t) = == £y(s), Ly(s),

where L is a linear differential operator with constant coefficients. From the relation (*+lx
= t£.

[***<*>(<)] - -

ktkxW(t)

we obtain th+1x(k+i\t)

=

L[y'(s)-ky(S)},

which proves the assertion The functional differential equation T Q'{t) Q'(t) = = AQ(t) AQ(i) + + BQ 5 QT(T ( r --t), -oo 0, Q(t) is a differentiable n x n matrix and QT(t) is its transpose, has been studied by Castelan and Infante [41]. Existence, uniqueness and an algebraic representation of its solutions are given. This equation, of considerable interest in its own right, arises naturally in the construction of Liapunov functionals for retarded differential equations of the form x'(t) — Cx(t)+Dx(t—r), where C, D are constant n x n matrices. The role played by the matrix Q(t) is analogous to the one played by a positive definite matrix in the construction of Liapunov functions for ordinary differential equations. It is shown that, unlike the infinite dimensionality of the vector space of solutions of functional differential equations, the linear vector space of solutions to (4.32) is of dimension n 2 . Moreover, the authors give a complete algebraic characterization of these n 2 linearly independent solutions which parallels the one for ordinary differential equations, in­ dicate computationally simple methods for obtaining the solutions, and allude to the variation of parameters formula for the nonhomogeneous problem.

4. REDUCIBLE FDE

232

The initial condition for (4.32) is

o

(4.33)

= K,

where K is an arbitrary n x n matrix. Eq. (4.32) is intimately related to the system Q'{t) = AQ{t) + BR(t), R'(t)

= -Q(t)BT

R(t)AT,

(4.34)


(4.35)

-

with the initial conditions

«©-*

For any two nxn matrices P, 5, let the n 2 X n 2 matrix P ® S denote the Kronecker (or direct) product [23] and introduce the notation for the nxn matrix S\*

*-<*>-(£)-

sn* where Sj* and Sj* are, respectively, the ith row and j t h column of S; further, let there correspond to the nxn matrix S the n2-vector s — (si*,... ,sn*)T. With this notation Eqs. (4.34) and (4.35) can be rewritten as

[r(t).

A®I

B®I

-I®B

-I® A

r(*).

and

?(0 =[M,..., MT, rg) =(£,..., tj/, which, with the obvious correspondence and for simplicity of notation, are denoted as p'(t) = CP(t),

V (0=^/2-

(4.36)

233

4.2. LINEAR EQUATIONS

Here p(t) is an 2n2-vector and C is a 2n2 x 2n2 constant matrix. The use of (4.36) permits to prove the following result. T h e o r e m 4.12. ([41]). Eq. (4.32) with the initial condition (4.33) has a unique solution Q(t) for —oo < t < oo. Examination of the proof in [41] makes it clear that knowledge of the solution to (4.36) immediately yields the solution of (4.32), (4.33). But (4.36) is a standard initial-value problem in ordinary diflFerential equations; the structure of the solutions of such problems is well known. Furthermore, since the 2n2 x 2n2 matrix C has a very special structure, it is possible to recover the structure of the solutions of Eq. (4.32). Let A i , . . . , Ap, p = 2n2, be the distinct eigenvalues of the matrix C, that is, solutions of the determinantal equation (4.37)

det[AJ - C] = 0,

each Aj, j = 1,... ,p, with algebraic multiplicity m,j and geometric mul­ tiplicities nrj, E*=i nTj = m}, Ej m.j = 2n2. Then 2n2 linearly indepen­ dent solutions of (4.36) are given by

T\\x(t-i/tri j ^r(*)=«ap(Ai(*-0)g - i\\ J> ' («-0

r

(4.38)

where q = 1 , . . . , nr,, and the 2n2 linearly independent eigenvectors and generalized eigenvectors are given by

[V-CJe'^-e};1,

e°jiS = 0.

A change of notation, and a return from the vector to the matrix form, shows that 2n2 linearly independent solutions of (4.34) are given by

(\ U YZrW

\\^ (*-r/2)'"i 4r

T

234

4. REDUCIBLE FDE

where the generalized eigenmatrix pair (L'jr,Mljr) eigenvalue Xj satisfies the following equations {\I-miT-BMir T

L)TB

=

associated with the

-VrT\

T

+ M}T(XjI + A ) = -M}-1.

(4.39)

The structure of these equations is a most particular one; indeed, if they are multiplied by —1, transposed, and written in reverse order, they yield

{-XjjI-A)M^ (-X I-A)Mftrr-BLf -BL% = M M{jrlT/ ir M'B^ + LfA-XjI + AT) L°jr = Mj = 0. But this result demonstrates that if Xj is a solu­ tion of (4.37), — Xj will also be a solution; moreover, Xj and — Xj have the same geometric multiplicities and the same algebraic multiplicity. Hence, the distinct eigenvalues always appear in pairs (Xj, —Xj), and if the generalized eigenmatrix pairs corresponding to Xj are (DjT,Mjr), the generalized eigenmatrix pairs corresponding to — Xj will be

((-l) i + 1 <,(-l) i + 1 L£). These remarks imply that if the solution (4.38) corresponding to Xj is added to the solution (4.38) corresponding to —Aj multiplied by (—l)?+z, the n 2 linearly independent solutions of (4.34) given by

zi(t) w]At)

--^-m:-^ +«p(^(*-j))£(l(f

satisfy the condition

*G)-«&'(i).

4r r/2Y

-("l)'+i

235

4.3. BOUNDED SOLUTIONS

But this is precisely condition (4.35); it therefore follows that the ex­ pressions eX i=i

(g - *) !

p( A i(*-j))4r

+ (-l)t*«p(-A i ( « - £ ) ) « £

(4.40)

are n 2 linearly independent solutions of (4.32). T h e o r e m 4.13. ([41]). Eq. (4.32) has n2 linearly independent solu­ tions given by Eq. (4.40) where the generalized eigenmatrix pairs

satisfy Eq. (4.39) for one of the elements of the pair (Xj, — \j), each of which is a solution of Eq. (4.37). Eq. (4.32) has been used by Repin [225] for the construction of Liapunov functionals and also encountered in a somewhat different form by Datko [62]. 3. Bounded Solutions for Differential Equations with Reflection of the Argument The existence of a unique bounded solution of the equation x'(t) = f(t,x(t),x(-t)),

teR,

(4.41)

where / G C ( R x R x t , l ) , has been studied by Aftabizadeh, Huang, and Wiener [3]. For results concerning bounded solutions of differential and functional differential equations we refer the reader to [2, 56, 57, 167, 168, 210]. Let us first consider a special case of Eq. (4.41), i.e. a! It) + ax(t) + bx(-t) = g(t),

b^0,teR.

(4.42)

4. REDUCIBLE FDE

236

LEMMA 4.1. Ifx(t) is a solution of Eq. (4.42) on R andy(t) = x(—i), then {x(t),y(t)} satisfies the system of ordinary differential equations

dx — = -ax -by + g(t) dt

= bx + ay-

g(-t).

(S)

Let x(t) be a solution of (4.42) on R, and

PROOF.

x(-t)

= y(t).

(4.43)

Then Eq. (4.42) becomes dx = -ax - by + g(t), dt and differentiating (4.43) gives dy_ = -x'(-t), dt which, by virtue of (4.42) and (4.43), implies dy — = bx + ay-

g(-t).

4.2. Let g(t) e C(R) and bounded on R, A2 = d2-b2, Then the general solution of the system (S) on R is given by LEMMA

A .. X — a >, + « -A< + X(t) x(t) = ci—;—e - c2' b b xt + c2e-Xt+Y(t), y(t) = cie

X> 0.

(4.44)

where

x{t)=

eM e_As((A

-k i r

a)9{s)+

~

w-w*

1 kA< /-co g A s ( ( A + a ^( s ) + 2A

b

9(~s))ds

(4.45)

237

4.3. BOUNDED SOLUTIONS

and Y{t)'t\ = -

=" ~k I6'* f 2A

>e XS b

~ ( 9(s) - (* + a)g(-s))ds 2A

e~Xt f

eXs(bg(s) + (A -

a)g(-s))ds

LEMMA 4.3. Let g(t) e C(R) and bounded on R, A2 = a2 - 62, A > 0. T/ien ewen/ solution of Eq. (4.42) is of the form (4.44), provided ci =

— c2.

(4.46)

P R O O F . Condition (4.46) implies that j/(i) = x(—f), where x(t) and y(i) are the components of the general solution of the system (S). Then it is easy to see that x(t) satisfies (4.42). ■

LEMMA 4.4. Under the conditions of Lemma 4.3, Eq. (4.42) on U. has a unique bounded solution given by (4.45), and moreover, sup \x(t)\ < M ± i M sup \g(t)\,

t e R.

(4.47)

Since we are looking for bounded solutions of Eq. (4.42), then in (4.44), c-2 = 0. The uniqueness follows from the fact that if u(t) and v(t) are two bounded solutions of (4.42) on K, then from (4.42) PROOF.

(u(t) - v(t))' + a(u(t) - v(t)) + b(u(-t) - «(-<)) = 0, and by (4.45), u(t) - v(t) = 0, or u(t) = v(t), t £ i .

■

REMARK 2. The estimate (4.47) is the best possible, as the only bounded solution of the equation x'{t) + 5x(t) - 3x(-f) = 1 is x(t) = 1/2.

4. REDUCIBLE FDE

238

Theorem 4.14. Consider the functional differential equation (4.41). Assume that f(t, 0,0) is bounded on R and

(i) r..x

^ - f (*. x> y)and

Y^*' x ' ^ e x i s t f°r ^'Xj ^

(n)

df df —— > —a and — > —6, ox ay

(iii)

5/ 3 / ^ q\22 df df — H—- < - Q^ 9x 3y a| + |b|

,.

i\

6 R3

'

0 < q < 1,

where a and b are constants and A2 = a2 — b2, A > 0. Then Eq. (4.41) has a unique bounded solution on M.. PROOF. Let E = [u€ C(R, R): u is bounded], with the norm

H=supK*)|. For u G E, define an operator T: E —* E by Tu = x where x is a solution of x'(t) + ax(t) + bx(-t) = f(t, u(t), u(-t)) + au(t) + bu(-t). (4.48) The right-hand side of Eq. (4.48) is bounded, because f(t, u(t), u(-t)) + au(t) + bu(-t) = au(t) + bu(-t) + f(t,u(t),u(-t)) f(t,0,u(-t)) + f(t,0,u(-t))-f(t,0,0) + f(t,0,0), and since f(t,u(t),u(-t))

- f(t,0,u(-t))

= jf1 f2(t,ru(t),u(-t))dr

• u(t),

f(t, 0, u(-t)) - f(t, 0,0) = £ f3(t, 0, Tu{-t))dT ■ «(-<),

239

4.3. BOUNDED SOLUTIONS

then from conditions (i-iii) and the fact that f(t, 0,0) is bounded we have

\f(t,u(t),«(-<))

+ au(t) + bu(-t)\ < -J^\u(t)\ \a\ + \b\

+ \f(t,0,0)\.

This implies that Eq. (4.48) is a form of Eq. (4.42) and therefore by Lemma 4.4, Eq. (4.48) has a unique bounded solution on R. For u and v € E, let Tu = x and Tv = y, then from (4.48) we have

(*(*) - !/(*))' + «(*(<) " V(t)) + K*("*) - *(-*)) = /(*,«(*),«(-<)) - /(*, u(f), K-<)) + a(u(<) - t;(f)) + &(i/(-*) - i;(-<)) (4.49) It is easy to show that the right-hand side of (4.49) is bounded. Indeed, R.H.S. of (4.49) < j - ^ L s u p | « ( < ) - v(t)\. Thus by Lemma 4.4 \\x(t)-y(t)\\
4.5. ([57]). Let x be a differentiable map from M. into R+,

such that x'(t) < w(x(t)), with w continuous from R

+

t G R,

(4.50)

into R, and such that

w(x) < 0

for x > M > 0.

Then any bounded (on R) solution of (4.50) satisfies x(t) <M,

teR.

REMARK 3. A dual lemma can be easily obtained when t is changed to -t.

240

4. REDUCIBLE FDE

Theorem 4.15. Consider the differential system x'{t) = f(t, x(t), x(-t)),

t G K,

(4.51)

where f(t,x,y) is almost periodic on t, uniformly with respect to x and y in any bounded set of R2n. If x(t) is a bounded solution of (4.51), then x{i) is almost periodic, provided (i)

(f(t,x,y)

- f(t, z,y), x-z)>

(ii)

K,\\x -

zf,

\\f(t,x,y)-f(t,x,z)\\
where K\ and Ki > 0, and K\ — Ki > 0. PROOF. Since x'(t + r) = f(t + r, x(t + r), x(-t - r)), then -\x(t

+ T) - x(t)] = f{t + r, x(t + r),x(-t

- r))

-f(t + T,x(t),x(-t-T)) + f{t + T, X{t),x(-t - T)) - f(t + T, X(t), x(-t)) + f(t + r, x(t), x(-t)) - f{t, x{t), x{-t)). Let u(t) - x(t + T) - x(t). Then (u',u) = ([/(* + r,x(t + T),x(-t

- T)) - f(t + T,x(t),x(-t

-

T))],U)

+ ([fit + T, X(t),xi-t - T)) - f(t + T, Xit), Xi-t))],u) + ([fit + T, Xit),xi-t)) - fit, X^), Xi-t))],u). Since 1 U .,

(«' ,u) =■ 2JtM

., n

■

then using conditions (i) and (ii) we have

—Nr^ifiHI'-^IHI' 2 at

+ ([f(t + T, Xit), Xi-t)) - fit, X^), Xi-t))],u),

4.4. EQUATIONS WITH ROTATION OF THE ARGUMENT

241

or

ft\\uf

> 2{KX - K2)\\uf +

T,x(t),x(-t)) -

f(t,x(t),x(-t))\\.

w(r) = 2{KX - K2)r2 -2r sup ||/(t +

T,x(t),x(-t)) -

f(t,x(t),x(-t))\\,

-2\\u\\s*p\\f{t Then using Remark 3, with

we obtain \\x(t +

r)-x(t)\\ < IP 77- SUP | | / ( * + T, X(t), x(~t)) i\\ — j\2 tew

- f(t, X(t), X(-*))||,

which shows that x(t) is almost periodic. Proof is complete. REMARK 4. The conclusion of Theorem 4.15 holds true if we change inequality (i) to (f(t,x,y) - f(t,z,y),xz) < -K^x - z||2, and in this case apply Lemma 4.5. Bounded solutions of differential equations with reflection of the ar­ gument have been also studied in [186] and [266]. 4. Equations with Rotation of the Argument An equation that contains, along with the unknown function x(t) and its derivatives, the value x(—t) and, possibly, the derivatives of x at the point — t, is called a differential equation with reflection. An equation in which as well as the unknown function x(t) and its deriva­ tives, the values x(e\t — a\),..., x(emt — am) and the corresponding val­ ues of the derivatives appear, where e i , . . . , em are m-th roots of unity and a i , . . . , am are complex numbers, is called a differential equation with rotation. For m = 2 this last definition includes the previous one. Linear first-order equations with constant coefficients and with reflection have been examined in detail in [223]. There is also an indi­ cation (p. 169) that "the problem is much more difficult in the case of

242

4. REDUCIBLE FDE

differential equations with reflection of order greater than one." Mean­ while, general results for systems of any order with rotation appeared in [276], [277], [50], and [278]. First, consider the scalar equation ik) tt"kx akxW(t) t bkX + tf(t), zemm = 11 kxW(et) (t) == £b W(et)+m,

t=0

(4.52)

*=0

x( * ) (0) = xxjJbb,

iitfc = 0 , . . . , n - l

with complex constants a*, 6*, and e, then the method is extended to some systems with variable coefficients. Turning to (4.52) and assuming that %l> is smooth enough, we introduce the operators ■■ dk

"

"

■■ dk

(j = 0 , . . . , m - 1), and apply A\ to the given equation A0x = (B0x)(et) + ip. Since Ai[(B0x)(et)] = (5lJB0x)(e2<) + (B0r/j)(et), we obtain AxA0x = {BxBax)(eH) + Atf + (B0ip)(et), and act on this relation by A2. From A2\{B,BQx){e2t)\

= (5 2 B 1 B 0 x)(e 3 <) +

A2[(Bo1>){et)] =

(BxBo^eh),

(AiBoMet)

it follows that A2AiA0x = (B2B1B0x)(e3t)

+ A2A^

+ (AiB0tl>)(et) + ( f l ^ X e 2 * ) -

Finally, this process leads to the ordinary differential equation

(4ra"1) - BfX))* = J:{A[m-x-^B^x^){eH) 3=0

(4.53)

4.4. EQUATIONS WITH ROTATION OF THE ARGUMENT

243

where A\j)

= AjAj.i ■■■Au 0)

B\j) = BjBj-i 1}

-Bi,

i< j

4 = 4" = /,

and / is the identity operator. Thus, (4.52) is reduced to the ODE (4.53) of order mn. To agree the initial conditions for (4.53) with the original problem, it is necessary to attach to conditions (4.52) the additional relations {AU)

_ ^O+I>B0OVfc)(*)L=o = £ e M i ~ 0 * r V * > ( ' ) L (4-54) t=0

(j = 0 , . . . , m — 2; k = 0 , . . . , n — 1). System (4.54) has a unique solution for x^k\0)(n < k < mn — 1), iff a{ # (€%„)*, (0 < i < m - 1, 1 < j < m - 1).

(4.55)

These considerations enable us to formulate: T h e o r e m 4.16. ([50]). If ip £ C ^ - 1 ) " and inequalities (4.55) ore fulfilled, the solution of ordinary differential equation (4.53) with initial conditions (4.52), (4.54) satisfies problem (4.52). T h e o r e m 4.17. ([50]). If e^l,

the substitution

at y = xexp ( [jZr€)\

transforms the equation Ay = exp(at)(By)(et)

+ if)

(4.56)

with operators A and B defined by (4.52) to Px = (Qx)(et) + Vexp (-YZr~)

'

where P and Q are linear differential operators of order n with constant coefficients pk, qk and pn = an, q„ — bn. COROLLARY 4.2. Under assumptions (4.55) and em = 1, (4.56) is reducible to a linear ordinary differential equation with constant coeffi­ cients.

4. REDUCIBLE FDE

244

REMARK 5. Conditions (4.55) hold if, in particular, |a„| ^ \b„\. The­ orems 4.16 and 4.17 sharpen the corresponding results of Bruwier [30] and Valeev [264] found for the homogeneous equations (4.52) and (4.56) by operational methods under the restriction |a„| > |6„|. EXAMPLE

10. The substitution y = xexpt reduces the equation [50]

y\t) = (5y(-t) + 2y\-t))

exp 2t,

y(0) = y0

to the form x'(t) + x(t) = 7x(-t) + 2x'(-t),

x(0) = y0.

Therefore, (4.53) gives for x(t) the ODE x" - 16x = 0 with the initial conditions x(0) = yo, x'(0) = — 6j/o- The unknown solution is y(t) - ^(5exp(-3<) - exp5i). The analysis of the matrix equation X'(t) = AX(t) + exp(at)[BX(et) X(0) = I

+ CX'(et)],

(4.57)

with constant (complex) coefficients was carried out in [276]. The norm of a matrix is defined to be ||C||=maxX:| C l j |, '

(4.58)

j

and / is the identity matrix. Theorem 4.18. ([276]). If e is a root of unity (e ^ 1), ||C|| < 1, and the matrix A is commuting with B and C, then problem (4.57) is reducible to an ordinary linear system with constant coefficients. The following particular case of Eq. (4.52) has been investigated by Mazbic-Kulma [187]. T h e o r e m 4.19. ([187]). Suppose we are given a differential equation with reflection of order n with constant coefficients £ [akxW(t) + hx^(-t)}

= y(t).

(4.59)

245

4.4. EQUATIONS WITH ROTATION OF THE ARGUMENT

We suppose that (a) a " - 6 ^ 0 , (b) aj_kak — bj-tbk ^ 0 for k = 0,1,...

(c) the polynomial

J2 Cjk Xj

,n and j = k + 1 , . . . , k + n,

J2 X^jt3 has simple roots uq only, where

for 0 < j


cjk = ( - l ) " + i - * ( a » - bl)(aj-kak

=

J2 Cjk for n<j<

-

bj_kbk),

2n.

[k—j—n

Then every solution of Eq. (4.59) is of the form x(t) = (-l)"(a 2 n - hi)'1

t

[(-l)mamx(t)

-

bmi(-t)}

m=0 k 2 + tt tt C / {a -- -he-W), he-W), + Ckkuuf (ateSS* keW

?=1 *=0

where the Ck are arbitrary constants and x(t) is a solution of the equa­ tion " ( d2

\

T h e o r e m 4 . 2 0 . ([50]). Suppose that the coefficients of the equation £ ak{t)x^\i) *=o

= x(et) + t/>(t),

z<*>(0) = xk, k = 0,...,

n- 1 (4.60)

4. REDUCIBLE FDE

246

belong to C ^ " 1 ' " , em = 1, a„(0) ^ 0 and Lj=te-jkak(eH)^, j = 0,...,m-l. at *=o Then the solution of the linear differential equation

(4.61)

Lt~l)x{t) = x(t) + E,(4m"1V)(«'-1*) + ^m~1t)

(4-62)

k=\

(L(rl)

= L m _,L m _ 2

■■■Lk,0
with the initial conditions x{k\0) = xk fc

(k = (t)

Z,0x< >(i)|i=0 = e*x«(0) + V (0),

0,...,n-l), lb = 0 , . . . , ra(m - 1) - 1

satisfies problem (4.60). PROOF.

Applying the operator L\ to (4.60) and taking into account

that (L0x){et) = x(e2t) + ip(et) we get LiL 0 x(/) = x(eH) + L^(t)

+ 4>(et)

and act on this equation by Li to obtain UUUx{t)

= x(e3t) + La-M(t) + (L^)(et)

+ ip(e2t).

It is easy to verify the relations {Ljx)(ejt) = x(ej+1t) + xp{ejt),

j = 0 , . . . ,m - 1.

In particular, (L m _ 1 z)(6 m - 1 *) = a ; (0 + V(em-1*)Thus, the use of the operator Lm_i at the conclusive stage yields equa­ tion (4.62) ■

247

4.4. EQUATIONS WITH ROTATION OF THE ARGUMENT

Theorem 4.21. ([50]). The system tAX'(t) + BX(t) = X(et)

(4.63)

with constant matrices A and B is integrable in closed form if em = 1, detA^O. PROOF. For (4.63) the operators Lj defined by formula (4.61) are d Lj = tA— + B. Hence, on the basis of the previous theorem, (4.63) is reducible to the ordinary system

(

A

\

m

*A- + Bj X(t) = X(t). (4.64) This is Euler's equation with matrix coefficients. Since its order is higher than that of (4.63), we substitute the general solution of (4.64) in (4.63) and equate the coefficients of the like terms in the correspond­ ing logarithmic sums to find the additional unknown constants. ■ EXAMPLE 11. We connect with the equation [50] tx'(t) - 2x(t) = x(et),

e3 = 1

(4.65)

the relation

The substitution of its general solution x(t)

= dt3 + i3/2 (c2 sin h@- Int\ + C3 cos (^Int)

)

into (4.65) gives C
(4.66)

248

4. REDUCIBLE FDE

with constant coefficients A and B, det A ^ 0 and em = 1 is integrable in closed form and has a solution X(t) = P(t)tA~lB

(4.67)

where the matrix P(t) is a finite linear combination of exponential func­ tions. P R O O F . The transition from (4.66) to an ordinary equation is realized by means of the operators

Lj = € j

' [AJt ~ tlB)

'

J=

0,...,m-1

in consequence of which we obtain the relation

(

j

v m

Aj-rlB\ X(t) = em(m-lV2X(t). Since e m ( m - 1 )/ 2 = ± 1 , it takes the form m

n \4--(e I k

(4.68)

+ rlB) X(t) = 0

dt

where ek are the m-order roots of 1 or —1. The solutions of the equa­ tions AX'(t) = (ekI +

rlB)X(t)

are matrices Xk(t) = exv(ektA-l)tA~iB,

k=

l,...,m.

Their linear combination represents the general solution of (4.66).

■

EXAMPLE 12. In accordance with (4.68), to the equation [50] tx'(t) = 3ar(t) + tx(-t) there correspond two ordinary relations x'{t) = (Zrl + t > ( i ) , x'(t) = ( 3 r ' -

i)x(t).

(4.69)

249

4.5. BOUNDARY-VALUE PROBLEMS

We substitue into (4.69) the linear combination of their solutions x(t) = t3(Ci exp(it) + C2

exp(-it))

and find C2 = iC\. A solution of (4.69) is x(t) =Ct3(sint

+ cost).

5. Boundary-Value Problems for Differential Equations with Relfection of the Argument Differential equations with involutions can be transformed by differ­ entiation to higher order ordinary differential equations and, hence, ad­ mit of point data initial or boundary conditions. Initial-value problems for such equations have been considered in numerous papers. The study of boundary-value problems for differential equations with reflection of the argument has been initiated by Wiener and Aftabizadeh [291]. Re­ cently, these results have been generalized by Gupta [104, 105, 106]. The purpose of this part is to discuss existence and uniqueness of solutions of y" = f(x,y(x),y(-x)),

(4.70)

where / £ C([—a, a] x l x l , l ) , a > 0 , with the following types of boundary conditions y(-a) = yo,

y(a) = Vi

(4-71)

y'{a) + ky{a) = 0,

(4.72)

or y'(-a)

- hy(-a) = 0,

where h, k > 0, h + k > 0. 5.1. Preliminary Results. First we prove a sequence of lemmas for the linear case, y"{x) = a{x)ij(x) + b(x)y(-x)

+ c(x),

which are needed in order to prove our results for the general equations of the form (4.70).

4. REDUCIBLE FDE

250

Before we proceed further, we present some results without proof, which help to simplify the proofs of our results. LEMMA

4.6. ([117]). Ify(0) = y(l) = 0 and y(x) G C ^ l ] , then jloy\x)dx<^jlQ[y'{x)fdx.

LEMMA 4.7. ([21]). Iff(t,x,y) is continuous and has continuous first partials with respect to x and y on [a, b] X P where P is an open convex set, then for (t,x,y),(t,x,y) G [a, b] x P , /(*, *> y) - f(t, *, y) = A(*, r(t), «(<))(* - «) + /»(*, r(t), »(*))(» - y), w/iere /a(t, r(<), *(*)) s / 0 ! / 2 (*, TZ + (1 - r)«, ry + (1 - r)y) rfr,

/ 3 (t, r(«), 5(*)) s j f

/S(«,

ra: + (1 - r ) s , ry + (1 - r)y)
are continuous functions on [a, 6] x P UM£/I s(<), s(£) between y and y, r(t), f(t) between x and x, and 0 < r < 1. LEMMA

4.8. The homogeneous boundary-value problem u" = 0,

together with u(a) = u(b) = 0, has the Green's function G(x, t), defined by (b-x)(t-a), G(x,t) = -

b — a (b - t)(x - a),

a
and the following estimates hold true:

£\G(x,t)\dt
\2

Jha\Gx{x,t)\dt<

b— a

251

4.5. BOUNDARY-VALUE PROBLEMS

Now consider the second order linear differential equation y" = a(x)y + b(x)

(4.73)

with y'(a) - hy(a) = 0,

y'(b) + ky(b) = 0,

h, k > 0, h + k > 0,

where a(x),b(x) e C([a,b],R). Then: LEMMA 4.9. Suppose a(x) > m > 0 on [a,b], then BVP (4.73) has a unique solution satisfying swp\y(x)\<—sup|6(x)|, x G \a,b\. m 5.2. Main Results. We are now in a position to state our results. LEMMA 4.10. Ify(a) = y{b) = 0, and y(x) e C^a.fc], then jby\x)dx<^^j\y\x)fdx. 7T^

Ja

PROOF.

Jet

This follows easily from Lemma 4.6.

LEMMA 4.11. Ify(a) = y(b) = 0, and y G C 1 [a, b], then sup|y(z)| <

y/b — a

1/2

J[y'{x)fdx

a < x < b.

Ja

PROOF. Since y(a) = y(b) — 0, then one has

2y(x) = fXy'(t)dtJa

[by'(t)dt, Jx

or

2|y(x)| < f \y'(t)\ dt + f \y\t)\ dt = f \y\t)\ dt, Jo,

t/u

Jx

or svvW)\<\lab\y'{t)\dt,

a<x
Using the Cauchy-Schwarz inequality we have -|l/2

1/2 r

su P |2/(x)|<- I dt

Ja[y\t)fdt

a < x < b.

252

4. REDUCIBLE FDE

Thus y/b — a

sup|y(x)| <

f[y'(x)fdx

1/2

,

a < x < b.

Ja

LEMMA

4.12. Consider Eq. (4.73) with the boundary conditions y(a) = y(b) = 0,

(4.74)

where a(x) >-a0>-

( ^ )

.

Then any solution of (4.73) and (4.74) satisfies n{b-a)2

sup|y(x)| <

2[TT2 - a0(b -

a)2]

sup|6(a;)|,

a < x < b.

(4.75)

PROOF. On multiplying (4.73) by y(x), and integrating the result from a to 6, we find, because of (4.74), / [y'(x)]2dx — — Ja

a(x)y (x) dx — / b(x)y(x) dx,

Ja

Ja

or / [y'(x)\2dx < ao / y2(x)dx + sup \b(x)\ I Ja

Ja

\y(x)\dx.

Ja

Applying Lemma 4.10 and the Cauchy-Schwarz inequality, we get

/V(*)] 2 ^<^^/V(*)] 2 ^ Ja

7T^

Ja

(b - afl"1sup|6(x)| jy{x)fdx + IT Ja

or J

-nlb-ayi* „ , X1 SU ^ ~9 71 ^ P \Kx)u a < x
jba[y\x)fdx

/2

1/2

253

4.5. BOUNDARY-VALUE PROBLEMS

LEMMA 4.13. If a(x) and b(x) satisfy all conditions of Lemma 4.12, then problem (4.73) with the boundary conditions y(a) = Vu

yQ>) = 2/2

(4.76)

has a unique solution. PROOF. First, we show the uniqueness. Suppose u(x) and v(x) are solutions of (4.73), (4.76). Let R(x) = u(x) — v(x), then R"(x) = a(x)R(x),

R(a) = R(b) = 0.

By Lemma 4.12, R(x) = 0, which implies u(x) = v(x). So prob­ lem (4.73), (4.76) has a unique solution. To prove the existence, let u(x) and v(x) be solutions of the following initial-value problems: (i)

u"{x) — a(x)u(x) + b(x),

u(a) = ui,

(ii)

v"(x) = a(x)v(x),

v(a) = 0,

u'(a) = 0; v'(a) = 1.

We note that u(x) and v(x) exist and are unique. Moreover, v(b) ^ 0, because if v(b) — 0, then from v(a) = 0, (ii), and Lemma 4.12 we have v(x) = 0 which contradicts v'(a) = 1. Therefore, by linearity . . , . 2/2— u(6) , . y(x) = u(x) + ——-—v(x) defines the solution of the problem (4.73), (4.76).

■

Let us now consider the second order linear functional differential equation y"(x) = a(x)y(x) + b(x)y(-x)

+ c(x)

(4.77)

where a(x),b(x),c(x) G C[—a,a],a > 0. We shall show that, under certain conditions on a(x) and b(x), Eq. (4.77) with a boundary condi­ tion has a unique solution on [—a, a] and obtain an estimate for such solution. LEMMA 4.14. Let a(x) > -m, \b(x)\ < n, b{x) ^ 0, x G [—a,a], and 4a 2 (m + n) < 7r2. Then any solution of Eq. (4.77) with the boundary conditions y(a) = y{-a) = 0,

(4.78)

254

4. REDUCIBLE FDE

satisfies sup \y(x)\ < - 1 — - Y — - — r sup \c(x)\, ■Ki — iaz(m + n)

-a < x < a. (4.79)

P R O O F . On multiplying (4.77) by y(x) and integrating the result from —a to a we have

fa[y'(x)}2dx

= -fa

J—a

a(x)y\x)dx J—a

- [" b(x)y(x)y(-x)

dx - f c(x)y(x) dx,

J—a

J—a

or / [y'(x)]2dx <m j J—a

y2(x)dx

J—a

+ nfa

\y(x)\\y(-x)\dx

+ sup\c(x)\ f"

j—a

\y(x)\dx.

J—a

Now, using Lemma 4.10, the Cauchy-Schwarz inequality, and the facts that

\y(x)\\y(-x)\<\[y2(x) + y\-x)), and / y2(x) dx = f t/ 2 (-x) dx, J—a

J—a

we obtain \T [y'i^dxY L/_ a WWl

J

2

<

„

; f/

-7r2_4a2(m

-sup|c(x)|, + n)

fl"WI.

-a<x
-(4gQ)

Applying Lemma 4.11 and inequality (4.80), we get inequality (4.79).

Having Lemma 4.14, we can prove the following theorem. Theorem 4.23. In addition to the assumptions of Lemma 4.14, sup­ pose that a(x) and 6(x) are even functions on [—a, a]. Then Eq. (4.77) with the boundary conditions y(-a) = yu

y(a) = y2

(4.81)

255

4.5. BOUNDARY-VALUE PROBLEMS

has a unique solution. PROOF. Uniqueness follows from the fact that if u(x) and v(x) are two solutions of (4.77), (4.81), then R(x) = u(x) — v(x) implies R"(x) = a(x)R(x) + b{x)R(-x), R(-a) = R(a) = 0. Hence, from Lemma 4.14, R(x) = 0 and u(x) = v(x). Now we show that problem (4.77), (4.81) in fact has a solution. Let u(x) = y(x) - y(-x).

(4.82)

Then u"(x) = y"(x)-y"(-x).

(4.83)

From (4.77) and (4.83) we have u"(x) = a(x)y(x) +

b(x)y{-x) — a(—x)y(—x) — b(—x)y(x) + c(x) — c(—x).

Since a(x) and b(x) are even, then u"(x) = a(x)[y(x) - y{-x)] - b{x)[y{x) - y(-x)]+c(x)

-

c(-x),

or by (4.82), u"(x) = [a(x) - b{x)]u(x) + c(x) - c(-x),

(4.84)

and u(-a) = t/i - j/2,

"(a) ~V2~ Vv

(4.85)

Problem (4.84), (4.85) is a form of (4.73), (4.76); then by Lemma 4.13, it has a unique solution u(x). Hence y(—x) is given by y(-x)

= y(x) - u(x).

This implies that y"{x) = \a(x) + b(x))y(x) + c(x) - b{x)u{x)

(4.86)

and y(-a) = yi,

y(a) = y2.

(4.87)

256

4. REDUCIBLE FDE

Again by Lemma 4.13, problem (4.86), (4.87) has a unique solution which is the solution of (4.77), (4.81). Proof is complete. Now consider the following second order linear functional differential equation y"(x) = a(x)y(-x)

(4.88)

+ b{x)

where a(x) ^ 0 on [—a, a]. By differentiation and algebraic elimination this equation can be reduced to the fourth order differential equation iA\x)

= A(x)y"'(x) + B(x)y"(x) + C(x)y(x) + D(x),

x G [-a, a], (4.89)

where A(x) =

2a'(x) a(x)

a'(x) a'(x) a(x) a(x) C(x) ~ a(x)a(—x),

B{x)

D{x) = a(x) b(-x) +

= 2a'(x) ~ 4 a 2 ( x ) '

a(x)

By a solution of (4.88), we mean a solution that is four times differentiable. We shall show that Eq. (4.89), with the boundary conditions y(-a) = Au

y(a) = A2,

y"(-a)

= Bu

y"(a) = B2

(4.90)

has a unique solution. We use the method given in [262]. First we need the following lemmas. LEMMA 4.15. Consider the fourth order linear ODE (4.89) with the boundary conditions

y(-a) = y(a) = y"(-a)

= y"(a) = 0,

(4.91)

257

4.5. BOUNDARY-VALUE PROBLEMS

where A € Cl[-a,a], \C(x)\
B,C,D

G C[-a,a],

\A'(x) - B(x) < m and

4m7r2a2 + 16na4 < TT4,

(4.92)

then any solution of (4.89), (4.91) satisfies sup \y(x) | <

8/ra4 sup|D(*)|, 7T — 4m7r2a2 — 16na4 4

(4.93)

— a < x < a. PROOF.

Let

(4.94) y" = z. On multiplying (4.94) by y(x) and integrating the result from —a to a, we have / b/{x)]2dx = - I

yzdx,

or

jaJy'{x)fdx

< [f_ay\x)dx]

11/2

[ £ / ( * ) dx]

1/2

Applying Lemma 4.10, we obtain

[/>v)P4/2<^[/>v)H'/2

(4.95)

Also, from (4.94) and (4.89), z" = A(x)z' + B(x)z + C{x)y + D(x),

(4.96)

z(—a) = z{a) = 0. In a similar manner, we get

! dx

[/>'M1

1/2

<

7r3(2a)3/2 sup|£>(*)|, 7T4 — 4ma27r2 — 16na4 — a < x < a.

Prom Lemma 4.11, (4.95), and (4.97) inequality (4.93) follows.

(4.97)

258

4. REDUCIBLE FDE

Theorem 4.24. Suppose all assumptions of Lemma 4.15 hold true. Then problem (4.89), (4.90) has a unique solution. P R O O F . Assume that there exist two distinct functions u(x) and v(x) satisfying (4.89) and (4.90). Then tp(x) = u(x) — v(x) satisfies

^ (4) (x) = A(x)iP'"(x) + B(xW(x)

+ C(x)i/>(x),

(4.98)

xjj(-a) = V(a) = V>"(-a) = tf'(a) = 0. Now, from Lemma 4.15 and (4.98) it follows that sup |^(a:)| < 0, which proves ip{x) = 0 and u(x) = v(x) on [—a, a]. This shows that prob­ lem (4.89), (4.90) has at most one solution. In order to prove that (4.89), (4.90) indeed has a solution, we define functions yi(x), i — 1,2,3,4, as solutions of the respective initial-value problems: (i) yj4) = A(x)y'C + B(x)y'{ + C{x)yx + D(x), = Ah y\{-a) = y'[{-a) = y'{'{-a) = 0; yi(-a)

(ii) yi4) = A(x)fl + B{x)y'i + C(x)y2, y'2{-a) = 1, y 2 (-a) = y%(-a) = y2"(-a) = 0; (iii) 2/i4) = A{x)y'i + B{x)y'i + C(x)yz, y'>(-a) = Bu y3(-a) = y'3(-a) = y3"(-a) = 0; (iv) y f = A{z)fl + B{x)y'{ + C(x)y4, y'4"(-a) = 1, y4(-a) = y'4(-a) = jtf(-a) = 0. Prom the continuity of A(x), B(x), C(x), and D(x) we are assured that unique solutions of these initial-value problems exist on [—a,a]. Furthermore, the function z{x) = 2/i (x) + sy2(x) + y3(x) + ty4(x), s, t being scalars, satisfies the initial-value problem zW = A(x)z"' + B(x)z" + C(x)z + D(x), z(-a) = Ai,

z'(-a) = s,

z"(-a) = Bu

z'"(-a) = t.

4.5. BOUNDARY-VALUE PROBLEMS

259

The function z(x) will be a solution of (4.89), (4.90) provided that s,t satisfy sy2(a) + ty4(a) = A2 - yi(a) - 2/3(0), n£(<0 + *2/4'(o) = B2-

y'[{a) - y»(a).

If A = 2/2(0)2/4(0) - 2/2(0)2/4(0) # 0, a unique solution of the preceding linear system can be found, and the corresponding function z(x) then is the unique solution of (4.89), (4.90). However, if A = 0, then 2/2(0) -TT\ 2/2(0)

=

2/4(o) ~T7-\ =P (constant). 2/4(o)

We can assume that p ^ 0, because if /> = 0, then 2/2(0) = 0 and by means of Taylor's formula it can be shown that the solution of y24)^A(x)y2"

+ B(x)y^ + C(x)y2,

2/2(0) = y'H-a) = y2"(-a) = y2{-a) = 0 has the property y'2(—a) = 0, contradicting the original assumption t/2(—a) = 1. Similarly, p cannot be unbounded. Thus it follows that 2/2(0) =py2(a),

p
(4.99)

Now using (ii), and the Taylor formula, we obtain 2/2(0) = 2a + 2-a\A{a)y'2"{a) + B(a)y'2\a) + C(a)y2(a)}, —a < a < a, 2

y'2'(a) = 2a [A(P)y2"(f3) + B(/3)jfi(P) + C(/3)W(/?)], -a < j3
(4.100)

On combining (4.99) and (4.100) we get

ap[A(P)y'2"(/3) + B(f3)y'M + C(JJ)M - ±a*{A(a)y2"(a) + B(a)y2'(a) + C(a)y2(a)] = 1,

260

4. REDUCIBLE FDE

for all Aix), Bix), C(x) G C[—a, a]. In an attempt to determine j/ 2 (a),

M, !#(«),tf(P),&(<*), l#(P), we choose Aix) Aix) Aix) Aix) Aix) Aix)

= 1, = 1, =-1, =-1, = 1, = 1,

Bix) Bix) Bix) Bix) Bix) Bix)

C(x) C(x) C(x) C{x) C(x) C(x)

= 1, = 1, = -1, = -1, =-1, = -1,

= 1, = -1, = 1, =-1, = 1, =-1,

giving the system 1 <*PbW) +1^09) + IftW)] - ^ 3 [ ^ " ( « ) + l £ ( « ) + W(«) <*[!/£"(/?) + !#(/?) - Sfe(/?)] - | a 3 [ # ( a ) + y2'(a) - ifc(a) a p h i W ) " !#(/*) + 2/2(/?)] " ^ 3 [ - y 2 " ( a ) " l£(«) + Ifc(«) a p h i W ) - 2/2(/*) - &({})) - ^ 3 [ - y 2 » - y2'(a) ap[y2"(/?) - y'HP) + w(/J)] - ^

3

b

2

ap[2/2"(/?) - t/2'(£) " WG9)] " \a*\tf(a)

w(a)

» - j/2'(<*) + » ( « ) - tf(«) -

w(a)

But this latter system in the unknowns J/2(Q;), 2/ 2 (/?),... is inconsist ent. We thus conclude that A cannot vanish and the proof of the theorem is completed. Now, it is easy to show that problem (4.88) with j/(—a) = j/(a) = 0 has a unique solution. Theorem 4.25. Suppose that aix) G C2[—a,a], bix) G C[—a,a], b(-a) = 6(a) = 0, [a'(x)/a(x)] 2 < m, |a(x)a(-x)| < n. / / 4m7r2a2 + 16na4 < 7r4,

261

4.5. BOUNDARY-VALUE PROBLEMS

then Eq. (4.88) with y(-a) = y(a) = 0,

(4.101)

has a unique solution. PROOF. Since Eq. (4.88) can be reduced to (4.89), then by Theo­ rem 4.24 problem (4.88), (4.101) has a unique solution. ■ Now we consider the general equation (4.70) and prove the following theorems. Theorem 4.26. Suppose f is bounded on [—a, a] x R x R. problem (4.70), (4.71) has a solution.

Then

PROOF. Problem (4.70), (4.71) is equivalent to

y{x) = o(yi + vo) + ^-(yi - z/o)* + f G(X, *)/(*, y(t), y(-t)) dt, 2 2a J-a (41Q2) where G(x,t) is given by Lemma 4.8. Let M be the bound of | / | on [-a, a] x R x R. Define a mapping T: E —> E by Ty{x) = ikz/i + j/o) + ^ ( y i - yo)x + f" G(x, t)f(t, y(t), y(-t)) dt, 2 2a a (4.103) where E = C([—a,a],R) is a Banach space with the norm ||2/||E = max|j/(a;)|,

-a < x < a.

Then from (4.103) and the estimates on G(x,t) and Gx(x,t), it follows that \Ty(x)\ < - | W + j/o| + -|z/i - yo\ + ^a2M,

(4.104)

and \T'y(x)\<^\yi-yo\

+ aM.

Hence, T maps the closed, bounded, and convex set

B = iy e E: \y(x)\
(4.105)

262

4. REDUCIBLE FDE

into itself. Moreover, since T'y{x) verifies (4.105), T is completely continuous on C([—a,a],R) by Ascoli's theorem. The Schauder's fixed point theorem then yields a fixed point of T, which is a solution of (4.70), (4.71), thus completing the proof of the theorem. ■ Theorem 4.27. Suppose f is continuous and satisfies a Lipschitz condition \f(x,y,z)

- f(x,y,z)\

< Lx\y -y\ + L2\z - z\,

(4.106)

for (x, y, z), (x, y, z) € [—a, a] x R x R, where L\ and L2 > 0. Then the boundary-value problem (4.70), (4.71) has a unique solution, provided a 2 (Lj + U) < 2.

(4.107)

P R O O F . Let B be the Banach space of functions y 6 C([—a,a],R) with the norm

\\y\\B = max|y(x)|,

-a < x < a.

Define the operator T: B -»• B by (4.103). Then for j/i(x), y2(x) G B, we have \Tyi(x) -

Ty2(x)\

< £ \G(x,t)\ \f(t,yi(t)M-t))

-

f(tMt)M-t))\dt.

Using the Lipschitz condition (4.106) we obtain \Tyi{x)-Ty2{x)\ < f_a \G(x,t)\[Li\yi(t)

- y2(t)\ + L2\yi(-t)

- y2(-t)\]

dt,

or \Tyi(x) - Ty2(x)\ <

a(Ll

+L2) 2

m a x | y i ( x ) - y2(x)\,

-a < x < a,

or

||ryi(x) - Ty2(x)\\B < ^l±M\\yi{x)

_ w(x) || B .

Also, \\T'yi(x) - T'y2(x)\\B < a(L, + L2)||j/1(x) - y 2 (x)|| B .

263

4.5. BOUNDARY-VALUE PROBLEMS

All of these considerations and inequality (4.107) show that T is a contraction mapping and thus has a unique fixed point which is the solution of (4.70), (4.71). The proof is complete. Theorem 4.28. Assume that there exist positive numbers m and r such that (i) sup|/(x,0,y)| < r(7r2 - Ama2)/(2na2), forx€ [-a, a], \y\ < r, (ii) f(x,y,z) has a continuous partial derivative f2{xiViz) with re­ spect to y on [-a, a] x R x R and -7T2

f2{x,y,z)

> - m > —-j, forx G [-a,a], y G R, \z\ < r.

Then Eq. (4.70) with the boundary conditions y(-a) = y(a) = 0,

(4.108)

has a solution. PROOF. Let B r = {y G C [ - a , a ] : \y\ < r}. For x G [-a,a] and y G B r , define a mapping T: C [ - a , a] —> C[-a,o] by (Ty)(x) = v(x), where v"(x) = f(x,v(x),y(-x)),

(4.109)

v(-a) = v(a) = 0.

(4.110)

and

Eq. (4.109) is equivalent to v"{x) = f(x, v(x), y(-x))

- f(x, 0,y(-x)) + /(x, 0, t/(-x)).

Then from Lemma 4.7, it follows that v"{x) = ( f1 f2(x, Tv(x),y(-x)) V°

dr) v(x) + f(x,0, j/(-x)). ' (4.111)

Applying Lemma 4.12 and condition (ii) we have sup|v(x)| < - j — - — ^ s u p / ( x , 0 , y ( - x ) ) ,

-a<x


264

4. REDUCIBLE FDE

or from (i), sup|t;(x)| < r,

—a<x
(4.112)

Hence, T maps the closed, bounded, and convex set B r into itself. Moreover, from (4.109), (4.110), and Lemma 4.8, v(x) = f

G(x,t)f(t,v{t),y{-t))dt,

J—a

or v'(x) = f

Gx(x,t)f(t,v(t),y(-t))dt.

J—a

Since \f(x,v(x),y(—x)\ then

< k, for x G [—a, a], \v(x)\ < r, \y(—x)\ < r, \v'(x)\ < ak.

All of these considerations show that T is completely continuous by Ascoli's theorem. Shauder's fixed point theorem then yields a fixed point of T, which is a solution of (4.70), (4.108). ■ T h e o r e m 4.29. Assume that there exist positive numbers m and r such that (i) sup \f(x,0,y)\< mr, forx£[-a,a], \y\ < r, (ii) f(x,y,z) has a continuous partial derivative f2(x,y,z) with re­ spect to y on [—a, a] x R x R and f2(x,y,z)

> m > 0,

for x e [-a, a], y £ R, \z\ < r.

Then the boundary-value problem (4.70), (4.72) has a solution. P R O O F . Let B r = {y e C[-a,a\: \y\ < r}. For y £ B r , define a mapping T: C[-a,a] —> C[—a, a] by (Ty)(x) = v(x) where

v" = f(x,v(x),y(-x)), v'(-a) - hv(-a) = 0, v'(a) + kv(a) = 0. Eq. (4.113) is equivalent to v" = U f2[x,Tv(x),y(-x)]dTjv(x)

+

f(x,0,y(-x)).

(4.113) (4.114)

4.6. PARTIAL DIFFERENTIAL EQUATIONS WITH INVOLUTIONS

265

Using Lemma 4.12 and conditions (i), (ii) we obtain sup |f (x)| < r,

—a<x
This shows that T maps B r into itself. Also from (4.113) we have v'(x) = v'(-a)+[X

f[t,v(t),y(-t)]dt. J—a

Since i/(—a) = hv(—a), then \v'(—a)\ < h\v(—a)\ < hr. Moreover, \f(x,y,x)\ < k for x G [-a,a], \y\ < r, \z\ < r. Then \v'(x)\ < hr + 2ak. Thus, T is completely continuous by Ascoli's theorem. Schauder's fixed point theorem then yields a fixed point of T, which is a solution of (4.70), (4.72). ■ 6. Partial Differential Equations with Involutions Some problems of mathematical physics lead to the study of ini­ tial and boundary-value problems for equations in partial derivatives with deviating arguments. Since research in this direction is developed poorly, the investigation of equations with involutions is of certain in­ terest. They can be reduced to equations without argument deviations and, on the other hand, their study discovers essential differences that may appear between the behavior of solutions to functional differential equations and the corresponding equations without argument devia­ tions. The solution of the mixed problem with homogeneous boundary con­ ditions and initial values at the fixed point to of the involution f(t) for the equations ut(t, x) = auxx(t, x) + buxx(f(t), x),

(4.115)

utt(t, x) = a2uxx(t, x) + b2uxx(f(t), x)

(4.116)

and can be found by the separation of variables method. Thus, for (4.115) the functions Tn(t) in the expansion oo

u(t,x)=

ZTn(t)Xn(x) 71=1

(4.117)

4. REDUCIBLE FDE

266

are determined from the relation Tn{t) = -XnaTn(t)

- X„bTn(f(t)),

Tn(i0) = C*„.

(4.118)

Its investigation is carried out by means of Theorem 4.6, according to which the solution of the equation - Xl(a2 - b2)f'(t)Tn(t)

TZ(t) = -A„a(l + f'(t))Z(t)

(4.119)

with the initial conditions Tn(*0) = C„,

Z(t0) = -Xn(a + b)Cn

satisfies Eq. (4.118). The following theorems illustrate striking dis­ similarities between equations of the form (4.115) and (4.116) and the corresponding equations without argument deviations. T h e o r e m 4.30. The solution of the problem ut(t, x) = auxx(t, x) + buxx(c - t, x), u(t, 0) = u(t, I) = 0, u(c/2, x) = <j>{x)

(4.120)

is unbounded as t —> ±oo if a — b ^ 0. If \b\ < \a\, b ^ 0, expan­ sion (4.117) diverges for all t ^ c/2. PROOF. By separating the variables, we obtain

rT fy=C c (c-t)), nw = —p-(arB(t) ++ bT Kr«(c-t)), "(f)= " 7r 2 n 2

Z(t) = -^(aTn(t)

n

n

n

(4.121)

The initial conditions for Eqs. (4.120) and (4.121) are posed at the fixed point of the involution f(t) = c — t. In this case, Eq. (4.119) takes the form

_i^V _~ aa )T O ) ==~^-(b rw ! ) (t), m 2

2

n

r„(f)=c», r.(f)--=*<.+.,*. The completion of the proof is a result of simple computations. De­ pending on the relations between the coefficients a and b, the following

4.6. PARTIAL DIFFERENTIAL EQUATIONS WITH INVOLUTIONS

267

possibilities may occur: 2

(1)

= cnl cos n'nWb

T»(t) ■■

- aHt - c/2) — -—-

p *>n>V»-a*(t-c/2)\

i+b 9

9

v

= c„( l - ^ - ( a + 6 ) ( t - 0 j ,

(2)

Tn(t) ■■

(3)

Tn(t) -

/

"2

C n

Ja2

1

'I);

(|a| = |6|);

/^n2

A2\

/

r\"

. I -1-7)-* (?*»=* (<-§), )

i/fl2

fi2\

/

TT2TJ2

/

♦('

i~\\ 1

(|6|<|a|). ■

T h e o r e m 4.31. 77ie solution of the equation utt(t, x) = a?uxx(t, x) + b2uxx(-t, x)

(4.122)

satisfying the boundary and initial conditions u(t, 0) = u(t, I) = 0, u(0, x) = (j){x), ut(0,x) = tp(x), is unbounded as t —> ±oo if a2 = 62. In the case a2 < 62 expan­ sion (4.117) diverges for all t ^ 0, if ip(x) ^ 0. PROOF. Separation of the variables gives for the functions T„(t) the relation „„. , 7r 2 a 2 n 2 m , , 7r262n2 . . n(t) = p-Tn{t) ~ —p-Tni-t), r„(0) = An, Tn(0) = Bn,

T

, (4.123)

268

4. REDUCIBLE FDE

by successive differentiation of which we obtain T(3)(i)

7r2a2n2

=

T^(t)

7r262n2__,,

.,

.

——r n (t) + —p-rn(-t), 7r2a2n2

=

7r2&2n2

,, .

,.

.

From Eq. (4.123) we find

7rzaln2 _ , p Tn(

Ki-t) =

. t)- "

p

Tn(t)

and also

ir2a2ri i a2 P -Tn(--t) =~~ 6 2

OH

2 4 :! 7ra n 2 2 -Tn(t). 6Z

Thus, Eq. (4.123) is reduced to the fourth-order differential equation m ,ii,

T?\t)

,

27r2a2n2 .. . 7r4(a4 — 64)n4 . . + -ir-T';(t) + K /4 > Tn(t) = 0

with the initial conditions Tn(0) = An,

K(0) = Bn,

T^O) = 7rV2 (a 2 -6 2;)n 2

^ 3, ), f t , ^ (0) =

/2

2 2 n?r[ a(a

+ b2)n2 ± 0)n An,

Bn.

It remains to consider various cases that may arise depending on the characteristic roots. (1). For b2 < a2, im\/a2

+ b2

Tn(t) = An cos (2). If a2 = b2, then

I

I . irny/a2 — b2 1+ -, Bn sin 1. 7rnva — o I

Tn(t) = An cos — - ^ — t + Bnt. (3). Finally, the inequality a2 < b2 leads to the result m u\ A im\/a2 r „ ( 0 = A n cos

+ b2 «

. 7rn-/6 2 - a 2 , 1+ £ n sinh 1 7rnvo — a * I

4.6. PARTIAL DIFFERENTIAL EQUATIONS WITH INVOLUTIONS

269

Of some interest is the equation ut(t, x) = Auxt (^~

, x\

(4.124)

with the hyperbolic involution jt — a having two fixed points *o =

a+A

a2 + /?7 = A2 > 0, 7 ^ 0

,

h =

a-A

. 7 7 The search for a solution in the region (a/7,00) X [0,1] satisfying the conditions u(i, 0) = u(t, I) = 0,

u(t0, x) = <(>(x)

(4.125)

leads to the relation

K(t) = ™~^-.(££). /2

J

"^7i_aJ'

<«*>

which is a generalization of Eq. (4.1). Differentiation changes (4.126) to the form (7< - «) 2 7:'(0 +

Tn{t) = 0.

The substitution \^t — a\ = exps allows us to integrate it in closed form. Omitting the calculations, we formulate a qualitative result. Theorem 4.32. The solution of problem (4.124), (4.125) is un­ bounded as t —► oo. For

•>H&)

1/2

the functions Tn (t) are oscillatory.

Biological models often lead to systems of delay or functional dif­ ferential equations (FDE) and to questions concerning the stability of equilibrium solutions of such equations. The monographs [60] and [188]

270

4. REDUCIBLE FDE

discuss a number of examples of such models which describe phenom­ ena from population dynamics, ecology, and physiology. The work [188] is mainly devoted to the analysis of models leading to reducible FDE. A necessary and sufficient condition for the reducibility of a FDE to a system of ordinary differential equations is given by Fargue [77]. His method is frequently used to study FDE arising in biological models. We omit these topics and refer the reader to the paper of Busenberg and Travis [31].

CHAPTER 5

Analytic and Distributional Solutions of Functional Differential Equations Recently there has been considerable interest in problems concerning the existence of solutions to differential and functional differential equa­ tions (FDE) in various spaces of generalized functions. Many important areas in theoretical and mathematical physics, theory of partial differ­ ential equations, quantum electrodynamics, operational calculus, and functional analysis use the methods of distribution theory. Research in this direction discovers new aspects and properties in the theory of FDE. It is well known that normal linear homogeneous systems of or­ dinary differential equations (ODE) with infinitely smooth coefficients have no generalized-function solutions other than the classical solu­ tions. In contrast to this case, for equations with singularities in the coefficients there may appear new solutions in generalized functions as well as some classical solutions may disappear [88]. Distributional solutions to linear homogeneous FDE may be origi­ nated either by singularities of their coefficients or by deviations of the argument. In [281] it has been discovered that the system x'{t) = Ax(t) + tBx(\t),

-1 < A < 1

has a solution in the class of distributions—an impossible phenomenon for homogeneous ODE without singularities. In fact, there are some other striking dissimilarities between the behavior of ODE and FDE which deserve further investigation. Thus, in [302] it was shown that a first-order algebraic ODE has no entire transcendental solutions of order less than | , whereas even linear first-order FDE may possess such 271

272

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

solutions of zero order [285, 286, 50]. Furthermore, it is well known that the solution of the initial-value problem for a normal linear ODE with entire coefficients is an entire function. Let in the linear FDE w'(z) = a(z)w(\(z))

+ b(z),

w(0) = w0

the functions a(z), b(z), and \(z) be regular in the disk \z\ < 1, and A(0) = 0, |A(z)| < 1 for \z\ < 1. Then there is a unique solution of the problem regular in \z\ < 1. In general, this solution cannot be extended beyond the circle \z\ = 1, even if a(z), b(z), and X(z) are entire functions. Thus the solution of the equation w'(z) = a(z)w(z2), where a(z) is an entire function with positive coefficients, has the circle \z\ = 1 as the natural boundary [227, 228]. The results of the previous chapter provide information on the struc­ ture of FDE with rather special transformations of the argument. How­ ever, they can be used as a starting point for the exploration of analytic solutions to FDE with linear argument transformations. In particular, the techniques that will be developed for the study of distributional so­ lutions can be applied to investigate entire solutions of equations with variable coefficients and deviations proportional to the argument. In­ tegral transformations establish close connections between entire and generalized functions [89]. Therefore, a unified approach can be devel­ oped for the study of both distributional and entire solutions to broad classes of linear ODE and FDE. 1. Holomorphic Solutions of Nonlinear Neutral Equations Wiener, Debnath, and Shah [300] conducted a study of local existence and uniqueness theorems for analytic solutions of nonlinear FDE of neutral and advanced types. We consider the equation w'(z) = F(z,w(z)MHz))MHz)))-

(5-1)

If X(z) has a fixed point ZQ, then an initial-value problem for (5.1) can be posed at ZQ in the same manner as for ordinary differential equations.

273

5.1. HOLOMORPHIC SOLUTIONS

We may always assume ZQ = 0, that is, A(0) = 0 and prescribe for (5.1) an initial value w(0) = w0. Putting z = 0 in (5.1) gives the equation u/'(0) = / ( 0 , w0,w0,w'(0))

(5.2)

for the unknown value u/(0). Eq. (5.1) is of neutral type because it contains the derivative w' = dw/dz at different arguments, z and X(z). T h e o r e m 5 . 1 . Assume for (5.1) the following (i) Eq. (5.2) has a solution w'(0) = w'0. (ii) The function f(z,w,wi,ui2) is holomorphic R : 1*1 < r0,

\w - w0\ < M 0 ,

where Mi >

hypotheses. in the region

\w{ - w0\ < M0,

M0 (- Iwnl, +

and

^0

df dw-)

\w2 - w'0\ < Mh

< 1 in R .

(iii) The function X(z) is holomorphic in the disk \z\ < ro and satisfies in it the inequality |A(z)| < \z\. Then in some disk \z\ < r there exists a unique holomorphic of Eq. (5.1) with the initial values VJQ,W'0.

solution

PROOF. We replace (5.1) by the integral equation w(z) = w0 + JQ f(s, w(s), w(X(s)), w'(X(s)))

ds

and introduce the operator Tg(z) = w0 + j * f(s, g(s), g(X(s)), g1 (X(s))) ds

(5.3)

on the space G of all functions g(z) holomorphic in the disk \z\ < r and satisfying the conditions <7(0) = w0,

g'(0) = w'0,

\g{z) - w0\ <

^M0.

The value of r is to be determined later. Clearly, the first restriction on r is r < r 0 . Since g'(z) — w'Q is the derivative of (g(z) — w0) — w'0z, we have <Mo

\g'(z) -w' 0\<^ W(z) --w'o\

ro

+ KI K| +

274

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

in \z\ < r. Taking in hypothesis (ii)

Mi>— + KI, we conclude that the function f(z,g(z),g(\(z)),g'(\(z))) phic (and bounded) in this disk. Let M(r) = max\f(z,g(z),g(\(z)),g'(\(z)))\,

is holomor\z\ < r.

Then from (5.3), \Tg(z) - w0\ < M(r)\z\ < rM(r). We choose r such that rM(r) < rMo/ro, that is, M(r) <

*

which is always possible to do. Now, we evaluate d

dTg(z)

- W'0 =

\f(z,g(z),g(\(z)),g'(X(z)))-w'0\

<\f(z,g(z),g(\(z)),g'(\(z)))\

+ \w'0\

<M(r) + K,|<— + K|<M!,

\z\
In R the function / satisfies a Lipschitz condition \f(z,w,wuw2)

- f(z,y,2/1,y2)|

< U\w - y\ + Li\wi -yi\

+ L2\w2 - y2\,

with L% < 1. We introduce a metric in the space G by the formula d(gi,92) = (L0 + Li)max\gi(z)

- g2(z)\ + L2max.\g[(z) - g'2(z)\,

\z\ < r.

275

5.2. NONLINEAR ADVANCED EQUATIONS

Then from (5.3) \Tgi(z) - Tg2(z)\ < L 0 r max ^ ( z ) - g2(z)\ + Lxr max \gi(X(z)) - g2(X(z))\ + L2r max \g[(X(z)) - g'2(X(z))\ < (L0 + Li)r max \gx(z) - g2(z)\ + L2rmax\g[(z) - g'2(z)\ and max\T9l(z)

-Tg2(z)\

< rd(gug2).

(5.4)

Furthermore, TT9l(z)

-

-Tg2(z) =

\f(z,9l(z),gi(X(z)),g[(X(z)))

- f(z,g2(z),g2(X(z)),g2(X(z)))\

<

d(9l,g2).

and max

£■»«-&«,)

<
(5.5)

Multiplying (5.4) by (Lo + L\) and (5.5) by L 2 and adding yields d{TguTg2)

< (r(L0 + L{] +

L2)d(gug2).

Finally, the condition r

\-L2 < 7

7~

shows that T is a contraction of the space G into itself, which proves the theorem. 2. Holomorphic Solutions of Nonlinear Advanced Equations The equation [247] w'(x) = aow(Xz) + a\zw\\z)

+ a2z2w"(Xz)

(5.6)

is of considerable interest. If the coefficients are real and 0 < A < 1, then for z > 0 it is of advanced type. Furthermore, it appears that ad­ vanced equations, in general, lose their margin of smoothness, and the

276

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

method of successive integration shows that after several steps to the right from the initial interval the solution may not even exist. Nonethe­ less, (5.6) admits analytical solutions. Namely, if 0 < |A| < 1, then the initial-value problem w(0) = WQ for the complex differential equa­ tion (5.6) with complex constants a,- and A has a unique holomorphic solution, and it is an entire function of zero order. In fact, substituting the series w z

( ) = Z) Wnz" n=0

in (5.6) yields Z(n + l)wn+lz"= n=0

ZaoXnwnzn n=0

oo

oo

n=0

n=0

+ £ (n + l)a>A"u;n+1z"+1 + £ (n + 2)(n +

l)a2Xnwn+2zn+2

and (n + l)wn+i = (a0A" + naiA" -1 + n(n - l)a2Xn~2)wn,

n>0.

Prom here, it follows that for large n,
(5.7)

f(z,w,W\,Wi,W2,...,ivn)

\w - w0\ < M 0 , \wx - w0\ < M0, (i = l , . . . , n )

\w{\ < M{

and X(z) is holomorphic in the disk \z\ < r<> and satisfies in it the inequality |A(z)| < \z\.

277

5.2. NONLINEAR ADVANCED EQUATIONS

Then in some disk \z\ < r there exists a unique holomorphic solution of the problem (5.7). PROOF. Replace (5.7) by the integral equation w(z) = w0+ r / ( * , w(s),w(\(s)),

sw'(\(s)),...,

snw(n\\(s)))

ds

and introduce the operator Th(z) = w0 + ff(s,h(s))ds

(5.8)

where (g(s),g(\(s)),sg'(\(S)),...,sngW(\(s))),

h(s) =

on the space H of all functions g(s) holomorphic in the disk \z\ < r and satisfying the conditions g(0) = w0,

\g(z) - w0\ < m,

M{

m < -~,

(» = 0 , . . . , n )

The first restriction on r is r < ro. Since g^\z) order i of the function g(z) — WQ, we have

I^WWI
is the derivative of

(i = l,...,n)

for \z\ < r. Therefore, the function f(z,h(z)) disk. Let M = m&x\f(z,w,w\,wi,...

is holomorphic in this

,wn)\

in R.

Then from (5.8), \Th(z)-w0\

<Mr,

and we choose r such that Mr < Mo, that is, r < Mo/M. Furthermore, d' r»w dz 7

^r/('.»(*»

—

yJ —1

and 2 1 '—TVz) < (i-l)IAfr.

dz

'

—

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

278

The requirement (i — 1)! Mr < M,- gives r <

Mt

(i-iy.M

for 1 < i < n. In R the function / satisfies a Lipschitz condition \f(z,w,w\,wu...,wn)

-

f(z,y,yx,yu...,y„)\ n

< L0\w -y\ + Lx\wx - yx\ + £ Li\u>i - Vi\«=1

We introduce a metric in the space H by the formula d(hi,h2) = (L0 + Lx)ma.x\gi(z) - g2(z)\ + ± L8-max |z*'(#(z) -
\z\ < r.

i=l

Then from (5.8), max\Thi(z) - Th2{z)\ < rd(huh2).

(5.9)

Furthermore, max

^A- (r/ll(z) _ r , l2(z)) d*-1

= max

zlj-z-xU{zMz))-tt*M{z)))

, i ( i - l ) i J ma.x\f(z,hi(z))Q nl'-l

< 7

(5.10)

f(z,h2(z))\dz

<(i-l)\r2d(huh2), (t = l , . . . , n ) . Multiplying (5.9) by (L0 + Lx) and (5.10) by L,- and adding all inequal­ ities yields d(ThuTh2)<

r(Lo + LA) + r 2 £ ( ; - l ) ! L , - d(hhh2).

5.3. ANALYTIC AND ENTIRE SOLUTIONS OF LINEAR SYSTEMS

279

Finally, if r(L 0 + LA) + r - 2 f : ( z - l ) ! L 1 < l ,

(5.11)

1=1

then T is a contraction of the space H into itself, which proves the existence and uniqueness for (5.7). In particular, to satisfy (5.11), we can require

r
r< \L0 + LX + £ ( i - l)!L,-j .

REMARK 1. Theorem 5.2 holds true if on the right of (5.7) the terms zj w^(X(z)) are changed to zkj w^(X(z)), with kj > j . 3. Analytic and Entire Solutions of Linear Systems The famous Izumi theorem [130] states that if in the equation u/W(z) + a 1 (^)^("- 1 )(A 1 (z)) + • ■ • + an(z)w(Xn(z))

= b(z)

a,i(z), b(z), Xj(z) are regular in the disk \z\ < 1 and A,(0) = 0,

\Xi(z)\ < 1,

for \z\ < 1 ,

there exists a unique solution with the given w^'\0) regular in the closed disk \z\ < 1. Cooke and Wiener [50] have generalized this result for linear neutral equations with infinitely many arguments. The norm of a matrix A — (a,j) is denned to be ||A|| = m a x £ | a y | . i Theorem 5.3. Suppose the system oo

X\t)

= E Ai(t)X(Xi(t)) .=o

oo

+ E Bi(t)X'(n(t)), i=o

X(0) = XQ (5. 12 )

in which A{, B{, and X are r x r-matrices satisfies the following hy­ potheses. (i) The coefficients and the deviations are regular in the disk \t\ < 1. (ii) A,-(0) = 0, m(0) = 0, \Xi(t)\
280

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

(iii) The series E £ 0 £ j = o P u l l and EfeoE^o \\Bij\\> where AH and B^ are the Taylor coefficients of Ai(t) and B{(t), respectively, converge and £ g 0 ||#i(0)|| < 1. Then the problem has a unique solution regular in \t\ < 1. P R O O F . Let OO

OO

x(t) = E xntn,

00

(Xi(t))n = E xinktk,

n=0

(/*,■(<))" = E /*.***•

t=0

t=0

Then oo

oo B X(Xi(t)) *(M*)) ==: E * *»(M0) «(*(*))" = 71=0 7i=0

oo

+ * '(«(*)) = *'(M»W) =: E E( (« n+

oo OO

oo oo

k

E*n E ^ ZEJ "ink A*.*t**,i n=0 1

n=0

l)Xn+l (fu(t))Bn l)^ n+ i(w(0)

nn=0 =0

k=0 k=0 oo oo

oo oo

E /****■ Vinktk. l ) *n+1 n + l£ = = E (n■++l)X

E(«

n=0 n=0

k=0

fc=0

Since A;(0) = 0, /x,-(0) = 0, we note that A;„t = 0 and //,-„* = 0 for k < n. Changing the order of summation in the latter equations gives OO

OO

X(Xi{t)) = E ^ E *iknXk, n=0

t=0

OO

OO

*'(M*)) = E *" E ( * + i ) ^ n ^ + i . n=0

i=0

However, now A,*tn = 0 and /z,tn = 0 for k > n. Therefore, oo

n

X(Xi(t)) = E tn E AanXt> n=0

fc=0

%W)=E'nE(Hi)ftbXHi n=0

*=0

and oo

n

}

Mt)x(Xi(t)) = E tn E A»-; E Aifcix*,

B^X'i^t))

n=0

j=0

fc=0

oo

n

j

= E *" E B*,»-i £ ( * + l)A*a;**+i. n=0

j'=0

t=0

5.3. ANALYTIC AND ENTIRE SOLUTIONS OF LINEAR SYSTEMS

281

Hence, the coefficients Xn of the unknown solution satisfy the equations oo oo

n n

Jj

(n + l)X l)X n+l = E E A*,»-j E A t t i X t n+l ii'=0 = 0 jj=0 =0

fc=0 oo oo n n

jj

i=0 1=0 j=0

ifc=0 =0

n">0. >0.

+ EV -Si.n-i ( f c l+W l)fHkjXk+h +E EE i E E( H j^w,

(5.13)

These relations can be written as (n + 1) 11- E /^inn-BtO ) X.+1 oo

n

i

= E E ^i.n-j E ^ikjXk :=0 j=0

ife=0

oo n—1

J

+ E E #»,n-j E ( f c + l)A»«ti^t+l i=0 j = 0 oo n— 1

ib=0

+ E£ioE(fc + i)M»*»**+i. i=0

*r=0

By virtue of the Cauchy inequalities laJ < i T " max j / ( r ) | for the coefficients o„ of a function f(t) analytic in the disk \t\ < R, we have \\ikj\ < max |(A,-(*))*| < qk,

M

< qk■

These relations together with inequality (iii) ensure the existence of the inverse matrices (7 — £ g 0 / W - B J Q ) - 1 f° r au< n : oo \ oo / oo \ ■f - E MnnBiO = E E ^inn-B.O i=0 / j=0 \ i = 0 / > -1 7 — E /^inn-BiO i=0

< (1-Ellflnf i=0

J

,

282

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

Hence, formulas (5.13) uniquely determine the coefficients X„ and yield the inequalities

( E E WM E v oo oo

oo

,'=0 j = 0

ife=0

k

oo oo oo \ + i=0j=0 EEII%IIE(*+ !)
where

M n =max|P0||.

(5.14)

Due to (iii) and the convergence of the series £
n*< Ci

Thus, starting with some N,

M

"

—-.

n+1

Thus, starting with some N, \\Xn+l|| < M n , Mn+l = M„, Mn = MN, and ||^n|| < CiM^r/n, which proves the theorem.

n>N

REMARK 2. If in (5.12) the number of deviations is finite, the milder restrictions |A;(i)| < 1, |/x,(f)| < 1 substitute for inequalities (ii). Following [285], we apply the method of the previous theorem to prove the existence of entire solutions of linear FDE with polynomial coefficients and to evaluate their order of growth. T h e o r e m 5.4. Suppose that the system oo P

X{p)(t) = ZEQiJ(t)xU\\ijt),

xU\0) = Xj,

(5.15)

t=0 j=0

(j =

0,...,p~l)

in which Qij and X are rxr-matrices, satisfies the following conditions: (i) Qij(t) are polynomials of degree not exceeding m;

5.3. ANALYTIC AND ENTIRE SOLUTIONS OF LINEAR SYSTEMS

283

(ii) X{j are complex numbers such that 0 < ?i < |A0-| < 1,

(J =

0,...,p-1),

0 < <72 < |A,p| < q3 < 1;

(iii) the series E Q^ converges, where Q^ = max,-* ||Qyjb|| and Qijk are the coefficients ofQij(t), and £ g 0 IIQipWII < *• Then the problem has a unique holomorphic solution, which is an entire function of order not exceeding m + p. PROOF.

The expansions m m

oo oo

fc=0 k=0

n=0

k Quit) x(t) = = £ Xnt" xntn Quit) = = £Z Qijkt Qijktk,, X(t) n=0

imply t h a t

X^(t) xW(*)==

£ n=0 n=0

{

i + Py. Xn+pt , n!-

^xn+pt»,

n

^(i3(Aiii) = £ ^ i T ^Ar j X„ + ^ n , n=0

Q«(*)*W(V)

"•

=£f £ n=0

Jt=0

(n

+ J ~ f A£-*Q^xn+J_t

\Tl — K)\

and yield the following recursion relations for the matrices Xn:

(r r\»n W v r f »!(" + i-*)' 7 i - E VtoJ*** - L E E j — j j (n _ ^ + S £ ( n + P)!("-fc)!

t

Q y *x B+i _ t n ^ °- <5-16)

Hypotheses (ii) and (iii) ensure the existence of the inverse matrices CO

I - E A^Q.-po i=0

284

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

for all n:

f'-EWpo)

oo / o o

\fc

i=0 \«=0

/

<(i-Enroll) -

(i-ZKQw)

Therefore, formulas (5.16) determine the coefficients Xn uniquely and, since n! (n + j — k)\ <(n + p)-\ (n + p)\(n-k)\ n! [n + p — k)\ <1. (n + p)\(n- k)\

0<j
we obtain, by virtue of (iii), •,

p—1 m

m

||x„+p|| < — E E \\xn+i-k\\ + &?," E ll^»+p-tlln

+ P j=0 Jb=0


For large n, there is 53 < (n + p)~ and m+p-l

| X n + p | | < cCn+p)" 1 E

||*»+fc-m||.

(5.17)

k=0

Here a, 6, and c are some positive constants. With the notation (5.14), it follows from (5.17) that *n+p|| <

c(m + p) Mn+p-i. n +p

Starting with some N, c(m + p) n +p

< 1,

\\Xn+P\\ < Mn+p_u

Mn+p = M„ + p _!

and Mn = MN,

n> N.

(5.18)

5.3. ANALYTIC AND ENTIRE SOLUTIONS OF LINEAR SYSTEMS

285

It remains to apply (5.18) successively to (5.17): \\XN+p+k\\ < ||^Ar+p+(m+p)+jfc||

Hy

<

c(m + p)MN N +p ' c2(m+p)2MN (N + p)(N + (m+p)+Py

n^

||A„ + p + 2 ( m + p ) + i || < {N

c\m + pfMN + p){N + im+p)+p){N

+

2{m+p)+py

(0 < k < m + p - 1 ) . Now it can be proved easily that, for all n, cn+\m +

\\XN+p+n{m+p)+k\\

< m=o{N

+ i{m +

p)^MN

p)+py

Thus, cn+\m \XN+p+n(m+p)+k\\

+ p)MN

<

n\ and the solution X(t) is an entire function whose order of growth does not exceed m + p. ■ T h e o r e m 5.5. The problem oo

F\z)

oo

= £ Ai(z)F(z - a{) + £ ^ ( ^ ( s - 6,-),

(5.19)

«'=0

i=0

lim

F(z) = F 0

ilez—>—oo

uni/i r x r-matrices A{, B{, and F has a unique holomorphic solution which is an entire function if: m

(i) Ai{z) = £ Aikek\

m

Bi{z) = £ ^ e f c z ;

fc=i

k=o

(ii) ai, 6,- are complex numbers such that 0 < Reai < Mi < oo,

0 < M2 < Reb{ < M3 < oo;

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

286

(iii) the series 52 A® and52B{i)e-Rebi A « = max \\Aik\\,

converge where

B® = max \\Bik\\,

k

k

oo

fle6i

and E 11^.(0)||e-


i=0

PROOF. The substitutions * = e z , a,- = e""', A = e~6i, F(z) = X(*) reduce (5.19) to (5.15) of the first order with the initial condition X{0) = F0. U We are concerned now with the system M N

W'{z) = E E Pi^W^iXijz),

W(0) = W0

(5.20)

i=0 j=0

in which Pij(z) and W(z) are r x r-matrices. The following proposition extends to (5.20) the conclusion about the solutions of Eq. (5.6). Theorem 5.6. Assume that Pij(z) are polynomials of degree not ex­ ceeding p: PM=J2Pi0tzk,

E Pijkzk,

Pij(z)=

(5.21)

k=j-\

t=0

(j>l,

P>N-l)

the complex numbers A,j satisfy 0 < |Ay| < 1, and the matrices M

fl

( +1)/

"="

N

Cn + lV

-SS(^7?i)!^ + ^-'

are nonsingular for all n = 0 , 1 , . . . , where I is the identity matrix. Then (5.20) has a unique holomorphic solution oo

W(z) = E Wnzn, n=0

and it is an entire function of zero order.

(5.22)

5.3. ANALYTIC AND ENTIRE SOLUTIONS OF LINEAR SYSTEMS

287

PROOF. From (5.22) and (5.21) we obtain

wu){z)

=

g &+llw n

n=0

^

f ^)^')(Ay

-

P

*)

n=0

= E

^

oo

n!

PijkZ

k

i

Wm+jzm

E

P t f (z)W«(V) = Jb=0 ±oo Pijkz E ^f^A^ n m=0 S

m + i

^

(m=0 n - +J)Kn*=0 "»! *p.. w S = E *"E — V" r n V* ( n (n ~ -s)\ + - ? J - \ n^- 8 1p ijs ''W n-■s+j ■ n=0 =o (n-s)\ n =o ss=0

Since (5.21) implies P ! ; s = 0, for s < j — 2, the index s in the last sum extends from j — 1 t o n . Hence, the substitution k = s — j + 1 leads to the equation

(» + i ) ^ i = EE E J , TlivA^ i=0j=0 Jt=0 l n ~ «

—

J +

^^•-1^"-t+1-

l

J-

From here, Af A M NT n-j+1

(n(n-k

k + lV lY

iL u_i -n

wn+l = 5 - 1 EE E , ( \ +ZMrk-1+1PiiMi-iWn-M, n i=o i=o 4=1 (,n — « — J-t-iJ: i=o j=o 4=1 (n — K

—

j -t i)i

n >>00 . Let ||W„|| = c„, then

»=oj=o t=i l n

*

J + -U! X

since P ^ = 0, for k > p. Furthermore, ( n - * + l)i (n-k-j + l)\-

< n

j '

ll-Py,*+j-l||cn-*+l>

288

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

and for large values of n we have (n-k+ 1)1 ,. ,„_A_i+i ; ( n - A; - j + l)! where |Ay| < q < 1. Also,

„

M N n!A"~-'+1

-l

lfl"»s^r i- SS^-j+i)!^-1 and for large n,

l«rMl< n "+ V with some constant ^. Therefore, p+i

cn+i < eg" £ c„_jt+i,

c = const.

(5.23)


Denote M„ = maxcjt,

0 < k < n,

then c„+i < c(p + l)g"M„. Since c(p + l)q" < 1 for large n, we have c„ +1 < M„ and Mn+\ = Mn. Hence, starting with some natural number m, Mn = Mm,

n>m.

Successively applying this result to (5.23) yields
l)\mqm^qm+2"Mm,

Cm+n(p+i)+k < cn+l(P+ l)" +1 g m (" +1 H"("+ 1 W 2 M m , (i < k

5.3. ANALYTIC AND ENTIRE SOLUTIONS OF LINEAR SYSTEMS

289

REMARK 3. The strict inequalities |A,j| < 1 cannot be replaced by |A,j| < 1. Indeed, the scalar equation w'(z) = w{z) + (2z - z2)w'(z),

w(0) = w0

is of type (5.20), with A = 1. However, its solution w(z) = w 0 e* / ( 1 _ l ) has a singularity at z = 1. The next theorem [50] generalizes the results of [277] which were obtained for the equation with a regular singular point tx'(t) = Ax(t) + tBx(Xt).

(5.24)

T h e o r e m 5.7. Suppose the system oo

tX'(t)

= AX(t)

+ t £ Ai(t)X(Xit), (5.25) ;=o in which A, A{, and X are r x r-matrices, satisfies the following hy­ potheses. (i) A is constant, Ai(t) are polynomials (i) A is constant, Ai(t) are polynomials

Ai(t) = E Mtk, (ii) (ii) (iii) (iii)

of the highest degree m: of the highest degree m:

m > 0.

fc=0 constant,

The parameters A; are 0 < A,- < 1. The parameters are constant, 0 < matrix A,- < 1.A None of any two A;eigenvalues of the None of any two eigenvalues of the matrix A integer. integer. (iv) The series converges, where A^ (iv) The series converges, where A^ Then there exists a matrix solution X{t) = P(t)tA,

P(0) = I

differ by a positive differ by a positive — max||.Aijfc||. — max||A'fc||(5.26)

with an entire function P(t) of order not exceeding m + 1 . / / the values A; are separated from unity: 0 < Aj < q < 1, the order of P(t) is zero. PROOF. The representation of the solution X(t) = P{t)tR,

P(0) = /

290

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

leads to the equation oo

tP'(t) + P(t)R = AP(t) + t £ Ai{t)P{\it)\f, »=o whence at t = 0 we have R = A, and the expansion oo

P(t) = £ P„*n n=0

yields the relations oo n—1

Pn(A + nl) - APn = £ £ A ^ ^ P f c A f Af, i=o t=o Po = / , »> 1 for the determination of the coefficients P„. Since >li]n_i_ifc = 0

(n - 1 - A; > m)

we get oo m

P„(A + nl) - APn = £ £ A - m _ t P n + t _ m _ 1 A ^ - m - 1 A ^ . (5.27) i=0 fc=0

If the matrices .4 + nl and A have no common eigenvalues for any natural n each of the Eqs. (5.27) has a unique solution Pn and all Pn can be found successively for all n > 1. Starting with some number the matrices A + nl have inverses oo

(A + nl)'1 = £(-l) i n-*'- 1 A«',

n > Nx.

i'=0

Therefore, oo

||(J4 + n / ) - 1 | | < £ n - ' - 1 p | | ' = (n-||A||)- 1 ,

n>N1,

(5.28)

i=0

Taking into account that 0 < A; < 1 it is easy to establish

lltf II < £ |lnA,.pltt = £(-l)i m i A,il^, i=o

r

j=o

||Af || < XJm.

j!

(5.29)

5.3. ANALYTIC AND ENTIRE SOLUTIONS OF LINEAR SYSTEMS

291

Prom (5.27) we obtain Pn —

(

I \APn

oo m

+ E E A>--kPn+k- -m-

\n+k- -m-

APn + »=0 E £=0 E Aitm_kPn+k_m_l\ni+k-m-l\f k=0 and (5.29), we find Hence, by virtue of»=0(5.28)

l

\\

-1

- VJ\(A(A ++ niynl)- . 1

)

Hence, by virtue of (5.28) and (5.29), we find \\Pn\\ < E Ar IMII A« E X^-^WPn+k-m-xUn i=0

- 2\\A\\)-\

n > Nt.

k=0

Since 0 < A; < A < 1 and 211^11 < Ni for large N\, the convergence of series (iv) implies H^H^A-tn-iV,)-1 E \\Pk\\k=n—m-l

With the notation M„ = o m a x | | n | we have ||P n || < n(m + l)A"(n - JVi)-1ATw_1 and from N > N\ onwards fi(m + l)A"(n - TV)"1 < 1,

||P„|| < M„_i,

Mn = M „ _ L

By employing the procedure of Theorems 5.4 and 5.6 we evaluate ||P* + 1 + B { m + i H *|| < Hn+i(m + 1)Xn^)(m+mMN_

(5 30)

When A < 1, an+1 ||i\+i+„(m+i)+t|| < (m + l)MN —~ n! Thus, P(t) is an entire function whose order of growth does not exceed m + 1 . If we supplement the conditions of the theorem by the restriction A < q < 1, it is obvious from (5.30) that P(t) is of zero order. ■

292

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

4. Finite-Order Distributional Solutions In this section, we seek solutions of linear homogeneous ODE and FDE in the form m

x(t) = £ Xk6w(t),

xm ± 0,

(5.31)

fc=0

where 8^ denotes the k-th. derivative of the Dirac delta function. The number m is called the order of distribution (5.31). Finite-order solu­ tions of ODE with regular singular points have been studied in [284], and an existence criterion of solutions (5.31) to any linear ODE was established for the first time in [287]. The Schwartz-Sobolev theory of distributions was discovered by Sobolev [255] in 1936, and then developed consistently and thoroughly by Schwartz [239] in 1950-51. To be sure, the delta function and its deriva­ tives had been used in the physical and engineering sciences for quite some time before the advent of distribution theory. Indeed, many physi­ cal situations involving short impulses are reasonably described by the 8 function. This description has the advantage that it is independent of the actual duration of the impulse (which is usually accidental) and that it involves only the integrated strength of the impulse. For in­ stance, we may consider the response of a system to an impulse that acts for a very short time but produces a large effect. The physical situation is exemplified by a lightning strike on a transmission line or a hammer blow on a mechanical system. To formulate the idea of an impulse, let e be a small positive constant and let 8e(t) be the function defined as 8((t) = -,

for 0 < t < e;

8((t) = 0

elsewhere

Although the function is nonzero only on a small interval, the area under the curve is the area of a rectangle with base e and height 1/e, which is 1. The Laplace transform of 8e(t) is

m(t)}=j:1-e~stdt J

o e

1

=

-^^,

es

5.4. FINITE-ORDER DISTRIBUTIONAL SOLUTIONS

293

and this tends to 1 as e —» 0. It is a conceptual aid to introduce an expression S(t) that describes the effect of 8e(t) as e —► 0 and to say that £>[<$(£)] = 1. The symbol 6(t) is called the Dirac delta function or the unit impulse. It acts as if it were a function with the two properties 6(t) = 0,

ioit^O,

[+°° 6(t) dt = 1.

and

The latter equation is a special case of the general formula +oo

/

^ 8(t-t0)
=
for every continuous function
£™6e(t)
= ~Mc) =
(0 < c < e)

and observing that (p(0) as e —► 0. The delta function is closely connected to the unit step function H(t), which is equal to 1 for every positive value of t and equal to 0 for every negative value of t. Thus, we can write m

=

g(«)-g(«-«)

and as e —» 0, from a descriptive point of view at least, the derivative of H would appear to be 5: w

w

\+oo,

for t = 0.

Although the unit step function is usually associated with the name of Heaviside, and the delta function with that of Dirac, both concepts can

294

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

be found earlier in the literature. Cauchy uses the unit step under the name "coefficient limitateur" and defines it by the formula

*w=i(1+£)Moreover, Cauchy in 1816 (and, independently, Poisson in 1815) gave a derivation of the Fourier integral theorem by means of an argument involving what we would now recognize as a sampling operation of the type associated with the delta function. And there are similar examples of the use of what are essentially delta functions by Kirchoff, Helmholtz, and Heaviside. But it is undeniable that Dirac was the first to use the notation 6(t) and to state explicitly and unequivocally the more important properties which should be associated with the delta function. In Principles of Quantum Mechanics (1930) Dirac refers to 8 as an "improper function," and makes it quite clear that he is defining a mathematical entity of a new type, and one which cannot be identified with an ordinary function [71]. Although Dirac and other scientists have used this function with success, the language of classical mathematics is inadequate to justfy the definition of 8{t — to) which "is zero everywhere, except at the point to, where it equals infinity, and has an integral that equals 1." Indeed, these properties are contradictory, because if a function is zero everywhere except at one point, its integral is necessarily zero, without regard to the definition used for the integral. For Dirac, it is the sampling property which is the central feature of his treatment of the delta function, and he derives this by means of a formal integration by parts. If ip(t) is continuously differentiate on the neighborhood of the origin, then £

(p(t)6(t) dt = £ =


= ip(e)-J*cp'(t)dt

= v(0 - W) - v(0)] = v(0) (although the critical importance of the order in which the operations

5.4. FINITE-ORDER DISTRIBUTIONAL SOLUTIONS

295

of integration and of proceeding to the limit are carried out is not made manifest). The delta function dates back to the nineteenth cen­ tury and a summary of its history is given by Van der Pol and Bremmer [265]. An elementary introduction to the theory of generalized functions (with interesting applications and historic remarks) is pro­ vided by Hoskins [123]. Schwartz's approach to distributions or generalized functions which is based on the theory of continuous linear functionals has become indispensable, in view of the large and ever increasing body of literature that uses this method. We have observed that a symbolic function such as 6(t) becomes meaningful if it is first multiplied by a sufficiently smooth auxiliary (test) function and then integrated over (—oo, +oo). To ensure convergence of the improper integrals, it is natural to require that each test function vanishes outside some finite interval. It is not required that all test functions be zero outside the same finite interval. Integrating by parts the product of the derivative f'(t) of a continuously differentiable function f(t) and a test function ip(t) gives

f+°° f'(t)
(5.32)

■/— oo

Therefore, it is natural to assume that each test function is infinitely differentiable. The space of test functions, which is denoted by D, consists of all complex-valued functions (p(t) that are infinitely smooth and zero outside some finite interval. In other words, T> is the space of functions with continuous derivatives of all orders and having compact support. An example of a test function in D is

**,«) = W ~ « £ ? ) ' 0,

l
'

(5.33)

|*| > a.

It can be shown that every derivative of this function exists and is zero at t = ±a. Hence, it has continuous derivatives of all orders for ev­ ery t, and they are all equal to zero for \t\ > a. It is important to note that any complex-valued function f(t) that is continuous for all t and zero outside a finite interval can be approximated uniformly by test functions. We say that a sequence of test functions {
296

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

to zero if the functions ipn and all their derivatives converge uniformly to zero and if all the functions <£>„ vanish identically outside the same finite interval. It is not difficult to show that D is complete with re­ spect to this definition of convergence. Although S(t) is not defined in a pointwise manner, it associates with each test function and any complex numbers c\ and C2, the following property is satisfied: c2ip2=)==c,(/, ci(f, yi> ipi)+ +c2(/,y»2>. c2(f, to ' converges to the distribution / G D' if >->>,

(pel).

In this case we write /„ —► / in V and refer to the convergence as weak convergence. A function f(t) is called locally integrable if it is integrable in the Lebesgue sense over every finite interval. One of the simplest examples of a distribution is offered by the functional generated by a locally integrable function. If f(t) is locally integrable, we can define a distribution / through the convergent integral oo

/

^ f(t)
(5.34)

Actually, the limits on this integral can be altered to finite values, since
5.4. FINITE-ORDER DISTRIBUTIONAL SOLUTIONS

297

functional on D. Distributions which are definable in terms of locally integrable functions according to formula (5.34) are called regular dis­ tributions. All other distributions are called singular distributions. We notice that if f(t) and g(t) are locally integrable functions that are equal almost everywhere, then the associated distributions / and g are the same. Conversely, if f(t) and g(t) are locally integrable and if their corresponding regular distributions agree, that is, (/,
/

-oo

f(t)ip(t)dt

=
for a locally integrable function f(t) and any test function ip(t). Then for the function
(5.35)

However, the integral on the left tends to zero as a —> 0, which contra­ dicts (5.35). The product of a generalized function f(t) and an infinitely smooth function a(t) is introduced by the formula

(af,v) = (f,a(f), and it is obvious that a

= 0,

whence t5(t) = 0. The operations of differentiation and integration that were originally developed for functions, can be extended to distribu­ tions. Formula (5.32) prompts to define the distributional derivative / ' of a generalized function / according to the equation } = -')•

(5-36)

298

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

This rule shifts the differentiation of a generalized function to the dif­ ferentiation of a test function that has derivatives of all orders. An ordinary locally integrable function may not have a derivative at cer­ tain points or, for that matter, anywhere at all. Formula (5.36) reveals a remarkable fact that every generalized function has derivatives of all orders that are again generalized functions. Thus, for the Heaviside function H(t) we obtain

(H'(t)Mt)) = -(H(t),
=
and conclude that H'(t) = 6(t). Repeatedly applying (5.36) leads to the formula (/("),y>) = (-l)" for higher-order distributional derivatives. For example,

<«<">(*), ?(*)> = (-i)B(*(*),*»w(<)> = (-i)V n ) (o). A comprehensive and rigorous exposition of Schwartz's theory of dis­ tributions is available in the books by Gel'fand and Shilov [88, 89, 90], Zemanian [306], Kanwal [134], and in a number of other works. Here,we develop the methods of study and state the main results for linear ODE in the space of finite-order distributions. The next four theorems establish necessary and sufficient conditions for the existence of solutions (5.31). T h e o r e m 5.8. // the equation £>(t)a:(<) = 0

(5.37)

with coefficients a,(i) £ c ^ m + n - ^ in a neighborhood of t = 0 has a solution of order m concentrated ont = 0, then: (1) a0(0) = 0, (2) m satisfies the relation — (m + n)a'0(0) + a}(0) = 0,

299

5.4. FINITE-ORDER DISTRIBUTIONAL SOLUTIONS

(3) there exists a nontrivial solution (xo,..., x m ) of the system m+n

min(i,n)

E **+/-« E (-ir*ap"°(0)(fc + J - 0 ! = 0 j'=0

i=0

(k = 0 , 1 , . . . , m + n). PROOF. The existence of a solution (5.31) to Eq. (5.37) leads to the conclusion that n m+n—i

E

E

i=0

j=0

m

...

M*'£**«<*+»-■>(*) = 0,

a0- = op>(0)

Jt=0

since in the Taylor expansions m+n—i

<*,■(*) =

E

a,^' + r,(i)

j=o

the remainders and all their derivatives up to the order m + n — i vanish at t = 0 and r,(*)x("-')(*) = 0, for any distribution (5.31). The formula {

t'6M(t) =

1}

(k-j)\

0,

'

~J k<j

(5.38)

gives the result n m+n-i

m

(U 4. rj — j'V

E E (-1)% E «»**+»-'>(*) (i ( ^Viv =0 which can be written as n m + n - i A(*) (/"\

m+n-i

E E - T H E (-iy(k+j)\aijXk+i+j.n = o. ft! .=0 t=o i=o Changing the order of summation we obtain

m+n fj(k)

n m+n—i

E T T £ k=o

K

E

- «'=o j=o

(-iy(k+j)\aijxk+i+j.n=0

300

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

and, consequently, n m+n—i

E

E (-iy(k

i=0

j=0

+ j)\aijxk+i+j-n=0,

k = 0 , 1 , . . . , m + n.

The replacement of i + j by j yields n m+n

E E (-l)i-i(fc + i - 0' oy-.-«t+y-« = 0 whence whence n-l n-l

i

. .

Y, Xk+j-n Ei C - 11 ) 7' '(k + J ~ 0 !! aaiJ~i E C - ) '(* + i - 0 iJ-i j£ = 0 Zfc+j-n i=0 j = 0 m+n /n+n

i=0 n n

+ E **+i-n E(-i) i _ i (* + i - 0!«u-«- = o,

+ j£= n% i - n Ei=0( - i r ( H i - i ) ! a t j - i = 0, j=n

i=0

aa system system identical identical solution, solution, therefore therefore

it = 0 , 1 , . . . ,m + n

* = 0,l,...,m + n

x aoo a^xmm

with = 0 0 has has aa nonzero nonzero with (3). (3). Its Its last last equation equation = aoo — 0. The penultimate equation is aoo — 0. The penultimate equation is (aio - (m + n)a0i)xm = 0,

which confirms (2).

■

Theorem 5.9. Eq. (5.37) has an m order solution with support t = 0, if the following hypotheses are satisfied: (i) For some natural N (0 < N < m + n) ai*~°(0) = 0,

i = 0 , . . . , min(iV, n);

(ii) m is the smallest non-negative integer root of the relation

U-l)™-**?+1_,)(0)(m + n - ■)! = 0, i=0

M = min(N +1, n) where N denotes the greatest integer for which (i) holds; (iii) there exists a nonzero solution of system (3) in Theorem 5.8.

5.4. FINITE-ORDER DISTRIBUTIONAL SOLUTIONS

301

PROOF. Any nontrivial solution {xk} of system (3) originates a dis­ tribution (5.31) that satisfies (5.37). If assumption (i) is fulfilled, the last equation of system (3) becomes AN(m)xm

= 0,

where AN represents the left side in (ii). By virtue of (ii), we can put xm ^ 0 and determine the unknowns xm_^ successively since all their coefficients Aff(m — k) are different from zero. ■ Theorem 5.10. If the equation £ t'ai(*)a:(0(<) = 0

(5.39)

t'=0

with coefficients a,-(t) € C m and a„(0) ^ 0 has a solution (5.31) of order m, then E(-l)*'a,-(0)(m + i)\ = 0.

(5.40)

»=0

Conversely, if m is the smallest non-negative integer root of rela­ tion (5.40), there exists an m-order solution of (5.39) concentrated on t = 0. PROOF. This proposition may be considered a corollary of the previ­ ous theorems, but since it constitutes the basis for the study of equa­ tions with regular singular points we sketch also a different approach. The Laplace transformation of the equation

E* , Eay* , * W (0 = 0, i=0

aij = aY\0)

j=0

yields n

m

T. E ( - l ) , + , a i i ( ^ ( * ) ) ( , + , ) = 0.

(5.41)

,=o j=o The necessary and sufficient condition for the distribution x(t) to have the order m is that its transform &(s) be a polynomial of degree m.

302

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

Differentiating relation (5.41) k times and putting s = 0 we obtain n

m

£ £(-i) i+i (* + i + *)! w + * =o.

* = o. i, • • • (5-42)

i=0 j = 0

where re* = L^k\Q)/k\. The requirement x* = 0, k > m reduces (5.42) to a finite triangular system of equations the last of which A(m)xm = 0 has a solution xm ^ 0, A{m) being the left side of (5.40). Hence, (5.40) holds and if m is the smallest non-negative integer zero of A(rn) the substitution of xm in Eqs. (5.42) allows one to find all Xk(k < m) since their coefficients A(k) ^ 0. ■ COROLLARY

5.1. Bessel's equation t2y" + ty' + (t2 - v2)y = 0

has a solution of order m with support t = 0 iff i/2 = (m + l ) 2 and it is given by the formula

IW?

V

(m-k)!

(m_2fc)

\t'o4^(m-2ik)r

W

C = const.

'

where [m/2] stands for the greatest integer < m/2. Theorem 5.11. The equation te<">(*) + £ a,^)x( n -'')(t) = 0,

a,- G C m + n - '

(5.43)

i=l

has a solution of order m iff (i) ai(0) — m + n, (ii) there exists a nonzero solution of the system (ai(0)kl-(k

+

l)l)xk+1,n

m+n

min(;',n) -2)! = 0

+ j£= 2* * + ; - »

.'=1 £

( - i r , a p - , ) ( 0 ) ( A ; + i - 0 ! = 0 (5.44)

(k = 0,... ,m + n — 1). (fc = 0 , . . . ,m + n — 1).

5.4. FINITE-ORDER DISTRIBUTIONAL SOLUTIONS

303

Now the last theorem will be used to find distributional solutions of some interesting second and third order differential equations, and we shall return to these results again in the next chapter. COROLLARY 5.2. The confluent hypergeometric equation tx" + (b- t)x' -ax

=0

(5.45)

has a finite-order solution iff a and b are positive integers and b > a + 1. This solution is given by the formula da~l

N, 6 -"- 1

(d

x=C ^ (i- - 1) dta~l \dt ) and its order is m = b — 2. PROOF.

6(t) w

Relations (i), (ii) of the previous theorem will be as follows

for (5.45): b = m + 2, n(b — n — l)x„_i + (n + 1 — a)x„ = 0, n = 0 , . . . ,m + 1, whence b > 2 is an integer and if a is not a positive integer < b — 1, all xn = 0. On the contrary, when the hypotheses are observed then Xk = 0, k < a — 2, and taking any i m ^ 0 we find

xm_k = C^O,

COROLLARY

(-i)^-ak-iym-ky.c, 0<Jfe<6-a-l.

■

5.3. The equation [133] tx" + ax' + btx = 0,

b± 0,

(5.46)

has a finite-order solution iff the coefficient a is a positive even integer. This solution is given by the formula x= C

(I J2 d2

2 \\ (a( a --2)/2 )/2

[dfi + t ) 1

6(t)

304

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

and its order is m = a — 2. PROOF.

For Eq. (5.46) conditions (i), (ii) of Theorem 5.11 become

a= = m + 2, l)z n _i == b(n b(n+ +l)x„+i, l)x n + i, (a -— nn-- l)x„_i

nn==0,. 0 , .... ., m m++1, 1,

from which it follows that x2fc_i = 0,fc> 1, and the order m is even. Choosing any i m ^ 0 we determine k m 2 xm_2k = Xm-2k =bb"( 1j; )(Tn-2k)\x ' 1 (m -Zftj. m, X m ,

COROLLARY

0
5.4. The equation [133] tx" + (t + a + b)x' + ax = 0

/ias a finite-order solution iff a andb are positive integers. This solution is given by the formula ,d vH da-\ 1 x= a *(*) dt and its order is m = a + b — 2. -

COROLLARY

)

'

■

5.5. TTie equation [133] to'" + (a + 6)x" - ta' - ax = 0

(5.47)

/ias a distributional solution, iff a is integral positive and b is even positive. This solution is given by the formula 1 d"da~l (d ( d2 2 X r 1 x = C C^ d^{d^dta~l ) and its order is

6/2_1 \\ 6 ,.

U H «*>•> m

cC = cconst. st

= ™-

m = a + b - 3.

(5.48)

PROOF. For (5.47), system (5.44) takes the form (1 - a)x 00 = 0, 0, (2 - a)xi = 0, _2 + (k 2 ,......,, m (* = 2, m)) (a + b-kb - k - l)xkfc-2 + (k + 1 -- a)xk = 0, (a + + 6b-m_] = 0 0,, - m - 2)x 2)z m _i (a + b-mb — m —3)x 3)x i(5.49) m m= 0

5.4. FINITE-ORDER DISTRIBUTIONAL SOLUTIONS

305

and (5.48) follows from the requirement xm ^ 0. The penultimate equation gives x m _i = 0 and (5.49) implies that x m _2t-i = 0 (k > 0). If the parameter a is not positive integer, all Xk = 0. On the contrary, when a is positive even, all X2k = 0 and the order m is odd. From (5.48) it appears that b is even and from (5.49) we have a < m + 1 ; thus, b > 2. If a is positive odd, all a^t+i = 0 and again b is even. Turning to the calculation of the coefficients xm_2*: we obtain Xm-2k

COROLLARY

= ( - l )

k

r

2

~ j X m ,

fc

= 0, 1 , . . .

■

5.6. The equation [133]

tx'" -(t + p)x" -(t-p-

l)x' + {t - l)x = 0

(5.50)

has a distributional solution, iff p is a negative odd integer, p < —3. This solution is given by the formula d2

\ -(P+3)/2

and its order is m = —p — 3.

(5.51)

PROOF. The coefficients Xk of (5.31) satisfy the equations *i = 0> (P + % o + xi - 2x2 = 0 -(p + k + l)x t _ 2 + (p + k + 2)xjb_i +fcxjfc- (k + l ) x t + i = 0, (fc = 2 , . . . , m - l ) -(p + m + l)x m _ 2 + (p + m + 2)xm_i + mxm = 0, - (p + m + 2)x m _i + {p + m + 3)x m = 0, (p + m + 3)x m = 0. (5.52) Condition (5.51) provides the possibility to put xm ^ 0 and to assume that xm-2„+i = 0,

I / m/2 \

xm_2(„_i) = ( - 1 ) " x I n _ 1 J x m ,

1 < n < k.

306

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

Prom (5.52), 2kxm_2k - (2k - l)x m _2t+i + (m - 2k + 2)xm-2k+2 -(m-2k + 3)zm_2jfc+3 = 0. Since xm_2it+3 = xm-2k+i = 0 we get, for all k > 0, , sk(m/2\ x m _ 2i = ( - l ) I k Um The equation (2k + 2)xm_2(it+1) - (2k + l)xm_2ifc_i + (m - 2k)xm-2k = 0 gives £m_2/t_i = 0 (A; > 0). Inasmuch as xi = 0, the order m is even.

■

Generalized-function solutions of some ordinary differential equations with polynomial coefficients were studied by Aliev [12]—[16]. Existence of finite-order distributional solutions for linear functional differential equations was explored in [284]. Theorem 5.12. A criterion for the existence of solutions (5.31) to the system N

tx'(t) = £ Aj(t)x(\jt)

(5.53)

with matrices Aj(t) £ C m in a neighborhood of t = 0 and constants Xj ^0 is that some roots fi of the equation det (E l A ^ r ' A J ^ O ) + (fi + l)l\

=0

(5.54)

be nonpositive integers. If m is the smallest of their absolute values there exists a solution of order m. PROOF. In the Taylor expansions m

Aj(t)=

Z Aijtk + Rjm(t) k=0

5.4. FINITE-ORDER DISTRIBUTIONAL SOLUTIONS

307

the remainders and all their derivatives up to the order m vanish at t = 0. Therefore, Rjm{t)x(t)

= 0,

for any distribution of the form (5.31), and the sets of solutions of order not exceeding m to (5.53) and to the system N

m

**'(*) = E E Ajktkx{\jt)

(5.55)

j=0 4=0

are identical. The Fourier transformation of Eq. (5.55) yields

- («*(*))' = E E M- 1 W * ^ ? * * ) f j=ok=o

•

(5-56)

\ A i/

The necessary and sufficient condition for the distribution x(t) to have order m is that its transform 3~(s) be a polynomial of degree m. Dif­ ferentiating relation (5.56) n times and putting s = 0, we obtain g IAJI^AJM*, + (" + 1)/) ^n

+ EEH) fc |^r I Aj"-%- t F n+Jt = 05 (n = 0,l,...), (5.57) j'=0fc=l

where F„ = 3"(")(0). The requirement F„ = 0, n > m, reduces (5.57) to a finite system of matrix equations, the last of which

£ \\j\-l>qmAiQ + (m + l)l\ Fm = 0, possesses a nontrivial solution Fm. Its substitution into the foregoing equations allows one to find Fn (n < m) successively since the matrices N

E \XJ\ xrAio+(n+I)A are nonsingular.

■

*» < m .

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

308

COROLLARY

5.7. The system tx'(t) = A(t)x(t) + £ Aj{t)x{\jt)

(5.58)

has a solution of order m with support t = 0, if Aj(0) = 0 (j > 1) and m + 1 is the smallest modulus of the negative integer eigenvalues of the matrix A(0). Corollary 5.7 can be used to give another proof of Theorem 5.10 by changing Eq. (5.39) to the system w'(t) = A(t)w(t). This will be shown in the next chapter. Theorem 5.13. The system

(5-59)

tx'(t) = £ A,(*)*(fc(*)).

i=o in which Aj(t) 6 Cm, 4>j(t) G C 1 , has a solution (5.31) of order m, if the following hypotheses are satisfied: (1) the real zeros tjn of the functions (f)j(t) are simple and form a finite or countable set; (2) AW(tjn) = 0 (k = 0 , . . . ,m), for tjn / 0; (3) m is the smallest modulus of the nonpositive integer roots of equa­ tion (5.54) with \j = <^(0). PROOF. From the representation (f>j(t) = (t — tjn)\j(t) it follows that <\,-(t/n) = 'j(tjn) 7^ 0. In some neighborhood 1>j„ of the point tjn that does not contain other points of {tj„}, 6^(Xj(t)(t

- tjn)) = 8^(\j(tjn)(t

But, for t = tjn, Xj(t)(t - tjn) = \j(tjn)(t

- 'in)) = 0,

t±

tjn.

— tjn). Therefore,

6^(j(t)) = 6W(<j>'j(tjn)(t - tjn)),

t e Vjn.

(5.60)

Since for the distribution (5.31) suppx(<j>j(t)) = {tjn} there will be, by virtue of (5.60),

^w)=E^ife»)(<-g). n

5.5. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS

309

It is easy to show that

WtffaXt - tjn)) = ^-Msgnc)>'j(tjn). Hence, m

tjn)

*(M*)) -= EE n

Owing to

sgn^(<>„).


Aj(t)6W(t - tjn) = 0,

tJn # 0, fc = 0 , . . . , m,

we make the conclusion that Aj{t)x{h{t))

= 0,

0 6 {tjn}

and the problem of existence of a solution (5.31) to system (5.59) is brought to the same question for (5.53) where Xj = j(0). ■ 5. Infinite-Order Distributional Solutions Now we shall study solutions of the form

x(t) = £ xj^(t)

(5.61)

n=0

for linear functional differential equations (FDE) in the generalizedfunction space (So)'- We know already that distributional solutions of linear homogeneous ODE are generated solely by singularities in their coefficients. For linear FDE, distributional solutions may be originated either by singularities of their coefficients or by argument deviations. Thus, it has been shown in [280] that under certain conditions, the FDE with polynomial coefficients oo

*'(*) = E^('M^) has a solution (5.61) in (SQ)' with arbitrary /? > 1, which is an impos­ sible phenomenon for linear homogeneous ODE without singularities in their coefficients. The generalized-function space (So)' is conjugate

310

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

to the space S{j of test functions <j>(t) that are infinitely smooth on (—oo, oo) and subject, in particular, to the restriction [89] \<j>{n\t)\
n = 0,l,....

To ensure the convergence of series (5.61), it is sufficient to require that for n —> oo the coefficients xn satisfy the inequalities ||x n || < acnn-np,

p>\.

(5.62)

In fact, let (/, (p) be the value of the functional / applied to
£M(n)(<)>*(*)>

£(-i)V B) (o)*„

n=0

n=0

[i

(n)

< £ l^ (0)|||xn|| n=0

oo

(cdn^-p)n < oo,


for (3 < p. If series (5.61) converges, its sum is the general form of a linear functional in (So)' with support t = 0 [197]. The norm of a matrix ||A|| = m a x £ | a y | i and I is the identity matrix. T h e o r e m 5.14. The system tx'(t) = Ax(t) + 1 Bx(Xt),

(A = const.)

(5.63)

with constant matrices A and B has a solution in the space (SQ)' con­ centrated on t = 0, if B is nonsingular and —1 < A < 1, A ^ 0. PROOF. The Fourier transformation changes (5.63) to the equation

-Wt

-,!*(.)

'W(i),

5.5. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS

311

from which there follow relations F B+ i - -*A B+1 |A|B- 1 (A + (n + l)I)Fn, Fn = 3
n = 0,l,...

and inequalities

H^+iii^iArv+g+ijii^ii-iiF,,!! that imply estimates ^ll^nll < l A r ^ ^ l l ^ i r ^ + ^HFoll,

(5.64)

where q is some fixed natural number such that ||.A|| < q. Since for the coefficients xn of series (5.61),

IKII = ^ll^ll the bounds (5.64) prove the theorem, as the condition |A| < 1 makes them more restrictive than (5.62). Relation (5.63) provides an interesting example of a system that may have two essentially different solutions in the space (So)' with support t = 0. According to the previous results, (5.63) has a solution of finite order if the matrix A assumes negative integer eigenvalues. At the same time there exists an infinite-order solution (5.61) if A / — nl, for a l l n > 1. The particular importance of the system £ £ {Ajk + tBjk)xM(\jt) = tx(Xt) (5.65) j=o t=o is that depending on the coefficients it combines either equation with a singular or regular point at t = 0 and in both cases there exists, under certain condtitions, a solution of the form (5.61). T h e o r e m 5.15. Let system (5.65) in which x(t) is an r-dimensional vector and Ajk, Bjk are constant matrices of order (r x r), satisfy the following hypotheses: (i) Xj and A are real numbers such that 0 < |A| < 1, |Ay| > 1, j > 0;

312

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

(ii) the series E \Xj\~lA^

and E \Xj\ B'J' are convergent, where j=o

j=o

A® = max \\Ajk\\,C| B « = max \\B3jk\\; 0<*<m"

oo

(iii) E Xf\\Bj0\\

■" "

0
"'

2

< A" .

3=0

Then in the space of generalized functions (S{j)' with arbitrary /3 > 1 £/iere exists a solution x(t), concentrated on t = 0. Substituting the unknown solution (5.61) in (5.65) we ob­ tain, by virtue of PROOF.

s^(Xjt) = iA,rv*(t), the equality oo

m

oo

E E E |A j |- 1 AT»-*(^. ta : n 5(" +A )(f)-(n + A:)5jfcrrn5(n+t-1)(<)) j=0 k=0n=0 oo

= - E nlAr^-"*^"-1^), n=l

from which it follows that oo m

(n + l)|A|-1A--1xB+1 = E DCn + l J I A i l - V 1 ^ * ^ ! - * i=o *=o oo m - E E | A ; r-U-n A-%fcx„_t, i=o t=o

n>0,

and, hence,

/-|A|A n+1 g|A i r 1 A7"- l 5 i0 k + i \Mxn+1\EE\xJrxjn-lBjkXn+1.k

=

\:=0k=i 1 n

oo m

\

E E lAyrv^^-t •

+ 1 i=o t=o

(5.66)

313

5.5. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS

Inequalities (i) and (iii) ensure the existence of inverse matrices to the coefficients of xn+\ for all n:

('-

oo

\

i=o

/

_1

oo OO

// OO oo

E l A l i A( n+ = E +l ) i

!=0 i=n

1

. rr l AA rr -- ll BB .. 00

\j'=0 \7=o

»-»

^-lAIA^ElA.rV" ^ V,

\

E II A E A

3=0

4/

f

/

oo

\ -l

-A 2 EA7 2 ||B i0 || 3=0

/

Therefore, formulas (5.66) determine the values xn uniquely with the exactness to an arbitrary vector XQ and because of conditions (i) and the convergence of series (ii) provide the bounds m

lkn+i||
0
(5.67)

fj, is some constant. We set Mn — max ||xi|[.

(5.68)

Then from (5.67), \\xn+l\\<»{rn

+

l)qnMn.

For large values of n, fi{m + l)qn < 1. Hence | | x n + i | | < M„ and Mn+\ = Mn. Thus, we arrive at the conclusion that, starting with some N, M„ = MN,

n>N.

(5.69)

314

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

The application of (5.69) to (5.67) successively yields ||ZJV+I||


\\xN+2\\
+

l)qN+1MN, l)qN+mMN.

Therefore \\xN+k+i || < fi(m + l)qNMN, Putting n = N + m + l,...,N obtain \\xN+l+{m+l)+k\\

0
(5.70)

+ 2m + l'w. (5.67) and using (5.70), we

< fi2(m + l)YqN+mMN,

0
m.(5.71)

By employing (5.71) we can establish that \\xN+l+2{m+i)+k\\

< /z3(m + l)3qNqN+mqN+2mMN,

0 < k < m,

and continuation of the iteration process allows one to assume that, for all n and 0 < k < m, | | ^ + 1 + n ( m + 1 ) + t | | < nn+l(m + i)»+ig»(»+DW2.

( 5 . 7 2)

Replacing n in (5.67) by N+l+n(m+l)+m,... ,N+l+n(m+l)+2m we find that (5.72) holds for ||;c#+i+(n+i)(m+i)+ifc||; i.e., these inequali­ ties have been established by induction. Inequalities (5.72) prove the theorem since the condition 0 < q < 1 makes them more stringent than (5.62). It remains to observe that Eq. (5.63) and x'(t) = Ax(t) + tBx(Xt) are special cases of (5.65). We continue with the existence theorems for FDE in the space The study of the system

E E Mt^iXijit)) = 0

(SQ)'.

(5.73)

i j=0

generalizes the corresponding results of [284] and [287]. The choice of the coefficients enables us to consider both equations with a singular

5.5. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS

315

or regular point and to show that distributional solutions of FDE may be originated by deviations of the argument. We also investigate the system oo m

M*) = £ £ M^iXijit)),

(5.74)

i=0 ;'=0

the particular cases of which tpx'(t) = A(t)x(t) and oo

t'>x'(t) =

'£Ai{t)x(\it) i=0

have been studied in [16] and [283], respectively. LEMMA 5.1. // the function X(t) G C 1 , A(0) = 0 and A'(0) = a ^ 0, then for the distribution (5.61), z«(A(t)) = £ \a\-1a-u-ixn6W(t)

(5.75)

n

in some neighborhood of the origin. PROOF. From X(t) = t{t) it follows that (f>(0) = a, and there exists a neighborhood T of t = 0 at all points of which (j>{t) ^ 0, since assuming the opposite we find a sequence tn —> 0 such that <j>(tn) —► 0; hence, A'(0) = 0. In T, 6W(t(p) = 6ln)(cd) = 0, for t ^ 0. But t = at, for t = 0, and the relation «W(A(t)) =

&n\at)

holds for all t G T. The functional 6 is the derivative of the function H(t) which equals 1 for t > 0 and 0 for t < 0. Differentiating n + 1 times #(atf) = #(<), a > 0 and #(«<) = 1 - #(*), a < 0, we obtain «M(A(t)) = |a|- 1 a- , *W(<). ■ Theorem 5.16. Lei (5.73), uni/i a finite number of argument devia­ tions, in which x is an r-vector and A^ are r x r-matrices, satisfy the following hypotheses.

316

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

(i) The coefficients Aij(t) are polynomials in t of degree not exceed­ ing p: Ay® = E ^***>

^oo = Af>,

p>l.

(ii) The real-valued functions \{j(t) G C 1 in a neighborhood of the origin, A,j(0) = 0 and 0<|a00|l, i + j>h (iii) The matrix A is nonsingular and

a«i = 4 ( ° ) -

c = laool-^HAll - E l « « . | - p - 1 | | ^ | | > 0. Then in the space of generalized functions (SQ )' with arbitrary (3 > 1 there exists a solution x(t) supported on t = 0. PROOF. By virtue of (5.75) and the formula (-l) f c n! S(n-k)(t) h

>k

n)

t 6( = • 0,

n < fc

we obtain the equation

E(-I)%* E

(

?n+^!|a£ a r ^ ^ H ^ o

for the unknowns i „ of the solution (5.61). The replacement of n + j —A; by n gives the relations E ( - l ) * ( n + *)! |a i i |- 1 ay-*A« t x I l + f c _ i = 0 ,

n> 0

which can be written as t

_ ^

^

(n+p)!|a,jl

a

"

A

«^x»+^

+ I E law I la ,o" PAioP) xn+P = 0.

5.5. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS

317

Since >looi = 0 (k < p) the first sum does not include terms with aooAccording to (iii) the coefficients Bn of xn+p are nonsingular matrices and \\B~l\\ < c||A|||aoo|n. Consequently, m+p-l

||*n+p|| < Mn+P £

||*-+*-m||,

0
(5.76)

where \i is some positive constant. Using the notation ^ n = o max||x 1 ||

(5.77)

we conclude from (5.76) that \\
ll-^n+p

For large n there is /j.(m + p)qn+p < 1. Hence, ||x„+p|| < M„ +P _i and Mn+P = M„ +P _i. Thus, starting with some N, Mn = MN,

n>N.

(5.78)

The application of (5.78) to (5.76) successively yields: \\xN+P+i\\ <

n(m+p)qN+pMN, p)Y+pqN+p+(m+p)MN,

||x^ + p + ( m + p ) + i || < n\m + \\xN+p+2{m+p)+i\\

p)Y+pqN+p+im+pY+p+2{m+p)MN,

< n\m + (0
m+

p-l).

The conjecture

||z 7 V + p + „ ( m + p ) + i || < »n+\m

+

pr+Y{N+p)+n{n+1)(m+p)/2MN (5.79)

may readily be ascertained by induction, for all n and the mentioned values of i, and proves the theorem since the condition 0 < q < 1 makes it more restrictive than (5.62). ■ Theorem 5.17. System (5.73) with a countable set of argument de­ viations has a solution (5.61) if, in addition to the conditions of Theo­ rem 5.16, there exists a neighborhood of the origin in which each func­ tion Xij{t) has the only zero t = 0 and the series Ej^i (xj Ai converges

318

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

where At = max \\Aijk\\, at = inf |ory|, ],k

i + j > 1-

(5.80)

J

T h e o r e m 5.18. Suppose that system (5.74) in which x is an r-vector and A{j are r x r-matrices, satisfies the following conditions. (i) The Aij(t) are polynomials in t of degree not exceeding p + j — 2:

M*)=

p+i-2

£ Aijktk,

p>2.

(ii) There exists a neighborhood T of the origin in which the realvalued functions Xij G C 1 have the only zero t = 0 and |a,o| > 1, inf |ag | > 1, for i > 0, j > 1, atj = A^(0). (iii) ITie series A = £ ~ 0 a i rlj 4f converges where a; = inf | a y | ,

A; = max ||Ayt||

TTien Mere is a solution of (5.74) in (SQ)' uritfJi «ome /? > 1 supported on t = 0. PROOF. According to (5.75), Eq. (5.74) has in T the same distri­ butional solutions as the similar system with constants ay instead of Ajj(i), and it is easy to obtain the relations (n + p)!x n + p _, = Z(-l)p-k(n

k)\\aij\-1ar?-kAijkxn+k-j

+

for the coefficients x„ of (5.61). Assumptions (ii) and (iii) imply that m

p+j-2

j=0

Jt=0

n

(n + p)Fn+p^
£

*W*-i,

F„ = ||x„||n!

where a = inf |a»y|, i, j > 0, and the procedure of Theorem 5.16 yields the inequalities F „ + n ( m + p _ 1 ) + i < ( m + l)n(m+p-

x AnFN

l)na-n(n-l)(m+P-l)/2

f[(N + 1 + (m +p - l)i),

(0 < k < m +p - 1)

319

5.5. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS

more stringent than (5.62), if a > 1. Hence, the space (S{f)' with arbitrary ft > 1 contains a solution of (5.74) concentrated on t = 0. For a = 1, ,, \\xN+n(m+p-l)+k\\

S TTT"

(m + 1 ) M " M J V 7 ; TTTTTi

f '

(JV + n[rn + p — 1) + k)\ n\ and applying Stirling's formula we get and applying Stirling's formula we get

||JC„||

< acvv-vr, u = N + n(m+p-l) p= l + (m+p-l)-1.

+ k,

Therefore, if inf |Qfi01 = 1, Eq. (5.74) has a solution (5.61) in (SQ)' with 1 < /3 < p. M We have mentioned that integral transformations establish close links between spaces of generalized and entire functions. A comprehensive and detailed investigation of this interaction is provided in the next chapter. The following theorems shed some light on the subject. Theorem 5.19. The system oo

x{p)(t)

p

- E EP«,(t)*w(V), i= °i=°

xW

(0)

= Xj,

j =

o,...,P-i (5.81)

with r x r-matrices P^ and X has a unique holomorphic solution, which is an entire function of zero order, if (i) Pij(t) are polynomials in t of degree not exceeding m; (ii) \{j are complex numbers such that 0 < |Ay| < q < 1; (iii) the series EPW converges where P(') = xaaxjjc ||Pijjfe|| and Pijk are the coefficients of Pij(t), and £So ||^<m(0)|| < 1. PROOF.

Under the conditions of Theorem 5.16 the system oo

m

E E AaWX^Xaqt) = 0 !=o j=o with real constants «,_, has a distributional solution (5.61) the coeffi­ cients Xn of which satisfy the inequalities (5.62) and are determined with the exactness to arbitrary XQ, ... ,XV^\. We apply to the last

320

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

equation the Laplace transformation assuming a^ positive and retain­ ing the same notation for X(t) and its transform:

x(p)

Gr) + ^lA_1 £ £(- 1 ) p " i a r 1 ^ (** ( f ) ) w = o.

\<W .+i>ojb=o V \ a 0 7 / (5.82) an The substitutions s/aoo = * d aoo/o'y = ^«j reduce (5.82) to the form (5.81). This proves the theorem since the transform of 8^n\t) is s" and the coefficients X„ satisfy the inequalities (5.79). These estimates use only the moduli of a,j, hence the parameters A,j may be complex. T h e o r e m 5.20. Under the assumptions of Theorem 5.16 there exists a polynomial Q(t) of degree p — 1 such that the system k

m

i=0 j=0

with positive constants a,j has a solution X(t) regular at t = oo and X(t_1) is an entire function of zero order. PROOF. We apply the unilateral Laplace transformation to (5.82). T h e o r e m 5.21. Assume Aij[t) are polynomials of degree not exceed­ ing p + j — 2 (p > 2) and a:^ > 1. Then there exists a polynomial Q(t) of degree p + m — 2 such that the system k

m

t"x'(t) = E E A o -(0* a W) + Q(t) i=0 j=0

has a solution X(t) regular at infinity and X(t~l) of zero order.

is an entire function

The deep study of differential equations with linearly transformed ar­ guments remains one of the major problems in the theory of functional differential equations, and there is an enormous literature in this and related areas. We would like to mention the famous paper of Kato and McLeod [138] and the significant contributions of the following authors: Antonevich [17], Bolkovoi and Zhitomirskii [27], Bruwier [30], Bykova [32], Carr and Dyson [35, 36], Chambers [42], Derfel' [63]-[66], Derfel' and Molchanov [67, 68, 69], Derfel' and Shevelo [70], Feldstein

5.6. INTEGRAL EQUATION

321

and Jackiewicz [78], Flamant [80], Fox, Mayers, Ockendon, and Tayler [81], Frederickson [82], Grebenshchikov [100, 101], Gross and Yang [102], Hahn [113], Izumi [130], Karakostas [135], Kato [137], Kuang and Feldstein [147], Lim [174, 175], McLeod [189], Mohon'ko [199], Mure§an [201], Murovtsev [202], Pandolfi [209], Pelyukh [214], Pelyukh and Sharkovskii [215], Polishchuk and Sharkovskii [220, 221], Robinson [227, 228], Romanenko [230, 231], Romanenko and Sharkovskii [232], Rvachev [234], Samoilenko and Mustafaev [237], Staikos and Tsamatos[256], Valeev [263, 264], Vogl [270], and Zima [309]. 6. An Integral Equation in the Space of Tempered Distributions Some developments in astrophysics, radiophysics, and other sciences pose new problems concerning the existence of distributional solutions to differential and integral equations. Thus, in [145] Kreinovic sug­ gested the following query. Is black-body 3°K radiation really of cosmological origin or is it a mixture of radiation of many bodies as some physicists suggest? Of course, since the particular law is currently known only approximately, we cannot answer for sure. But in case we know precisely that the spectrum is subject to Planck's law, will it mean that the second case is disproved? In mathematical terms: (1) If Vu, > 0 (/0°° A(P)(eP" - l ) - 1 d/3 = (e**- - I ) - 1 ) and A(p) > 0, is A(/3) equal to 8((3 — PQ)! A positive answer to this question will follow, if one can prove that (2) If Vw > 0 (/0°° A(p)(e^u - l ) - 1 dp = 0) then A(p) = 0. Denoting B = In/?, W = lnw and turning to Fourier transforms, this can be reduced to the question (3) Is the Fourier transform of (exp(exp z) —1)_1 everywhere different from zero? As it appeared from the replies presented to [146] the conjecture in statements (2) and (3) had been discussed and proved in [208, p. 41] un­ der certain assumptions on A, e.g., if e~c^A(P) is integrable on (0,oo) for some c > 0. However, since the solution of (1) is not an ordi­ nary function, it seems appropriate to consider the problem from the standpoint of distribution theory. Our purpose is to establish a general

322

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

theorem [284] which includes the foregoing results as a particular case. Throughout the exposition we employ distributions of slow growth aris­ ing naturally in the generalized Fourier and Laplace transformations. Let t € E 1 be a one-dimensional real variable; S is the linear space of all functions (f) that are infinitely smooth and are such that, for any m > 0, k > 0, H m ^ ™ ^ / ) = 0.

(5.83)

The elements of S are called test functions of rapid descent. If <j> is in S, every one of its derivatives is again in S. A sequence of functions ipn £ S is said to converge in S, if for each set of non-negative integers m and k the sequence { l i l " 1 ^ ' ^ ) } converges uniformly over all of Ei. A distribution / is said to be of slow growth if it is a linear functional on the space S. Such generalized functions are also called tempered distributions. The space of all distributions of slow growth is denoted by S' and (/, <j>) is the value of the functional / applied to £ S. The support of a test function (j>(t) is the closure of all points where <j>{t) is different from zero. Two disributions / and g are said to be equal over an open set G if

(f,) =

M)

for every test function (t) whose support is contained in G. The sup­ port of a distribution / is the smallest set outside of which / equals zero. If a set X contains the support of a distribution / , it is said that / is concentrated on X. We denote by S + the space of all functions denned on t £ (0, oo) which are infinitely differentiate and satisfy (5.83), for t —v +00, and S'+ is the space of all tempered distributions concentrated on (0,oo). Now we introduce the following: Definition 5.1. Let <j> £ S + , / £ S'+. Then . for /A /A

= /, = 0,

A < t < co, t < A.

(5.84)

5.6. INTEGRAL EQUATION

323

T h e o r e m 5.22. The equation {f(t),4>(t,u))=4>(to,u), has a unique solution

(0 < u 0)

(5.85)

f(t) = 6(t - t0) if the following conditions are satisfied: (i) oo

(t,w) = '£ane-"x"t,

t>0,

(5.86) (5.86)

n=l

with positive parameters an, A„ such that an+i < an,

A„ < A n+ i,

Jirr^—
(ii) e~ctf(t) e S'+? for some c> 0; (iii) the Laplace transform F(p) = (f(t),e~pt) real half-axis c < p < oo.

is non-negative on the

PROOF. Eq. (5.85) can be written as A lim + {/ A ,0)

= (t0,uj)

and, by virtue of hypothesis (i), series (5.86) converges on any interval 0 < A < 2 < o o i n the sense of S. Hence oo

lim + ) = lim £ an(fA(t), e"^"') = (t0)u).

A—+0

A—^0"*"

(5.87)

n=i

Considering the series in (5.87) and taking into account (iii), one con­ cludes that, for A sufficiently small and LJ > c/\\, all its terms are non-negative and continuous with respect to u. Since the sum ( / A J ^ ) is also continuous, this series converges uniformly and oo

oo

£ a„ lim + (/A,e- uA »') = £ a„(/,e- A »') = #<„,«). n=l

A

-0+

n=l

Thus, oo

Z anF(uj\n) = <j>(t0,u). n=l

(5.88)

324

5. ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

The function F(p) of the complex variable p is analytic in the halfplane Rep > c as the Laplace transform of the distribution f(t) and, in view of uniform convergence, the left side of (5.88) is analytic for ui > c/Ai. Since <j>(to,w) is analytic for all u > 0, Eq. (5.88) is continued analytically onto (0,oo). Moreover, inasmuch as the right side of oo

£ an(F(tj\n)

- e-"A"<°) = 0

n=l

exists also for u> = 0 and the convergence in this relation is uniform, one may differentiate it and approach w to zero: oo

£ an\kn(F^(0+)

- H o ) * ) - 0,

k = 0,1,... .

n=l

Therefore fW(0+) = (-
F{p) = e-^,

Rep>0,

and

(/(0,e-*') = <
£ anF(uj\n) = 0 n=l

and hence F^(0+) = 0,k>0. The result F(p) = 0 implies f(t) = 0. Obviously, series (5.86) transforms into (eut —1)_1 when an = 1, An = n and t, u> > 0.

CHAPTER 6

Coexistence of Analytic and Distributional Solutions for Linear Differential Equations

We have mentioned already that recently there has been considerable interest in problems concerning the existence of solutions to linear ODE and FDE in various spaces of generalized functions due to increasing applications of distribution theory in many important areas of theoret­ ical and mathematical physics. Since integral transformations create close connections between entire and generalized functions, a unified treatment may be used in the study of both distributional and analytic solutions to some classes of ODE and FDE. This approach is employed here to explain the observation of some authors, in particular Littlejohn and Kanwal [177], on striking similarities between distributional and analytic solutions of linear ODE and FDE. Theorems are proved on the existence of finite-order distributional, rational, and polynomial solutions of linear ODE, with applications to important classical equa­ tions. We also investigate distributional solutions presented as infinite series of the delta function and its derivatives. Particular attention is given to the confluent hypergeometric equation. Existence and nonexistence theorems in spaces of infinite-order distributions are obtained for linear equations with polynomial coefficients and used to explore their entire solutions. These problems were studied by Wiener and Cooke in [294] and by Wiener, Cooke, and Shah in [295]. The variable t is real in the case of distributional solutions and complex for analytic solutions. 325

326

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

1. Distributional, Rational, and Polynomial Solutions of Linear ODE Littlejohn and Kanwal [177] investigated distributional solutions of the confluent hypergeometric differential equation and presented some interesting glimpses into the general hypergeometric equation as well. Thus, it is easy to verify that 6(t — 1) satisfies the equation t(l - t)x"(t) + (1 - 3t)x'(t) - x(t) = 0 and that (1 — t ) _ 1 is its classical solution. These functions exhibit in­ triguing similarities: (1—t)_i has a pole of order 1 and the distributional solution 6(t — 1) also is a simple pole. Furthermore, 6'(t — 1) — S"(t — 1) is a distributional solution of the equation *(1 - t)x"(t) + (1 - 5t)x'(t) - ix(t) = 0 and (1 + t)(l — t)~z is its classical solution. Again, we find that both these solutions have a pole of order 3. The following theorem shows that these features are not incidental. It establishes necessary and sufficient conditions for the simultaneous existence of solutions to linear ODE in the form of rational functions and finite linear combinations of the Dirac delta function and its derivatives, that is, m

x = £ xk6W(t),

xm + 0,

(6.1)

fc=0

where m is called the order of the distribution x(t). Theorem 6.1. If the equation

E
(6.2)

with polynomial coefficients qi(t) admits a rational solution m

x=J2(-l)kk\xkrk-\

xm^0

(6.3)

then it also has a distributional solution (6.1) or order m. Conversely, if (6.2) admits a distributional solution (6.1) of order m, then there

6.1. DISTRIBUTIONAL, RATIONAL, AND POLYNOMIAL SOLUTIONS

327

exists a polynomial q(t) such that the equation

iqi(t)x^i\t)

= q(t)

(6.4)

has a solution (6.3). P R O O F . First assume that (6.1) is a solution of Eq. (6.2). Then the (generalized) Laplace transform L[x] = F(p) of (6.1) satisfies the equation n

- > ) == 0. - > This implies that (6.5) admits a polynomial solution i=0 ■ ( -

(6.5)

m

F(P) = E *kPk

(6.6)

it=0

since JL[6W(t)) =pk. Settings > 0 and applying the right-sided Laplace transformation to (6.5) yields the equation

E(-ir i ft(-% ( n -°(s) =

(6-7)

where y(s) — fL[F(p)] and q(—s) is a polynomial whose coefficients include certain derivatives of F(p) at p = 0. The substitutions s = — t and y(s) = x(t) reduce (6.7) to (6.4). Since k\ s~k~l is the Laplace transform of p , we conclude that (6.3) is a solution of (6.4) On the other hand, if (6.3) is a solution of (6.2), then the func­ tion y(s) = x(—s) satisifies the homogeneous equation corresponding to (6.7). This means that Eq. (6.5) has a polynomial solution (6.6) which, in turn, proves that (6.1) is a solution of Eq. (6.2). ■ REMARK 1. If N is the highest degree of the polynomials qi(t), then the degree of q(t) in (6.4) does not exceed N — 1. EXAMPLE

1. It has been proved in [284] that Bessel's equation t2x" + tx' + (t2 - u2)x = 0

has a distributional solution (6.1) of order m iff

u2 = {m + \f

328

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

and it is given by the formula [m/2]

x = C £ ( m ~*)4-*«< m - 2 t >(<).

C = const.

k=Q.

Indeed, the Laplace transform F(p) of solution (6.1) satisfies the equa­ tion (p2F)" - (pF)' + F" -u2F

=Q

differentiating which k times at p = 0 leads to the relations \v2 - (k + l)2]Fk = Fk+2, [y2 - m2)Fm_x = 0,

k=

0,...,m-2

\v2 - (m + l)2]Fm = 0,

(Fk = F^k\0) = k\xk) the last of which has a nonzero solution Fm. Substituting it in the foregoing equations enables us to find all Fk (k < m). There exist constants CQ and c\ such that the equation t2x" + tx' + (t2 - v2)x = c0 + c\t has a solution

x=

[ c

z\-i)H-k{m~k)lt-m-l+2k

Theorem 6.1 describes the relationship between distributional and rational solutions of linear ODE. The proof makes use of the fact that the existence of a distributional solution to Eq. (6.2) implies that (6.5) has a polynomial solution. This can also be used to establish a corre­ lation between the existence of polynomial and distributional solutions to linear ODE. This direction is of considerable interest for many im­ portant equations of mathematical physics which admit polynomial or distributional solutions. Theorem 6.2. // Eq. (6.2) with polynomial coefficients qi(t) of the highest degree N admits a polynomial solution, then Eq. (6.5) has a

6.1. DISTRIBUTIONAL, RATIONAL, AND POLYNOMIAL SOLUTIONS

329

finite-order distributional solution. Furthermore, there exists a polyno­ mial q(t) of degree not exceeding TV — 1 such that the equation

t
= q(t)

(6.8)

has a rational solution (6.3). PROOF. Assume that Eq. (6.2) written in terms of variables p and F, tqi{P)F^-i\p)

= 0,

(6.2')

i=0

has a polynomial solution (6.6), with xm ^ 0. Since (6.6) is the Laplace transform of (6.1), then

£*(^)[(-*r ! y(s)] = o, the Laplace-transformed equation of which is (6.2'), has a solution m

y(s) = £

xk6^(s).

The substitutions s — —t, y(s) = x(t) and the formula $(*)(-*) = (_i)*tf(*)(i) show that the equation 71

gEft» ( -•1)(| ) (**"**) == 0° i=0

("

•**) =

(6"5')

admits a distributional solution

s=E(-i)*s** (i) (0>

Xm T^O.

k=0

Eq. (6.5') coincides with (6.5) written in variables t and x. Setting p > 0 and applying the right-sided Laplace transformation to (6.2') yields Eq. (6.8), where the coefficients of the polynomial q(t) depend on the values of F(p) and its derivatives at p = 0. Since &\p"] = fc! f_i_1, the rational function (6.3) is a solution of (6.8). ■

330

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

EXAMPLE 2. The equation PF'(p)

= F(p)

has a solution F = p and is the Laplace-transformed relation of tx'(t) =

-2x{t).

Hence, the latter equation admits a distributional solution x = 6'(t). It has also a rational solution x = t~2. The equation tx'(t) =

-x(t)

has a distributional solution x = 8(t) and a rational solution x = t Therefore, the equation

.

(*5-0(4+1)—■ that is, t2x"(t) + tx'(t) - x(t) = 0 has three types of solutions: polynomial x = t, rational x = t~ , and distributional x = 8(t). 2. Application to Orthogonal Polynomials It has been proved in Theorem 5.10 that if the equation

ttiqi(t)x^(t)

=0

(6.9)

»=0

with coefficients qi(t) € C m and qn(0) ^ 0 has a solution (6.1) of order m,then E ( - i y f t ( 0 ) ( m + t)l = 0.

(6.10)

i=0

Conversely, if m is the smallest non-negative integer root of Eq. (6.10), there exists an m-order solution (6.1) of Eq. (6.9). Assume now that the coefficients qi(t) in (6.9) are polynomials or holomorphic functions in the neighborhood of t = 0 and denote Qi{t) =

fq^t).

6.2. APPLICATION TO ORTHOGONAL POLYNOMIALS

331

The differential equation

t Qi (-~ ) [ p * > ( p ) ] i=0

= 0

(6.11)

\

is obtained by applying the Laplace transformation to (6.9). Hence, we can formulate the following theorem which finds useful applications in the theory of orthogonal polynomials. T h e o r e m 6.3. Assume that the coefficients g,-(<) are polynomials or holomorphic functions in the neighborhood of t = 0 and g„(0) ^ 0. If Bq. (6.11) has a polynomial solution (6.6) of degree m, then rela­ tion (6.10) takes place. Conversely, if m is the smallest non-negative integer root of (6.10), there exists a polynomial solution to (6.11) of degree m. REMARK 2. The condition qn(0) ^ 0 is equivalent to Q^(0) ^ 0. T h e o r e m 6.4. The equation t2x" + 2tx' - [t2 + v{y + l)]x = 0

(6.12)

has an m-order solution (6.1) if and only if i/(i/ + l) = m ( m + l ) .

(6.13)

It is given by the formula _ K?] (-i)*(2m-2lb)! {m_2k) m W - to 2 £! (m - *)! (m - 2k)\° ' ^ V whose coefficients coincide with the corresponding coefficients of the Legendre polynomial Pm(t). X[t)

PROOF. Eq. (6.10) corresponding to (6.12) is (m + 2)(m + 1) - 2(m + 1) - u{u + 1) = 0 and coincides with (6.13). Hence, condition (6.13) is necessary and sufficient for the existence of an m-order solution to (6.12). Substitut­ ing (6.1) in this equation and taking into account the formulas tS(k+l)(t) = -(k + tm

k+2

\t)

l)5^(t),

= (k + 2)(k + l)8W(t),

332

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

gives m

m—2

£ [k(k + 1) - 1/(1/ + l)]*t*<*>(<) - £ (* + 1)(* + 2)xk+26^(t)

=0

fc=0 fc=0

whence [v(v + 1) - fc(* + l j j n = - ( * + l)(fc + 2)a*+a, * = 0,..., m - 2 [1/(1/ + 1) - m{m - l)]x m _i = 0, [v{y + 1) — m(m + l)]x m = 0. If (6.13) holds true, we choose xm ^ 0 and successively find all xk (k < m). We have x m _i = 0 and [m(m + 1) - *(* + l)]Fk = -Fk+2,

Fk = k\ xk

and multiplying these relations for k = m — 2, m — 4 , . . . , m — 2j, we get 2jj\ (2m - l)(2m - 3) • • • (2m - 2; + l)F m _ 2 i = ( - l ) j F m . From here, m 2j

^ -

(-l)im!(2m-2j)! (2m)!j!(m-i)! ' m '

or Xm 2j

~

(-lV(m!) 2 (2m - 2j)\ (2m)! j ! (m - j)l (m - 2§)fm'

(6.15)

We can require that the coefficients xm_2j of the distributional solu­ tion x(t) coincide with the corresponding coefficients of the Legendre polynomial Pm(t). Indeed, applying the Laplace transformation to (6.12) produces the differential equation (p2 - l)F" + 2pF' - v(y + \)F = 0 for the Legendre polynomials. The usual way of finding their coefficients is by means of Rodrigues' formula [169] m 1 dm 2 , (t ■- l ) m . Pm(t) = m 2 m\dtmK

333

6.2. APPLICATION TO ORTHOGONAL POLYNOMIALS

It is easy to see t h a t here the coefficient xm of tm is (2m)! Xm

r«i^ (6 16)

= 2 ^ "

-

Therefore, equating xm in (6.15) to the value (6.16) yields the coeffi­ cients _ Xm—2j

(_1)i(2m-2j)! 2 j\(rn-j)\(m2j)\ m

of the distributional solution (6.14), where [m/2] denotes the largest integer < m / 2 . Furthermore, there exist constants CQ and c\, such that the equation t2x" + 2tx' - [t2 + m(m + l)}x = c 0 + erf has a rational solution

x(t) =

["!/?] iyi (2m-2k)\ (2m-2fc)!

+2k 1+2k

t02™k\(m-k)\

T h e o r e m 6.5. The equation tx' + (j

+ v+ 1j x = 0

(6.17)

/ia5 an m-order solution (6.1) «/ and on/y z/ ^ = m.

(6.18)

This solution is given by the formula

x(t) =

[m/2] / _ -i\knm-2k [W2] i ) * 2 - " m !\ (

to

k\{m-2k)\

W

whose coefficients coincide with the corresponding Hermite pohjnomial Hm{t).

' coefficients

of the

334

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

PROOF. In the case of (6.17), Eq. (6.10) takes the form (6.18), which implies that (6.18) is a necessary and sufficient condition for the exis­ tence of an m-order solution (6.1) to (6.17). Substituting (6.1) in (6.17) leads to the relations 2(v - k)Fk =-Fk+2, fc (i/ - m + l)F m _! = 0 ,

= 0,...,m-2 (i/ - m)Fm = 0

for the coefficients Fk = k\xk. If v = m, we choose Fm ^ 0 and find -2k Xm •Em~2k

(-l)*m> ( _ ! ) * m! = Xm = 4*k\(m-2k)\ 4*&! (m --2*)! - xm.

(6 20)

-

The coefficients xm-ik of the distributional solution x(t) can be chosen to coincide with the corresponding coefficients of the Hermite polyno­ mial Hm(t) denned by the formula [169] Hm(t)

m : = ("- l ) e '

idm

-t 2 m

dt

It is easy to see that the leading term of Hm(t) is (2t)m. setting xm = 2 m in (6.20) yields the coefficients Xm 2k

~

Therefore,

_ (-l)*2 m - 2 *m! " Jfc!(m-2A)!

of (6.19). They are identical to the coefficients of Hm(t) because apply­ ing the Laplace transformation to (6.17) yields the differential equation F" - 2pF' + 2uF = 0 for the Hermite polynomials. Furthermore, there exist constants CQ and c\ such that the equation

ta'. + (t* (— + m + 1J\ x = c0 + c\t

tx + I — + m + 1 I x = c0 + c\t has a rational solution

x{t)~-= 2 m ro! r-m- 1

[m/2]

E *=0

jfc!

'

6.2. APPLICATION TO ORTHOGONAL POLYNOMIALS

335

where the sum on the right is the Taylor sum of order [m/2] of the function el / 4 . ■ T h e o r e m 6.6. The equation t2x" + 3tx' - (t2 + v2 - l)x = 0 2

(6.21) 2

has an m-order solution (6.1) if and only if v = m . This solution is given by the formula m [m/2]

*(*) =

. (m — k

£(- -iy k\ (m —2fc)! y k=0

-2i^(m-

"»)(*)

(6.22)

whose coefficients coincide with the corresponding coefficients of the Chebyshev polynomial Tm{t) = cos(marccosi). PROOF.

In the case of (6.21), Eq. (6.10) takes the form (m + 2)(m + 1) - 3(m + 1) - (v2 - 1) = 0,

that is, v2 = m 2 , which implies that this condition is necessary and sufficient for the existence of an m-order solution (6.1) to (6.21). Sub­ stituting (6.1) in (6.21) gives m

£ [(* + 2)(fc + 1) - 3(* + 1) - v2 + l]x t «W(f) i=0

- "Z(k + 2)(k + l)xk+26(k\t) fc=0

that is, m

.

D (k2 - v2)xk6^{t)

m—2

- £ (* + 2)(* + l ) i w « ( " ( t ) = 0.

Therefore, (m2 - i/2)xm = 0,

[(m - l ) 2 - !/ 2 ]i ra _i = 0,

(ifc2 - u2)xk = (k + \){k + 2)xk+2, 2

k = 0 , . . . , m - 2.

2

If v = m , we choose z m 7^ 0 and from the relations (m 2 - fc2)^ = -Fk+2,

k= Fk = k\xk

m-2,m-i,...,m-2j

= 0,

336

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

find (-l)jm(m-j-l)!_ m 2

=

,

.

(6<23)

* " ' 4ii!(m-2j)! ^ The constant be selected so that the coefficients xm_2_,- of the distributional solution #(£) coincide with the corresponding coefficients of Tm(t). Indeed, if m is even, we take 2j = m and get x0 = 2 1 - m ( - l ) m / 2 x m . On the other hand, x0 = Tm(0) = ( - l ) m / 2 , hence, r

= 2m_1

and substituting this value in (6.23) yields the coefficients of (6.22). They are identical to the coefficients of Tm{t) because applying the Laplace transformation to (6.21) produces the differential equation (p2-l)F"+pF'-m2F = 0 for the Chebyshev polynomials. In the case of odd m, the same result follows by employing the derivative T^(0). Furthermore, there exist constants Co and C\ suth that the equation t2x" + 3tx' - (t2 + m2 - l)x = c0 + cit has a rational solution p»—2*

■M-TS*^©' k=0

T h e o r e m 6.7. Bessel's equation of imaginary argument t2x" + tx' - [t2 + (p + l)2]x = 0 has an m-order distributional solution (6.1) if and only if {u + l)2 = (m + l)2.

(6.24)

337

6.2. APPLICATION TO ORTHOGONAL POLYNOMIALS

This solution is given by formula x(t) =

^(-l)k(m-k)\nm_2k,2k)

t0

W

kl(m-2k)\

'

(6.25)

whose coefficients coincide with the corresponding coefficients of the Chebyshev polynomials of the second kind Um(t). PROOF. Substituting (6.1) in (6.24) yields the relations [(m + l ) 2 - (u + lf\xm

= 0,

[m2 - [y + l) 2 ]x m _i = 0,

[(k + l) 2 - (j/ + l)2]x* = (fc + l)(fc + 2)x*+2,

fc=0,...,m-2. 2

2

If (i/ + l) = (m + l) , we take xm ^ 0 and find (-l)*(m-Jfc)!

I

- " = Uum-M)!*-

,„„„, <6 26)

'

The differential equation (p2 - 1)F" + ZpF' - v{y + 2)F = 0 for the polynomials Uv(p) is the Laplace-transformed relation of (6.24). These polynomials are generated by the expansion [169] —±—= tum(t)xm, 1 - Ltx 2+ xz m=0 hence, Um(0) = ( - l ) m / 2 , for even m. Putting k = m/2 in (6.26) gives x 0 = (-l) m / 2 2- m x m = ( - l ) m / 2 , whence xm = 2 m . This value, together with (6.26), completely deter­ mines the coefficients of solution (6.25). Also, there exist constants Co and c\ such that the equation t2x" + tx - [t2 + (m + l) 2 ]x = c0 + cxt has a rational solution m-2k

■w-E^G)'

338

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

Theorem 6.8. The equation t2x" + 2(1 - j)tx' - [t2 + 2 7 + 1/(1/ + 2 7 + l)]x = 0

(6.27)

has an m-order distributional solution (6.1) if and only if i/(i/ + 2 7 + l) = m(m + 27 + l).

(6.28)

This solution is given by the formula _ K?) ( - i ) * 2 m - « r ( m + A - t ) {m_m , {th (6 29) ~h k\{m-2k)\T{\) whose coefficients coincide with the corresponding coefficients of the Gegenbauer polynomial C^(t), where F denotes the gamma function and X{t)

A = 7 + ~ > 0.

(6.30)

PROOF. Substituting (6.1) in (6.27) gives (m - v)(m + v + 27 + l)a;m = 0, (m - 1 - v)(m + v + 27)x m _i = 0, (1/ - *)(* + 1/ + 2 7 + 1)F* = -Fk+2, (Fk = k\xk, k=0,...,m-2). If (6.28) holds true, we can take xm ^ 0 and find all xk (A; < m) from the equations 2(2m + 2A - 2)F m _ 2 = - F m , 4(2m + 2A - 4)F m _ 4 = - F m _ 2 , . . . , 2fc(2m + 2A - 2fc)Fm_2* = - F m _ 2 t + 2 , whence (-l)km\F(m + \~k) k ~ ~ 4 k\(m-2k)\r{m + X)Xm' where A is defined by (6.30). Again, we can normalize xm so that the coefficients xm-ik coincide with the coefficients of the Gegenbauer polynomials C^(t) generated by the expansion [143] Xm 2k

(l-2tx + x2)-X=

00

E * m=0

m

.

6.2. APPLICATION TO ORTHOGONAL POLYNOMIALS

339

Differentiating with respect to t gives £ ^ x m=0

m

= 2Az(l - 2tx +

x2)~x-x

<" C»

OO

= 2A £ Ci + 1 (t)« m + 1 = 2A £ C2+\(*)«m=0

m=0

and

Repeatedly applying this procedure yields the equation d'C*, _ dP and since Cg = 1, we obtain dmCxm dtm

n,r(A+i)rA+i

T(A) o m r(m + A)

r(A) •

Therefore, we set

„mr(m + A) %m ~

m!T(A) and write the distributional solution in form (6.29), whose coefficients really coincide with the coefficients of C^(t), because applying the Laplace transformation to (6.27) leads to the differential equation (p2 - 1)F" + 2(1 + j)PF' - v{y + 2 7 + 1)F = 0

(6.31)

for the Gegenbauer polynomials. It remains to note that (6.31) gener­ alizes Chebyshev's equations of the first and second kind, which follow for 7 — —1/2 and 7 = 1/2, respectively. If 7 = 0, we get Legendre's equation. ■ Theorems 6.4-6.8 show, in particular, that the study of polynomial solutions to some important classes of linear ODE with several singular points can be reduced to a technically easier task (which is impor­ tant also in its own right) of exploring distributional solutions (6.1) of Eq. (6.9) with the only singular point t = 0. This approach will be

340

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

extended later to the study of analytic solutions of linear ODE and FDE. 3. Interesting Properties of Laguerre's Equation We continue the study of finite-order distributional and polynomial solutions for some classes of linear ODE. Theorem 6.9. The equation (t22 + - a)t + t)x' t)x' + [(1 -a)t ++ (u (v + + l)]x l)]x = 0,

aa >> - -l1

(6.32)

has an m-order distributional solution (6.1) if and only if (6.18) holds true. This solution is given by the formula

_ Am

*(*) =

k=0

(-l)*T(m + qa + 1) (-lfflm l) (k)() -fiW(t), fc!(m-ifc)!r(Jb + a + 1)

,

,

where T denotes the gamma function and the coefficients coincide with the corresponding coefficients of the Laguerre polynomial L^(t). PROOF. Eq. (6.10) corresponding to (6.32) is - ( m + l)! + (i/ + l)m! = 0, which shows that (6.18) is a criterion for the existence of an m-order solution (6.1) to (6.32). Substituting (6.1) in (6.32) gives m—1

m

£ (* + 1 + a)(k + l)xk+i6W(t) fc=0

+ £ (i/ - k)xk6W{t) = 0, A:=0

whence (u - m)xm = 0,

(y - k)xk = -(k + 1 + a)(k + 0 < k < m- 1.

l)xk+u

If v = m, we take an arbitrary xm ^ 0 and find (-l)m-km\r(m + a + l) k = TTT T^FT, TVxmk\(m- k)\T(k + a + l)

x

(6.34)

v

;

6.3. INTERESTING PROPERTIES OF LAGUERRE'S EQUATION

341

We can normalize xm so that the coefficients of the distributional so­ lution (6.1) coincide with the coefficients of the Laguerre polynomial L%Sf) defined by the formula [169] OO

= £ L°m(t)xm,

(i - x)-°-ie-t*m-*)

|*| < I.

m=0

At t = 0 we have the expansion OO

1

(1 - x)—

= £ L%(0)xm m=0

and differentiate it m times at x = 0. Consequently, „ r ( m + <* + l) r » m Lm(0) = = *° m!r(a + l ) On the other hand, from (6.34) we have _ ( - l ) m r ( m + q-|-l) Xo Xm ~ r(« + i) ' m and putting x m = (—l) /m! in (6.34) delivers the coefficients of (6.33). They coincide with the coefficients of L^(t) because applying the La­ place transformation to (6.32) produces the differential equation PF"

+ (a + 1 - p)F' + uF = 0

(6.35)

for the Laguerre polynomials. Furthermore, there exist constants Co and c\ such that the equation (t2 + t)x' + [(1 - a)t + (m + l)]x = c0 + cxt has a rational solution m

X(t):

k=o(m

T(m + a + l) -k)\T(k + a + l)

k_r

which is the m-th Taylor sum of the function t~m-1 (l+t)m+a, In the case a = 0, Laguerre's equation pF" - (p - 1)F' + mF = 0 is the Laplace-transformed relation of

(t2 + t)x' + (t + m + l)x = 0,

for |<| < 1.

342

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

which admits a distributional solution f--h*mi

■«-55^r*» and a rational solution

x(t) = ph)rk-l

= t-m-l(i + ty

Eq. (6.32) has two singular points, t = 0 and t = —1. For t = —1, Eq. (6.10) becomes m + l + a + u = 0ora = —v — m — 1. Let i/ = n be a non-negative integer, then the equation (t2 + t)x' + [{m + n + 2)t + (n + l)]x = 0

(6.36)

has two finite-order distributional solutions: one of order n with support t = 0, and the other of order m concentrated on t = — 1. Substituting in (6.36) the unknown solution n

*«(*) =

rt^W,

k=0

it is easy to find n

*»(*)

:

/m + n --*> 1 ^ W 7 1 1 t=o I m j K!

For the special case m = 0, we have *.(<) = t

^

-

This implies that Laguerre's equation corresponding to (6.35), pF" -(p + m + n)F' + nF = 0, has a polynomial solution F

n(P) = £

*=o V

m

(6-37)

6.3. INTERESTING PROPERTIES OF LAGUERRE'S EQUATION

343

and the equation PF"

- (p + n)F' + nF = 0

(6.38)

has a polynomial solution *=o*! which is the n-th partial sum of ep. Now, we recant Eq. (6.36) as [(* + l) 2 - (« + l)]z' + [(m + n + 2)(t + 1) - (m + l)]x = 0 and substitute in it the distributional solution m

x(t) = £ ^ ( i ) (* +1), which leads to the relations (m — m)cm — 0,

(m — A;)c<; = (k + l)(k — m — n)c).+i, (0 < k < m - l ) .

Letting cm = 1/m! yields

^)-£(-i)*fM+"-*)^r^, n 4=0 V and for the special case m = 0, we have and for the special case m = 0, we have

/

fc!

(6.39) x(t) = 8(t + l). (6.39) This implies that Eq. (6.38) has a solution This implies that Eq. (6.38) has a solution F(p) = epp. F(p) = e . Hence, we have shown that (6.38) has the two solutions F(p) = ep = ET=oPk/kl and its partial sum Fn(p) = E?=oP*/^!- This remarkable property of Eq. (6.38) has been first noted by Hoyt [124]. Leighton [170] has also discussed (6.38), and Newton [207] has extended these results to third-order equations with polynomial or analytic at t = 0 coeffi­ cients. Further generalizations for broader classes of linear ODE having

x(t)-- = 6{t + l).

344

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

££LoC**4 and Et=oC*f<: as solutions have been made by Zeitlin [305]. On the other hand, expanding (6.39) in a series oo

6(t + 1) = =

E

fc=0 fc=0

shows that the equation

Jbl

(6.40)

/C!

(i2 + t)x' + [(n + 2)t + (n + l)]x = 0 has a formal distributional solution (6.40) whose rc-th partial sum (6.37) is also a distributional solution Theorem 6.10. Eq. (6.9), with qt(t) G C m and qi(t)

pi(t) e C m

= (t- ayPi(t),

has the distributional solutions ««.(*) = E ^ *=o

^

K!

W

(6-41)

and

x(t) = 6(i - a) = E S r - ^ C )

(6-42)

i/ and only if

E(-1) ! '*!E fm ;=o

+

.,_i)am-i*,*-i=0,

(6.43)

l

i=o \ / (k = 0 , . . . ,m)

and E ( - a ) ! # ( a ) = 0,

(6.44)

i=0

uAene ^ ^ = af _i) (0)/(fc - j)I. PROOF. Eq. (6.9) and the truncated equation n

m m

i=0


Ef'* E f *(( 0 W E ? a t * = 0 i'=0

(6.45)

6.3. INTERESTING PROPERTIES OF LAGUERRE'S EQUATION

345

have the same set of m-order solutions (6.1). Let F(j>) be the Laplace transform of (6.1); then the existence of an m-order solution (6.1) to (6.45) is equivalent to the existence of a polynomial solution of degree m to the equation n

m

£(-!)•' £ ( - l ) W > ) ( l + f c ) = 0, i=0

(6.46)

fc=0

which is obtained by applying the Laplace transformation to (6.45). Differentiating (6.46) j times at p = 0 leads to the equation

t(-mt(-i)k(i i=0

k=0

+J + k

. )mFk+j = o

\

l

I

for the coefficients Fk = F^k'(0) — (—a)k of the distributional solu­ tion (6.41). Since Fk = 0 for k > m, we have m—k

n

E(-i)'^E(-i)J'

/,' i „' r

.

t\

UjFk+J = o,

(* = o,...,m)

i=o j=0 V * / and make the substitution k + j —+ m — j , to obtain

t(-m

I? (-l)m-k-j (m

+ l

~ A qi.m-k-jFm-j = 0.

;=o i=o V % I Changing m - k to k and putting Fm_j = (—a)m_J yields (6.43). Fur­ thermore, we substitute (6.42) in (6.9) and get i i i tt Pi(t)8®(t-a) t (t-a) t'(t p-a) - i(t)8^(t-a)

= =Q0,)

i=0 i=0

whence n

i i -a)- = £(-l)-1)H i\t p\fi(t)8(t-a) = 0. Pi(t)6(t-

£(-

i=0

i=0

Expanding fpi(t) in the neighborhood of t = a, fPi(t)

= o'pf(o) + [a*p-(a) + za!'-1p!(a)](* - a) + • • •

and observing that *!Pi(a) = ( f t a ) ,

( t - a ) ' ' f i ( t - a ) = 0,

> = 1,2,...

346

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

produces (6.44).

■

6.1. Assume that the coefficients g,(<) are polynomials and q\ (a) = 0, for j < i. Then differential equation (6.11) admits the solutions fc L-ap) COROLLARY

=E

^m(p) =

fc=0

it!

and

if and only if hypotheses (6.43) and (6.44) hold true. z/ COROLLARY 6.2. Assume that in the equation tqi(t)x'(t) + q0(t)x(t) = 0

(6.47)

m

qo{t), <7i(0 £ C and 51(a) = 0. Eq. (6.47) admits distributional solu­ tions (6.41) and (6.42) if and only if qok = (m + l)qlk ~ a

E oPqij

(6.48)

(k = 0, . . . , m ) and q0(a) = aq[(a), tuftere g0- = q\j)(0)/j\,

(6.49)

i = 0,1-

EXAMPLE 3. Consider the differential equation t(t + l)x'(t) + q0(t)x(t) = 0,

(6.50)

where qo(t) is a polynomial. For (6.50) we have a = -l,

qi(t) = t + l,

gio = l,

gn = 1,

qij=0,

and conditions (6.48) become q00-m

+ 1,

901 = m + 2,

90* = 0,

2 < fc < m.

j>2

6.4. THE HYPERGEOMETRIC AND OTHER EQUATIONS

347

Condition (6.49) is q0(-l) = - 1 . Hence, (6.50) has distributional so­ lutions (6.37) and (6.40) if and only if qo(t) = n + l + (n + 2)t + rn(t), where rn(t) is any polynomial of degree greater than n containing a factor t + 1 (because r„(—1) = 0). 4. The Hypergeometric and Other Equations Theorem 6.1 opens an easy way of constructing examples of linear ODE having finite-order distributional solutions. Indeed, take a linear ODE with constant coefficients (D-k1)(D-k2)---(D-kn)y(s)

= 0,

where some of the numbers k{ are negative integers and D = d/ds, and make the substitutions es = t, y(s) = x(t). Then Euler's equation

(•!-*■) ( 4 - * ) •■•(«*-*.)*w-o has a rational solution (6.3) and hence, admits also a finite-order dis­ tributional solution. It has been proved in Theorem 6.1 that the existence of a rational solution (6.3) to Eq. (6.2) implies the existence of a distributional so­ lution (6.1). A question arises: when is the converse true? In other words, when is q(t) = 0 identically in (6.4)? Theorem 6.11. If Eq. (6.2) with polynomial coefficients qi(t) admits a distributional solution (6.1) of order m and if degqi(t) < n — i,

i = 0 , . . . ,n

(6.51)

then it also has a rational solution (6.3). P R O O F . The Laplace transform F(p) of solution (6.1) to (6.2) sat­ isfies Eq. (6.5). Setting p > 0 and applying the right-sided Laplace transformation to (6.5) yields Eq. (6.7), with q(s) = 0 identically. In­ deed, by virtue of (6.51), the order of the differential operator q^—d/dp) in (6.5) does not exceed n — i. Hence, the function pn~'F(p) and all its derivatives of orders up to and including degg,- — 1 equal zero at p = 0. ■

348

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

EXAMPLE

4. The equation t2x'" + (at — 6)x' + ax = 0,

a = const.

admits a distributional solution x = 5(t) and satisfies (6.51). Hence, it has also a rational solution x = t . The equation tx" + (t + 3)x' + x = 0 admits a distributional solution x = 8(t) + 6'(t) and also a rational solution x = t~l — t~2. REMARK 3. The conditions of Theorem 6.11 are sufficient but not necessary for the existence of a rational solution to Eq. (6.2). Thus, it has been shown above that the equation

(t2 + t)x' + (t + m + \)x = 0 admits a rational solution (6.3) but does not satisfy (6.51). The class of equations tx^(t)

+ £ afflxt^it)

= 0,

at E C m + n -''

(6.52)

has been studied in Theorem 5.11, where it is proved that (6.52) has a distributional solution of order m if and only if ai(0) = m + n and there exists a nonzero solution to the system (ai(0)k\ - {k +

l)\)xk+l_n m+n m+„

min(j,7i) min(j»

.

+ j=2 E^+i-n t'=l E (-lrM^oXfc + i--0! = 0 i=\ j=2 (k — 0,... ,m + n — 1). Since the verification of this condition is often difficult, we consider the particular case Z(ait + bi)x^n-i\t)

= 0,

(6.53)

which admits simple methods of study and yet remains sufficiently im­ portant.

6.4. THE HYPERGEOMETRIC AND OTHER EQUATIONS

349

Theorem 6.12. Eq. (6.53) with constants a,-, &,-, and a0 = 1, &o = 0 has a finite-order distributional solution if and only if all poles pi of the function

R(p) = t(hP - (n - zKy1-'-1 / £ a^ i=0

/ :=0

are distinct, all residues ri = res p=Pi R(p) are non-negative integers, and the residues of the complex conjugate poles are equal. The solution is given by the formula " /d \r' x = Cli[ — -pi) 6(t), ,_o \dt )

C = const.

(6.54)

and its order is n

m = J2 r{. i=0

/ / a„ = 0, there exists also a solution

*- c fl(5-") V

(6 56)

'

PROOF. Eq. (6.53) has a finite-order solution if and only if the equa­ tion F'(p) - R(P)F(p)

(6.56)

obtained by applying the Laplace transformation to (6.53) admits a polynomial solution. Hence, the integral of R(p) must contain only logarithms of non-negative integer powers of real linear factors or irre­ ducible quadratic trinomials, which is ensured by the conditions of the theorem. Furthermore, (6.54) follows directly from (lnF)' = £ri(p-pi)-1.

(6.57)

350

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

Applying the Laplace transformation to (6.56), with p > 0, and taking into account an = 0, we get the relation

t(-l)"-i(aiS-bi)x^-i\s)

=0

i=0

which can be reduced to (6.53) by the substitution s — —t. Since £j[6] = 1 and £[1] = s _ 1 , expression (6.55) is a solution of (6.53). ■ REMARK 4. If an ^ 0, there exists a constant CQ such that (6.55) is a solution of the equation ±{ait + bi)x^-i\t)

= cQ.

i=Q

COROLLARY 6.3. The confluent hypergeometric equation tx" + (6 - t)x' - ax = 0 has a finite-order solution iff a and b are positive integers and b > a + 1. The solution is given by the formula [287] v 6-a-l

x-Cd°~l

a l

dt ~

(d

■

\dt - 1 )

6(t)

(6.58)

and its order is m = 6 — 2. There exists also a solution r1.

* = Cy—T(T--I)

dP-1 \dt ) 5. The equation [133] 5. The equation [133] tx" + ax1 + btx = 0, b^0 has a finite-order solution iff the coefficient a is a positive even integer. This solution is given by the formula EXAMPLE EXAMPLE

/ J2

c

\ (°-2)/2

+,

'- (dF J

*W

and its order is m = a — 2. There exists a constant Co such that x =C

(/ dJ22

[dl?

+

-2)/2 («-2)/2 \\ (a»)

1

r -i

6.4. THE HYPERGEOMETRIC AND OTHER EQUATIONS

351

is a solution to tx" + ax' + btx = Co. EXAMPLE 6. The equation [133] tx'" + (a + b)x" - tx' - ax = 0 has a distributional solution iff a is a positive integer and b is even positive. This solution is given by the formula daa~1l (d2 2 X = C~d - r ( d dta~*

1 \\

t/2 1 t / 2"_ 1

,,

U" ) ««

and its order is m = a + 6 — 3. There exists also a solution da-l

x = c-—r

a l

EXAMPLE

/ d2

N*/2-l

--1 2

dt ~ \dt 7. The equation [133] tx'" - (t + a)x" -{t-a-

r1.

j l)x + (t - l)x = 0

has a distributional solution iff a is a negative integer, a < —3. This solution is given by the formula -(a+3)/2

\\ -(«+3)/2 (/ dJ22 *(*) = c\dfi' - 1 ) and its order is m = —a — 3. There exists also a solution X -

i =

/ ,22 \ /
W*

/

-(a+3)/2 -("+3)/2

r1.

Theorem 6.13. Eq. (6.53) has a polynomial solution of degree m if and only if the following conditions hold true: (i) a,- = 0, i = n,n — 1,... ,n— N; a„_./v_i ^ 0 (ii) bi = 0, i = n, n — 1 , . . . , n — N + 1 (iii) bn_N/an-N-i = - ( m - N), for an integer N such that 0 < N < min(m,n — 1).

352

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

PROOF. Eq. (6.53) written in terms of the variables p and F is the Laplace-transformed relation of

xx-ir-v"-'') *'(*)

i=0

+ ( i X - l V W n - 0 + M)*"-'-1] x(t) = 0. (6.59) Hence, (6.53) has a polynomial solution of degree m if and only if (6.59) admits an m-order distributional solution (6.1). If an ^ 0, then t = 0 is a regular point of Eq. (6.59), and in this case (6.59) has no singular distributional solution. Therefore, the condition an = 0 is necessary for the existence of a solution (6.1) to (6.59). Assuming a„ = 0, we write (6.59) as (n-\

■ A

i »'w

n t* (E(-l),-l)'0£t -' 1) **(*) 'a.-*n"*'" V»=o -' fn-\

. ,

"

.

.>

i

1-11 + ( E (V- Ii W M "» --- i)*"~' O*"^" £(-i)'"Mnn_, *(*)==0o ++E(-i)'M - 1s(t) + \«=0

\i=0 »=o i=0 y/ which is of the form (6.9). Now, assume that (6.59) admits a distribu­ tional solution (6.1) of order m. If a„_i ^ 0, then according to (6.10), we have

( - l r - ' o n - l + (-l)nbn

- (-l)n-\m

+ 1 K _ ! = 0,

that is, bn —

-man_\.

If a n _i = 0 and an_2 ^ 0, then (6.59) takes the form

t2 (xi-iYaif-*-*) At) 2

(-l)''o,(n - i)*"-'-2 U(i) + ME(-lWn-«K""

+ (E(-i) , V-')^) = o,

6.4. THE HYPERGEOMETRIC AND OTHER EQUATIONS

353

and since t2x' and tx are both distributions of order m — 1, we conclude that bn = 0. Therefore,

t2 ixi-iYoit"-*-2) x'(t) (n-1

n-1

■1A

+ <E(-iNi-o^'^+EC-iyM"t'=0 \«=0

)x(t) = 0, 1

and the substitution y = tx changes this equation to

t (gC-iyM"-*-2) y'(t) + fBff(-l)*«.-(n - «' - 1)<"",_2 + X) (-l)'M"-^ 1 ) »(*) = 0, \z=0

>'=0

/

which is of form (6.9). Since i(<) is an m-order distribution, j/(i) is of order m — 1. Hence, by virtue of (6.10), we have (-l)"- 2 a„_ 2 + ( - 1 ) " - V i - (-l)"- 2 ma n _ 2 = 0, that is, bn-i = ~{m - l)a„_ 2 . Assuming ai = 0,

i = n,n — 1 , . . . ,n — N;

a n -iv-i ^ 0

implies &j = 0,

£ = n, n — 1 , . . . , n — N + 1

and -N-1 +1 /n-N-1

i1 i n , Ar +^ r^Vi)^.^ - ir-^- +"E (-i) M - - ) x(o=o.

*" [" £

. „

-l)'a,-t"-'- ■JV-

\

(•

'(')

t=0

+tN

/n-N-

\ \

E

i=0 i=0

(-l)'ai(n - i)* B -

i-AT-

n-AT

-' + E(- -i)V '=0 i=0

i-JV |

1 x(t) --= 0.

/

354

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

The substitution y = tNx reduces this equation to (n-N-\

\

t [ E (-lYa^-^j y'(t) + l"~z\-^i(n \

- i - N)^-"-1

+ "JZi-iybif1-*-") y(t) = 0,

«=0

:=0

/

and since y(t) is a distribution of order m — TV, we obtain, by virtue of (6.10), "

( - l ) " - " - 1 ^ ^ + (-iy-Nbn_N _ ( _ 1 ) n _ ^ _ 1 ( m _N

+ 1)fln _ Ar _ i = 0 )

or K-N = -{m - N)an_N-\.

(6.60)

This proves the conditions of the theorem are necessary. They are also sufficient because hypothesis (iii) ensures that Eq. (6.60), which represents relation (6.10) for (6.59), has a unique (hence, the smallest) non-negative integer solution m. ■ T h e o r e m 6.14. The differential equation (1 -

2 P

)F"(p) + [/3-a-(a

+ f3 + 2)p\F\p) + jF(p) = 0, (6.61)

where 7 is a parameter, has a polynomial solution if and only if 7 is of the form 7 = m(m + a + (3 + 1),

m = 0,1,2,... .

(6.62)

PROOF. Eq. (6.61) is the Laplace-transformed relation of t2x" - (a + 0-

2)tx' - [t2 + (a-P)t

+ (a + (3 + -y))x = 0.

This equation is of type (6.9) and it admits a finite-order distributional solution (6.1) if and only if the corresponding Eq. (6.10), (m + 2)(m + 1) - (2 - a - f3)(m + 1) - (a + /? + 7) = 0, that is, m(m + a + (3 + 1) - 7 = 0,

355

6.4. THE HYPERGEOMETRIC AND OTHER EQUATIONS

has a nonzero integer root m. This is so if 7 is chosen according to (6.62). A classical proof of this theorem, without the use of dis­ tributions, may be found in [259]. Eq. (6.61), with 7 defined by (6.62), is the differential equation for the Jacobi polynomials. ■ Theorem 6.15. The hypergeometric equation p{\ - p)F"(p) + [7 - (a + (3 + l)p]F'(p) - a/3F(p) = 0 (6.63) has a polynomial solution of degree m if and only if a = —m

or

f3 = —m.

This solution is given by the formula =l { - ^ P k , fc=o (7)fc«!

F(a,P,r,P)

(6-64)

where (A)0 = l,

(A)i = ^

^

= A(A + l ) - - ( A +

fe-l).

PROOF. Eq. (6.63) is the Laplace-transformed relation of t2x" - [t + (a + (3 - 3)]tx' + [(7 -2)t + (a(3 + l - a - f3)]x = 0. (6.65) Substituting (6.1) in (6.65) gives m

±\(k + a)(k + f3)xk -(k + 1)(* + 7)z*+1]5(*)(*) + (m + a)(m + f3)xm8(m\t)

= 0,

whence (m + a)(m + /3)xm = 0, (k + a)(k + P)xk -(k + l)(k + 7)x* + i = 0 ,

k = 0 , . . . ,m - 1.

The condition (m + a)(m + /?) — 0 enables us to choose Xm = ( a ) m ( / ? ) m / ( 7 ) m m !

356

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

and find the coefficients of (6.64). The substitution p = 1 - 2s in (6.61) changes the Jacobi equation to the hypergeometric equation *(1 - s)F"(s) + [a + 1 - (a + (3 + 2)s]F'(s) + m(m + a + /? + l)F(s) = 0 and leads to the important formula [259] for the Jacobi polynomials and leads to the important formula [259] for the Jacobi polynomials normalized by the condition PM{1) normalized by the condition

=

fm + a\

{ m m )'

e«(D = ( r)- ■ Theorem 6.16. The hypergeometric equation (6.63) has a distribu­ tional solution (6.1) of order m if and only if (i) 7 = m + 2 and (ii) a or (3 is an integer j — 1 , . . . , m + 1 . This solution written in terms of the variables t and x is given by the formula X{t)=

t

( m + 1 7 a W ) / r ( f c + 2-/?),

(6.66)

where a is either the single integer or the greater of the two integers satisfying (ii). PROOF. Substituting (6.1) in Eq. (6.63) written in terms of t and x leads to the relations (m + 2 - j)xm = 0, {k + 1 - a){k + 1 - p)xk = (m + 1 - *)**-i, k = 0,... ,m; x_\ = 0.

(6-67)

Since xm ^ 0, hypothesis (i) is a necessary condition for the existence of an m-order solution (6.1) to (6.63). The equation for XQ is (1 - a)(l - (3)x0 = 0,

6.4.

THE HYPERGEOMETRIC AND OTHER EQUATIONS

357

and if a ^ 1, /? / 1, then xQ = 0. Furthermore, if (ii) is not satis­ fied, (6.67) implies that Xk = 0, for all k = 0 , . . . , m. On the other hand, if a is an integer such that 1 < a < m + 1 and /? is not, then xk = 0,

k=

0,...,a-2

and 1 • (a + 1 - f3)xa = (m + 1 - a)xa..i, 2 • (a + 2 - /3)x a+ i = (m - a)x a , j • (a + ;' - /?)z a+ j_i = (m - a - j + 2)x a + i _ 2 , (0 < j <m + 1 - a). Multiplying these relations yields (6.66), with the exactness to a con­ stant factor, and this result remains valid also when both a and d are integers such that 1 < a,/? < m -f- 1, a > /?. ■ COROLLARY 6.4. Under the conditions of Theorem 6.16, i/ie hypergeometric equation has a rational solution

i

("=j_1(-i»'(m^r)*,t"*-'/r't+2-«-

T h e o r e m 6.17. T/ie hypergeometric equation (6.63) written in terms of t and x has a distributional solution of order m with support t = 1 if and only if (i) ai + (3 — j — m + 1 and (ii) a or ft is an integer j — 1,.,. ,m + l. This solution is given by the formula *(<) =

t fc=a-l

(-1)* (m \

+ l a

~)

TU-

K

**>(* - l ) / r ( * + 2 - /?), (6.68) )

I

where a is either the single integer or the greater of the two integers satisfying (ii).

358

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

P R O O F . The substitutions p = 1 - t, F(p) = x(t) change (6.63) to the hypergeometric equation

<(1 - t)x"(t) + [(a + (3 -

7

+ 1 ) - (a + (3 + l)t]x'(t) - af3x{t) = 0,

and it remains to apply Theorem 6.16 to the latter equation. For­ mula (6.68) follows from (6.66) and from the equality k k

-0 == ( - -l) # \t-

«<*>(! -

- ! ) ■

Under conditions (i) and (ii), there exists also a solution X(t)=

£ ( m + 1 7 a W - l ) - * - y r ( f c + 2-/?).B(6.69)

* =a _i \

m-k

/

I

EXAMPLE 8. For the equation [177] t{\ - t)x" + (1 - 3t)x' - x = 0, with a = /? = 7 = 1, the condition a + /? — 7 = m + 1 implies m = 0. Hence, there exists a distributional solution x{t) = 6(t — 1) and a rational solution x(t) = (t — 1) _ 1 . For the equation [177] t(l - t)x" + (1 - 5t)x' - Ax = 0, with a = /? = 2 , 7 = l, the same condition implies m = 2. For­ mula (6.68) yields the distributional solution x(t) = 8'(t — l) — 6"(t — l), and (6.69) provides the rational solution x{t) = (t-

I)" 2 + 2(t - I ) " 3 = (t + l)(t - I)" 3 .

It has been indicated above that applying the Laplace transformation to first and second-order equations of type (6.9), whose coefficients are such that (6.10) admits a non-negative integer root, generates the most important linear ODE for orthogonal polynomials. New classes of linear ODE with polynomial solutions may be produced by applying the Laplace transformation to higher-order equations (6.9), in particular, to

f,ti(ait + bi)x®(t)=0 i=0

6.5. THE CONFLUENT HYPERGEOMETRJC EQUATION

359

and

£ti(ait2 + bi)xW(t) = 0. i=0

This approach is based on Theorem 6.3. Of course, the parameters cij, bi should be selected in such a way that Eq. (6.10) has a non-negative integer root. This condition is very easy to satisfy. Indeed, it suffices to consider the non-negative integer m and all coefficients of (6.10), except one, as given numbers, and to find this unknown coefficient. For instance, recently Everitt and Littlejohn [76] have studied the Legendretype polynomials that satisfy the fourth-order equation (p2 - 1)2F<4> + 8p(p2 - 1)FW + (4a + 12)(p2 - 1)F" + SaF' + PF = 0, (6.70) which is the Laplace-transformed relation of + 8t3x™ + 2[(2a + 6) - t2]t2x" + 8(a - t2)tx' + (/? - t2)x = 0. Simplifying the corresponding Eq. (6.10), (m + 4)! - 8(m + 3)! + (4a + 12)(m + 2)! - 8a(m + 1)! + /?m! = 0 yields the criterion /? = -m(m + l)(m 2 + m + 4a - 2) for the existence of a polynomial solution of degree m to (6.70). 5. The Confluent Hypergeometric Equation Solutions of the form * = 5>,-$ w (t)

(6.71)

for the confluent hypergeometric equation tx" + (b- t)x' - ax = 0

(6.72)

have been presented in [177] and used to exhibit their interplay with re­ lated results in the theory of ordinary differential equations (ODE). For

360

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

instance, sums of some hypergeometric series have been found with the help of these distributional solutions. Another motivation for studying solutions of type (6.71) to ODE is given in the work of Littlejohn [176] who has shown that weight distributions for a certain class of orthog­ onal polynomials have the form (6.71) and simultaneously satisfy a system of ODE. The authors of [177] first found the formal distribu­ tional solutions of infinite order to (6.72) and then used the Fourier transformation to get for them a better expression as a sum of a clas­ sical function and a finite-order distribution. The approach, which is based on the direct application of the Laplace or Fourier transformation to the differential equation, yields these and more general results in a considerably simpler fashion. As shown in Corollary 6.3, for a, b G N and b > a, the distributional solution to (6.72) is of finite order. We shall illustrate the method for a, b g N and b < a. T h e o r e m 6.18. / / a and b are positive integers and b < a, then Eq. (6.72) admits a formal distributional solution »=a-l V ~ a + V P R O O F . Formally applying to (6.72) the distributional two-sided La­ place transformation L[x] = F(p) yields the equation -(p2F)'

+ bpF + (pF)1 -aF

= 0,

that is, (pi-p)F'

=

[(b-2)p-(a-l)]F,

with the general solution F(p) = Cpa-1(P-l)b-a-\

C = const.

The particular solution Fo(p) corresponding to C = (—l) a

l a+b

F0(p)=p -\l-p)- -

.

oo

= EP\ »=0

is (6.75)

Differentiating the series

(I-P)-1

(6.74) a_t+1

bl
6.5. THE CONFLUENT HYPERGEOMETRIC EQUATION

361

a — b times and multiplying by p " - 1 gives i+b

FO(P) = £ [J

h)p

-\

which is changed to i=a-i \* - a + 1/ by the substitution i + b — 1 —> z. This proves the theorem. The distribution (6.58), ,b-a-l

x-Cda~l a l (d dt ~ \dt which is a solution of (6.72) for a, b £ N and b > a, is a continuous linear functional on the space of all infinitely differentiable functions. By considering certain test functions, it is easy to see that this is not so for the formal distributional solution (6.73). It is natural to ask whether (6.73) has any extensions to the space of distributions of slow growth, that is, continuous linear functionals on the space of infinitely smooth functions ^k'(t) = 0 as t —y ±oo, for any i > 0, k > 0). The answer in affirmative was given in Theorem 2 [177], and the following theorem proves the same result more simply. T h e o r e m 6.19. Assume a, b are positive integers and b < a, then

1 "ir^w (-D-1b gE I(r!:!W) ("- 1 ) -

i=a-\ \i-a + lj = E t1 T ^f-^e'Hi*) f-^e'Hi-t)/{a ■«) Ao -- b6 --01 - i)\

■S('T

-g(:

:*:2>

( , ,

«

(6.76)

1

where H{—i) = 1 — H(t) and H(t) is the Heaviside function which equals 1 for t > 0, and 0 for t < 0.

362

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

PROOF. Putting C = 1 in (6.74), consider the function

F,{p)=Pa-\p-l)b-a-1 and compare it with (6.75), whence

Flip) = ( - l r ^ o G O .

(6.77) (6-77)

Furthermore, P-M(P

-1) + IP"1 = £ ( a ~ *) (P -1)''

and la-l\

In the second sum on the right, we make the substitution b+i—a—1 —► i, then expand (p — 1)' into powers of p, and use the identity a— 1 ( ) \j -b + a + l)

to obtain

Thus,

( a- 1 \ - 2

u g<-*«.-;-J-

-i]

fa-i2\ \b-i-2J'

*«-g^*s(::.':.V

Since l/(p — l ) a

b 1+1

is the bilateral Laplace transform of

- * a - 6 - V i 7 ( - * ) / ( a - 6 - i)! (we must reject the functional ta~l>~letH(t)/(a — b — i)\ as being un­ bounded on some functions of rapid decay) and p* = £ [8^ (<)], it follows that — F\(p) is the Laplace transform of the right-hand side of (6.76).

363

6.5. THE CONFLUENT HYPERGEOMETRIC EQUATION

Taking into account (6.77) and the fact that F0(p) is the formal Laplace transform of (6.73) concludes the proof. REMARK 5. The right-hand side of (6.76) consists of a locally integrable function concentrated on the left semi-axis and of a distribution of order m — b — 2. Another way to find this distributional part is by writing = P»-- = ( , F 1 Cp)=^(l-i) Flip)

,

then differentiating a — b times (1 — u) _ 1 — l + u + u2-\ u = p~l. Consequently, i=o V aa -- bb IJ i=0 V

i=b_2 i=b-2

(N > (bl > i). I)-

and putting

c^y-

\ a-b

)

The substitution b — 2 — i —> i in the first sum produces the Laplace transform (with the opposite sign) of the distributional part in (6.76). Retracing the proof of Theorem 6.12, we can establish the following general result. Theorem 6.20. Assume the equation £{ait + bi)xln-i\t)

=0

(6.78)

i=0

with constant coefficients ai, hi, and a$ = 1, bo = 0 satisfies the hy­ potheses: (i) All poles pi of the function Rip) = £(kP - (n -

OOOP"-'"1

!=0

/ E aiPn'{ '

(6.79)

i=0

are distinct, all residues n = res p=Pi R(p) are integers, and the residues of the complex conjugate poles are equal.

364

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

(ii) The poles with positive real parts have negative residues, and the poles with nonpositive real parts have non-negative residues. Then Eq. (6.78) admits a formal distributional solution (6.71) which can be represented as a sum of an m-order distribution with support t — 0, where n

m=52 rh i=i

and of a locally integrable function concentrated on the left semi-axis, which is a finite sum of products of power functions with non-negative integer exponents and exponential functions or sines and cosines. P R O O F . The Laplace transform F(p) of a formal distributional solu­ tion to (6.78) satisfies the equation F'(p) = R(p)F(p), (6.80) whence F(WP) =

^ N(P)'

M{p) and N(p) are polynomials. According to (ii), p ~ 0 is not a pole of F(p), hence this function can be expanded into a power series in a neighborhood of the origin which is the formal Laplace transform of a distribution (6.71). On the other hand, F(p) is the sum of a polynomial of degree m and of a proper algebraic fraction, the first being the Laplace transform of a distribution of order m, and the second of a classical function described in the theorem. Of course, if F(p) is a proper fraction, the solution is a classical function; and if F(p) is a polynomial, the solution is a finite-order distribution. ■ 6. Infinite-Order D i s t r i b u t i o n a l S o l u t i o n s R e v i s i t e d Solutions (6.71) in the generalized-function space (So)' for linear ho­ mogeneous ordinary differential and functional differential equations (FDE) have been studied in Chapter 5. Disributional solutions of lin­ ear homogeneous ODE are generated solely by singularities in their coefficients. For linear FDE, distributional solutions may be originated either by singularities of their coefficients or by argument deviations.

6.6. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS REVISITED

365

It has been shown that, under certain conditions, the FDE with poly­ nomial coefficients oo

*'(*) == oo £ ■Pj(t)x(\st) 3=0 3=0

has a solution (6.71) in (SQ)' with arbitrary /? > 1, which is an impossi­ ble phenomenon for linear homogeneous ODE without singularities in their coefficients. As indicated above, the equation oo

x'(t) = £E Pj^xiXjt), PjiMXjt), tnx'{t)

n n>2 >2

J=0

and, in particular, tnx'(t) = P(t)x(t) admit distributional solutions (6.71) in (So)', with some /? > 1. Thus, the equation t2x'(t) = x{t)

(6.81)

has a solution

x(t)--

oo

*w(0 :!(i + l)!

in (So)', for 1 < /? < 2. However, this is impossible for Eq. (6.78), unless the distributional solution is of finite order. LEMMA 6.1. The Laplace transform oo

F(p) =

ZxiPl t"=0

of each distribution x(t) G (SQ)' of type (6.71), with (3 > 1, is an entire function whose order of growth does not exceed (5~ . P R O O F . We act with distribution (6.71) on a test function (j>(t) 6 S5 such that {-iyW(0)xi > 0 and

1^(0(0)1 = 6 ^ ,

366

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

then oo

(x(t),(t)) =

.

biEd\i'l}\xi\.

i=0

Since the latter series must converge, we have |xt'| < C\d~['i~lP,

C\ = const.

which implies that F(p) is an entire function. Its order of growth /x is determined by the formula i In i fi = Jim sup ; ,, '-►oo — In|x,| and since In|a;,| < lnci — i\nd\ — pilni, then \i < /3 _ 1 . ■ T h e o r e m 6.21. Eq. (6.78), with constant coefficients a,-, &,• has no infinite-order solutions (6.71) in (SQ)', with /3 > 1. P R O O F . The Laplace transform F(p) of a distributional solution to (6.78) satisfies (6.80), whence

F(p)=Cexp(jR(p)dp), where R(p) is a rational function defined in (6.79) and used in The­ orem 6.12, which states a criterion for the existence of a finite-order distributional solution to (6.78). If R(p) does not satisfy the conditions of Theorem 6.12, then F(p) is an analytic function with singularities in the finite plane. Here the condition bo = 0 is unessential, because bo 7^ 0 implies that F(p) grows in some directions at least exponentially as p —> oo. ■ EXAMPLE 9. The Laplace transform F(p) of a formal distributional solution x(t) to the equation tx" + (ait + bi)x' + (a2< + b2)x = 0 satisfies the equation -(p2F)'

-
that is, (p2 + alP + a2)F' = [ft - 2)p + (b2 - a{)]F.

6.6. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS REVISITED

367

If the roots Ai and A2 of the equation p2 + aip + a2 = 0 are distinct, then F(p) = C(p-\l)ri(p-\2y\

C = const.

where r

^(b1-2)Xl

+

(b2-al)

Ai — A2

(b1-2)X2 + (b2-a1) — -. A2 - Ai If Ai and X2 are real and r\ and r2 are non-negative integers, then F(p) is a polynomial and x(t) is a finite-order distributional solution. Other­ wise, F(p) has singularities (poles and/or branching points), and in this case the given differential equation has no infinite-order solutions (6.71) in (S(j)'. If Aj and X2 are complex, then F(p) has singularities in the finite plane unless 61 is even positive and 2b2 = aibi, but in this case r\ = r2 is a non-negative integer which implies that F(p) is a poly­ nomial and x(t) is a finite-order solution. Finally, if Ai = X2 = X, then F(P) = C(p - A) 6 '- 2 exp[(-A(6 1 - 2) - (b2 - a i ))(p - A)"1], r2 =

and F(p) has an essential singularity at p = X. Theorem 6.22. The equation £ eqi(t)xM(t)

=0

(6.82)

«=0

whose coefficients qi(t) are polynomials and qn(t) = 1 has no infiniteorder solutions (6.71) in (So)', with (3 > 1. PROOF. We define a vector w(t) with components Wi. -=^ f - y * '--11^^)),, w

ii == l,...,n. l,...,n.

(6.83) (6.83)

368

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

If x(t) satisfies (6.82), then differentiating (6.83) leads to the system tw'i = (i — l)wi + W{+\,

i = 1,..., n — 1

t< = ~ E qi(t)wi+i - (g»-i(t) - (n - l))«/„

(6.84)

t'=0

and it is obvious that the distributions «;,-(<) and x(<) have the same order. Let Q(t) denote the matrix of system (6.84). Then tw'1 = tw = Q(t)w

(6.85)

or (

k

\

tw'=('ZQ tw' = £ Qj* jAw, ™,

(6.86)

\i=o I where the Qj are n x n constant matrices and k is the highest degree of the polynomial entries of the matrix Q(t). Substituting series (6.71) with vector coefficients x, in (6.86) for w and taking into account the formula j

t 5^(t) =

j J j \(-l) Ht) f (-1)i\S^»-! g^ft)

(i-JV[0, [o,

{>j

' i >j i<j

give the equation

- E(i + iM')(t) = E Q> E (

t=o which can be written as

y=o

i=i

V '

W v!

(*-J)

- g(. + iM«>(<) = £ «»w E b a M M l ^ i , i=0

i=0

j=0

l

-

From here, E(-l) J 'QiC i + J - = - ( i + l)c i7 where

i = 0,1,...

(6.87)

6.6. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS REVISITED

369

If the norms |c,-| of the vectors c,- are unbounded as i —► oo, then the norm \F(p)\ of the Laplace transform oo OO

F(p) =■Ex ipl »=0 i=0

oo OO

/.J

i=0 1=0 i\l-

is at least of exponential growth. In this case, according to the above Lemma, the formal infinite-order distributional solution x(t) of (6.82) cannot belong to (S^)', with j3 > 1. If the norms |c,| are bounded, then from (6.87) it follows that c , c = const. i+1 which implies |c t | —> 0 as i —> oo. On the other hand, the inequality c,- <

1 i+1

1< \d E IICJII ■|ci+j| M<7^TI;IK?JIH*+;I * + J- ?'=0 j=0

is impossible for large i because the right-hand side tends to zero faster than the left as i —> oo. Therefore, (6.87) has no bounded solution, ex­ cept (co, c\,..., c ra , 0 , 0 , . . . ) , and in this case the distributional solution of (6.82) is of order m. ■ REMARK 6. The system (6.85) has a distributional solution of order m if m + 1 is the smallest modulus of the negative integer eigenvalues of the matrix Qo — (5(0). Indeed, the requirement Q = 0, i > m, changes (6.87) to a finite system of matrix equations the last of which (Qo + (m + \)I)cm = 0 has a nontrivial solution c m . Substituting it in the preceding equations of (6.87), we can successively find all c, (i < m), since the matrices Qo + (i + 1)J are nonsingular, for i < m. For Eq. (6.82), we compute the matrix of system (6.87) and observe that in this case the equation det(Q(0) + (m + 1)7) = 0 coincides with (6.10).

370

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

Theorem 6.23. The differential-difference equation t(ait

+ bfixt*-*}® = t(cit

i=0

+ di)x^-'Xt

+ U)

(6.88)

i=0

with constant coefficients a{, b{, c,-, d{ and real constant deviations ti has no infinite-order solutions (6.71) in (SQ)', with /3 > 1. PROOF. In accordance with the commonly accepted notations in the theory of functional differential equations, x^"~t'(t + ti) means the value of the derivative x^n~l\u) at u = t + t{. Assume that £[x(i)] = F(p) is the two-sided Laplace transform of x(t). Then L[x(n-i\t

+

ti)}=p"-ie'*
and L [txM(t

+ ti)] = -j-

[p^e^Fip]]

.

Hence, the Laplace transform F(p) of a distributional solution to (6.88) satisfies the equation

± [-a,(p n -'F)' + bvT-'F] = t [-Ciip^e^F)' t=0

+

d^e^F),

i=0

that is, n

n

4

IXce* - a^-'F'ip) = £ K- - Citfo*- ** - (ae* - a,)(n - z)p n ~ i_1 - &tf"_i F(p). If all c,- = 0, then F'(p)/F(p) grows exponentially in some directions as p —► oo; hence, if F(p) is entire, its order of growth is infinite. If some c,- 7^ 0, then F'(p)/F(p) may grow in some directions either exponentially or like a polynomial; and if F(p) is entire, its order of growth is not less than 1. ■

6.6. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS REVISITED

371

It has been mentioned above that Eq. (6.81) has an infinite-order solution in (Sg)'. This is not true for the similar differential-difference equation t2x'(t) = x(t + c) or, more generally, for the equation t2x'(t) = x(Xt + c).

(6.89)

T h e o r e m 6.24. Eq. (6.89), with constants c ^ 0 and A ^ 0, has no solution concentrated on t = 0 in (SQ)', (3 > 1. PROOF. Assume that £[#(£)] = F(p) is the two-sided Laplace trans­ form of x(t). Then £[*V(t)] = \pF(p)]", L[x{Xt + c)} = and according to (6.89),

\X\-1e^xF(p/X),

\pF(p)}" = \X\-lec"^F(p/X).

(6.90)

Assuming that F(p) is entire of order fi < 1 leads to a contradiction because in this case the left-hand side of the latter equation is of order fi and the right-hand side is of order 1, by virtue of the factor ecplx. ■ Since integral transformations establish close links between entire and generalized functions, the results on distributional solutions of linear ODE and, especially, FDE can be applied to the study of entire solu­ tions, and vice versa. Research in this direction, still developed insuffi­ ciently, discovers new aspects and properties in the theory of ODE and FDE. As mentioned, there are some striking dissimilarities between the behavior of ODE and FDE which deserve further investigation. Thus, disributional solutions to linear homogeneous FDE may be originated either by singularities of their coefficients or by argument deviations. As indicated above, some normal linear FDE with polynomial coeffi­ cients and arguments proportional to t have distributional solutions, which is impossible for ODE without singularities. Furthermore, it has been shown in [302] that a first order algebraic ODE has no entire tran­ scendental solution of order less than ^, whereas even linear first order

372

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

FDE may possess such solutions of zero order. We shall demonstrate now how the knowledge of distributional solutions to linear ODE and FDE can be used in the study of their analytic and, in particular, entire solutions. First, consider series (6.71) without any restrictions on the norms of its coefficients and substitute this series and its term-by-term derivatives for x(t) in the equation ±qi(t)x(n-i\t)=0 (6.91) ;=o with polynomial (or holomorphic at t = 0) coefficients. Equating to zero the coefficients at all derivatives S^'\t) produces recursion relations for £,. If the coefficients x,- of (6.71) satisfy these relations, we call (6.71) a formal distributional solution of Eq. (6.91). We may also consider formal solutions oo

x = 52(-l)ii\xiri-1

(6.92)

and obtain along the lines of Theorem 6.1 the following result. Theorem 6.25. If Eq. (6.91) with polynomial coefficients of the high­ est degree N admits a formal solution (6.92), then it also has a formal distributional solution (6.71). Conversely, z/(6.91) admits a formal dis­ tributional soltution (6.71), then there exists a polynomial q{t) of degree not exceeding N — 1 such that the equation ±qi(t)x(n-i\t)

= q(t)

(6.93)

i=0

has a formal solution (6.92). Theorem 6.26. / / Eq. (6.91) with polynomial coefficients qi(t) ad­ mits a formal distributional solution (6.71) and if deg qi(t) < n — i,

i = 0,..., n

then it also has a formal solution (6.92). EXAMPLE 10. The confluent hypergeometric equation (6.72) is the Laplace-transformed relation of

(t2 + t)x' = [(b - 2)t + (a- l)]x

(6.94)

6.6. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS REVISITED

373

Substituting (6.71) in (6.94) gives

£(»' + l)(t + 6)W (0 (*) = E(» + «)^ (i) (0, i=0

i=0

whence x x,

r(« + a)r(6) - r ( i + 6)r(a)i!

(6 95)

-

if a and 6 axe not negative integers or zero (here T denotes the gamma function). Therefore, (6.72) has a solution oo

M(a,M) = 2>>*', i=0

with coefficients (6.95), which is an entire function of order 1, type 1 (an alternate notation for M(a,b,t) is \Fi(a,b, t)). Indeed, its order is z In t z In t fi = lim sup —— = Jim sup , ■ .. = 1, t—oo ln(i!) »->oo ln(i'e ') and to find its type a we use the formula (aeti)1^ = lim sup (i1/"\xi\lfi)

,

V

I—tOO

that is, ae = lim sup(i(i!) -1 /') = lim sup(^'(^'e_,)_1/,) = e. «'—»oo

i—»oo

If 0 < a < b, then M and each M^ are all close-to-convex in the unit disk \t\ < 1 and hence univalent in this disk [242]. Furthermore, for a, b € N and b < a, Eq. (6.72) has a formal distributional solution (6.73) and therefore admits also a formal solution OO

x(t) =

E (--l)'i j=a-l

!

C:

-a + lj

Although this solution diverges for all finite t, it represents an asymp­ totic series as t —» oo for the integral too

(_l)"-i f00 e - ' V - ^ u + I) 6 - 0 " 1 du,

374

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

which is the Laplace transform of the solution ( — l ) 0 _ 1 u a - 1 ( u 4- l ) 6 _ a _ 1 of Eq. (6.94). EXAMPLE 11. Looking for a formal distributional solution (6.71) of the generalized Bessel equation t2x" + tx' + J2 (-l)kaktkx

= 0,

ak=

const., n > 2

(6.96)

leads to the equations £ akci+k + [oo + (t + l)2]c,- = 0,

(6.97)

k=\

Ci = i\xi,

i = 0,1,... .

If ao = — (m + l ) 2 , where m is an integer, we can choose cm ^ 0, and in this case (6.96) has an m-order distributional solution. If an ^ 0 and ao ^ — (i + l ) 2 , for any integer i, then we can choose CQ, ..., c n _i arbitrarily and successively find from (6.97) all coefficients x,- = cj/t! of a formal distributional solution to (6.96). The estimates for c,- found in Theorem 6.22 lead to the conclusion that the coefficients of any infiniteorder distributional solution (6.71) to Eq. (6.96) cannot tend to zero faster than cji\ (c = const.). T h e o r e m 6.27. If Eq. (6.91) with polynomial coefficients of the high­ est degree N admits a solution x{t) in the form (6.92), which is regular at t = oo, and x(t~{) is an entire function of finite order \i > 0 and finite type, then (6.91) also has a solution (6.71) in the generalized func­ tion space (So)', with 1 < j3 < 1 + (JT1. Conversely, if (6.91) admits a distributional solution (6.71) in ( S Q ) ' , with /3 > 1, then there exists a polynomial q(t) of degree not exceeding N — 1 such that Eq. (6.93) has a solution (6.92), which is regular at t = oo and xft-1) is an entire function of order fi < {(5 — 1) . PROOF. The arguments of Theorem 6.1 show t h a t if (6.91) admits a solution (6.92) it also has a distributional solution (6.71). Since x(t~l) is entire of order fi > 0, we write for its coefficients i\ x,- the estimates i\ \xi\ < Cii -1 /' 1 ,

c\ = const.

6.6. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS REVISITED

375

and Stirling's formula yields \xi\ < acii, - ( l + l//0»' for sufficiently large i. These inequalities of type (5.62) with p = 1+fi'1 prove that solution (6.71) belongs to (S^)', 1 < /? < 1+fi'1. Conversely, if (6.91) admits a distributional solution (6.71) in (So)', then N < cxd^i-W,

(3 > 1

and there exists a polynomial q(t) of degree < N — 1 such that (6.93) has a formal solution (6.92). For its coefficients we have

i\\x{\


as i —» co, which proves that (6.92) is regular at t = oo and x(i _ 1 ) is an entire function. If /f is its order of growth, then for a sufficiently small e > 0 there exists an infinite sequence of values of i such that iln i - ln(i! N)

>

fi-e,

whence

i! N

>

» ■

-i/0«-«)>

Therefore, 1

0- I < and I M<

1

. ■

EXAMPLE 12. The equation [46] pF" -F = 0

(6.98)

is the Laplace-transformed relation of (t2x)' - i = 0

(6.99)

376

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

which is changed by the substitution t2x = y to

*V - y = 0, with a solution f

/ 1\ °° ( - t ) ^exp^g—. Hence, (6.99) admits a solution -l)H-> - i\

CO

f_

=E

—2

(6.100)

t=0

in form (6.92) and a distributional solution

,= £» i=0i!(i

Therefore, (6.98) has a solution OO

OO

F-F = Y"l —

(6.101)

+ l)! p'+l jjj+l

!(* + !)!

i=0

which is an entire function of order /z = 1/2, as can be shown by apply­ ing Stirling's formula. The function x(i _ 1 ) corresponding to (6.100) is entire of order fi — 1, and distribution (6.101) belongs to the space (So)', with 1 < P < 2. The equation pF^(p)-aF{p)

= 0,

n>2,

a = const.

is the Laplace-transformed relation of (tnx)' = (-l)"ax, with a solution X

—

r"exp ( l

eXP

n-1

)

oo

( _ l ) ( » - l ) ' a « r " ' - •n+i

£

(n-l)*»!

i'=0

(6.102)

m

6.6. INFINITE-ORDER DISTRIBUTIONAL SOLUTIONS REVISITED

377

in form (6.92) and a distributional solution ai6(n-l)(i+l)^

* =,_o£(n — l)'i! (ni — i + n — 1)! Hence, (6.102) has an entire solution a y»-i)('+i) F= E ,_0 (n — l)'i! (ni — i + n — 1)! of order \i = (n — l ) / n . The functional differential equation t

^

mm

a iF(\ p), pF^ \p)-- == £ T,a*F(A*p), pFW(p) i k

A*t < < 11 --1 1 < < A

(6.103)

jfc=0

(Ai = const. 7^ # 00) ) (A* is the bilateral Laplace-transformed relation of m / / \ ajfelAtl" (*■ x(t))' = (--1)"

£

fc=0

n

Mi)

which is changed by the substitution t x = y to m

/ £ \

n 1 t
r)-

. (6.104) .**/ According to Theorem 5.18, Eq. (6.104) admits a distributional solu­ tion (6.71) whose coefficients satisfy inequalities (5.62),with n P = n n—1" Therefore, (6.103) has an entire solution of order *=o

ilni

_i

fj, < lim sup ——-—: r,— = P , i-*oo pi In i - I n a - i l n c that is, fi < (n - l ) / n . For its coefficients x{ = c,/i!, computations produce the formulas cKn-1)+r = cT U ( £ <*A?"- 1 ) + j / {j(n - 1) + r ) , (1 < r < n- 1).

378

6. COEXISTENCE OF ANALYTIC AND DISTRIBUTIONAL SOLUTIONS

For n = 2, some properties of Eq. (6.103) are similar to those of (6.98). In particular, (6.103) has two linearly independent solutions, one of which, F\(p), is entire and the other, F^ip), is represented in the form F2(p) = G(p) + F1(p)lnp, where G(p) is an entire function. Solutions of functional differential equations which tend to zero more rapidly than any exponential as t —> +oo, but are nonzero at arbitrar­ ily large times, are called small solutions. Henry [119] proved that any such solution of a linear autonomous FDE must vanish identically after a certain time. In the nonautonomous case this is no longer true as follows, in particular, from the example x'(t) — (sini)x(t — 2n). For nonlinear autonomous scalar equations Cao [34] showed that if the lin­ earized equation has no small solutions, then the nonlinear has no small solutions. Recently, Cooke and Lunel [48] have studied distributional and small solutions of linear time-dependent delay equations and the relation between these two types of solutions. First they show that the existence of a small solution implies the existence of an infinite order distributional solution, and then they prove that a certain con­ dition implies the nonexistence of this type of distributional solutions. This leads to a nonexistence theorem of small solutions for linear delay systems with real analytic bounded matrices. In conclusion, we mention that singular integral equations in spaces of generalized functions have been studied by Kosulin [142], Rogozhin [229], Estrada and Kanwal [73, 74], and others. However, it should be noted that, perhaps, the first work of this kind was done by Horvath [122].

Open Problems The following topics which are, in our opinion, of considerable interest either have not been explored at all or deserve deeper study. (1) Boundary-value problems for EPCA: here one may start with twopoint BVP for first-order equations with two piecewise constant arguments. (2) Stability results based on the theory of discrete Liapunov func­ tions for first-order nonlinear EPCA with a nondegenerate linear part. (3) Oscillatory and asymptotic behavior of the equation x"(t) = ax(t) + bx([\t/h]h). (4) Boundary and initial-value problems for the wave equation with the argument [Xt/h]h. (5) Asymptotic and oscillatory properties of first-order neutral dif­ ferential equations with arguments of the form [Xt/h]h and their comparison with the properties of the same equations with lin­ early transformed arguments. (6) Stabilization of solutions and periodic behavior for control prob­ lems with piecewise constant feedback delay. (7) Partial differential equations with both constant and piecewise constant arguments. (8) Boundary and initial-value problems for PDE with alternately retarded and advanced piecewise continuous arguments. (9) Parabolic PDE of neutral type with piecewise constant time.

379

380

OPEN PROBLEMS

(10) Bounded solutions of nonlinear parabolic equations with piecewise continuous arguments. (11) Bounded solutions of nonlinear hyperbolic equations with the ar­ gument [Xt/h]h. (12) Proofs of existence-uniqueness theorems for functional differential equations by using piecewise constant arguments. (13) Cauchy-Kovalevsky type existence-uniqueness theorems for par­ tial differential equations with the argument 0 < X(t) < t by using piecewise constant delays. (14) Resonance in linear EPCA with periodic nonhomogeneous terms. (15) Eq. (1.140) with two step functions. (16) Discontinuous solutions of differential equations with piecewise continuous arguments and connections between EPCA and im­ pulsive equations. (17) New classes of EPCA exhibiting chaotic behavior. (18) Loaded partial differential-difference equations. (19) Integral equations with piecewise continuous arguments. (20) Loaded integral equations. (21) Systems that include both differential and difference equations. (22) Boundedness, stability, and oscillation of differential equations with involutions: one of the approaches is to reduce the equation to a higher-order ODE keeping in mind that the derivative of a strong involution is negative. (23) Systems of integro-differential equations reducible to ordinary dif­ ferential equations. (24) Distributional solutions concentrated on the fixed point of the involution for reducible differential equations. (25) Finite-order and infinite-order distributional solutions of linear PDE with singular points and their relationship with rational, polynomial, and entire solutions. (26) Infinite-order distributional solutions of PDE with linearly trans­ formed arguments and their interaction with entire solutions, es­ pecially of zero order.

Bibliography [1] J. Aczel, Lectures on Functional Equations and Their Applications (Academic Press, New York, 1966). [2] A. R. Aftabizadeh, Bounded solutions for some gradient type systems, Libertas Math. 2 (1982), 121-130. [3] A. R. Aftabizadeh, Y. K. Huang, and J. Wiener, Bounded solutions for dif­ ferential equations with reflection of the argument, J. Math. Anal. Appl. 135 (1988), 31-37. [4] A. R. Aftabizadeh and J. Wiener, Oscillatory properties of first order linear functional differential equations, Applicable Anal. 20 (1985), 165-187. [5] , Oscillatory and periodic solutions of an equation alternately of retarded and advanced type, Applicable Anal. 23 (1986), 219-231. [6] , Differential inequalities for delay differential equations with piecewise constant argument, Appl. Math. Comput. 24 (1987), 183-194. [7] , Oscillatory and periodic solutions for systems of two first order linear differential equations with piecewise constant argument, Applicable Anal. 26 (1988), 327-333. [8] A. R. Aftabizadeh, J. Wiener, and J.-M. Xu, Oscillatory and periodic solutions of delay differential equations with piecewise constant argument, Proc. Amer. Math. Soc. 99 (1987), 673-679. [9] R. P. Agarwal, Difference Equations and Inequalities (Marcel Dekker, New York, 1991). [10] W. G. Aiello, The existence of nonoscillatory solutions to a generalized, nonautonomous, delay logistic equation, J. Math. Anal. Appl. 149 (1990), 114-123. [11] M. A. Aleksidze, Fundamental Functions in Approximate Solutions of Bound­ ary Value Problems (Nauka, Moscow, 1991). [12] F. S. Aliev, The fundamental system of solutions of Euler's equation in gen­ eralized functions, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (1964), 7-14.

381

382

[13]

BIBLIOGRAPHY

, The general solution of Euler's equation in generalized functions, Dokl. Akad. Nauk AzSSR 20 (1964), 9-13. [14] , On the solutions in generalized functions of some ordinary differential equations with polynomial coefficients, Dokl. Akad. Nauk SSSR 167 (1966), 259-262. [15] , On the solutions in generalized functions of ordinary differential equa­ tions with polynomial coefficients, Dokl. Akad. Nauk SSSR 169 (1966), 991994. [16] , On the solutions of certain systems of ordinary differential equations in the space of generalized functions, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (1973), 3-10. [17] A. B. Antonevich, Linear Functional Equations: An Operator Approach (Universitetskoe, Minsk, 1988). [18] 0 . Arino and I. Gyori, Necessary and sufficient condition for oscillation of a neutral differential system with several delays, J. Differential Equations 81 (1989), 98-105. [19] O. Arino, G. Ladas, and Y. G. Sficas, On oscillations of some retarded differ­ ential equations, SIAM J. Math. Anal. 18 (1987), 64-73. [20] D. D. Bainov, A. D. Myshkis, and A. I. Zahariev, Necessary and sufficient conditions for oscillation of the solutions of linear functional-differential equa­ tions of neutral type with distributed delay, J. Math. Anal. Appl. 148 (1990), 263-273. [21] J. W. Bebernes and R. Gaines, Dependence on boundary data and a generalized boundary-value problem, J. Differential Equations 4 (1968), 359-368. [22] J. Belair, Sur une equation differentielle fonctionnelle analytique, Canad. Math. Bull. 24 (1981), 43-46. [23] R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960). [24] R. Bellman and K. L. Cooke, Differential-Difference Equations (Academic Press, New York, 1963). [25] P. I. Bodnarchuk and V. Ya. Skorobogat'ko, Branching Continued Fractions and Their Applications (Naukova Dumka, Kiev, 1974) in Ukranian. [26] Yu. S. Bogdanov, On the functional equation xn = t, Dokl. Akad. Nauk BSSR 5 (1961), 235-237. [27] S. N. Bolkovoi and Ya. I. Zhitomirskii, The Cauchy problem for evolutionary functional equations, DifferentsiaPnye Uravneniya 10 (1974), 2021-2028. [28] V. M. Borok and Ya. I. Zhitomirskii, The Cauchy problem for a certain class of loaded equations, Uspekhi Mat. Nauk 34 (1979), 221-222. [29] H. Bremermann, Distributions, Complex Variables, and Fourier Transforms (Addison-Wesley, Reading, Massachusetts, 1965).

BIBLIOGRAPHY

383

[30] L. Bruwier, Sur I'application du calcul symbolique a la irresolution d'equations fonctionnelles, Bull. Soc. Roy. Sci. Liege 17 (1948), 220-245. [31] S. Busenberg and C. Travis, On the use of reducible-functional differential equations in biological models, J. Math. Anal. Appl. 89 (1982), 46-66. [32] L. Ya. Bykova, On almost-periodic solutions of systems of differential equations with deviating argument, Issled. Integro-Diff. Uravn. v Kirgizii 8 (1971), 306318. [33] Y. Cao, The discrete Liapunov function for scalar differential delay equations, J. Differential Equations 87 (1990), 365-390. [34] , Non-existence of small solutions for scalar differential delay equations, Preprint, Univeristy of Georgia, 1991. [35] J. Carr and J. Dyson, The functional differential equation y'(x) = ay(\x) + by(x), Proc. Roy. Soc. Edinburgh, Sect. A 74 (1974/75), 165-174. [36] , The matrix functional differential equation y'(x) = Ay(Xx) + By(x), Proc. Roy. Soc. Edinburgh, Sect. A 75 (1975/76), 5-22. [37] L. A. V. Carvalho, On the stability of discrete equations and ordinary differ­ ential equations, in Delay Differential Equations and Dynamical Systems, eds. S. Busenberg and M. Martelli (Lecture Notes in Math. 1475, Springer-Verlag, 1991) pp. 88-97. [38] L. A. V. Carvalho and K. L. Cooke, A nonlinear equation with piecewise con­ tinuous argument, Differential and Integral Equations 1 (1988), 359-367. [39] , On dichotomic maps for a class of differential-difference equations, Proc. Roy. Soc. Edinburgh, Sect. A 117 (1991), 317-328. [40] L. A. V. Carvalho and R. R. Ferreira, On a new extension of Liapunov's direct method to discrete equations, Quart. Appl. Math. 66 (1988), 778-788. [41] W. G. Castelan and E. F. Infante, On a functional equation arising in the sta­ bility theory of difference-differential equations, Quart. Appl. Math. 35 (1977), 311-319. [42] L. G. Chambers, Some functional differential equations, Quart. Appl. Math. 32 (1974/75), 445-456. [43] N. N. Chaus, On the uniqueness of the solution of the Cauchy problem for systems of differential equations with constant coefficients, Ukrain. Mat. Zh. 17 (1965), 126-131. [44] N. G. Chebotarev, Theory of Continued Fractions (Kazan', 1938) in Russian. [45] Q. Chuanxi, S. A. Kuruklis, and G. Ladas, Oscillations of linear autonomous systems of difference equations, Applicable Anal. 36 (1990), 51-63. [46] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955). [47] K. L. Cooke and I. Gyori, Numerical approximation of the solutions of delay dif­ ferential equations on an infinite interval using piecewise constant arguments,

384

BIBLIOGRAPHY

Preprint Series 633, Inst. Math, and Its Appls., University of Minnesota, 1990. [48] K. L. Cooke and S. M. V. Lunel, Distributional and small solutions for linear time-dependent delay equations, Differential and Integral Equations, to appear. [49] K. L. Cooke, J. Turi, and G. Turner, Stabilization of hybrid systems in the presence of feedback delays, Preprint Series 906, Inst. Math, and Its Appls., University of Minnesota, 1991. [50] K. L. Cooke and J. Wiener, Distributional and analytic solutions of functional differential equations, J. Math. Anal. Appl. 98 (1984), 111-129. [51] , Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99 (1984), 265-297. [52] , Stability regions for linear equations with piecewise continuous delay, Comput. Math. Appl. 12A (1986), 695-701. [53] , An equation alternately of retarded and advanced type, Proc. Amer. Math. Soc. 99 (1987), 726-732. [54] , Neutral differential equations with piecewise constant argument, Boll. Un. Mat. Ital. 7 (1987), 321-346. [55] , A survey of differential equations with piecewise continuous arguments, in Delay Differential Equations and Dynamical Systems, eds. S. Busenberg and M. Martelli (Lecture Notes in Math. 1475, Springer-Verlag, Berlin-HeidelbergNew York, 1991) pp. 1-15. [56] C. Corduneanu, Bounded and almost periodic solutions of certain nonlinear elliptic equations, Tohoku Math. J. 32 (1980), 265-278. [57] , Bounded and almost periodic solutions of certain nonlinear parabolic equations, Libertas Math. 2 (1982), 131-139. [58] C. Corduneanu and V. Lakshmikantham, Equations with unbounded delay, Nonlinear Anal. 4 (1980), 831-877. [59] C. Corduneanu and H. Poorkarimi, Qualitative problems for some hyperbolic equations, in Differential Equations, eds. I. W. Knowles and R. T. Lewis (North-Holland, New York, 1985) pp. 107-113. [60] J. M. Gushing, Integrodifferential equations and delay models in population dynamics, in Lecture Notes in Biomathematics 20 (Springer-Verlag, Berlin, 1977) . [61] R. Datko, An algorithm for computing Liapunov functionals for some differen­ tial difference equations, in Ordinary Differential Equations (1971 NRL-MRC Conference, Academic Press, 1972) pp. 387-398. [62] , A procedure for determination of the exponential stability of certain differential equations, Quart. Appl. Math. 36 (1978), 279-292. [63] G. A. Derfel', A class of differential-functional equations with linearly trans­ formed argument, Dokl. Akad. Nauk SSSR 83 (1976), 561-563.

BIBLIOGRAPHY

[64]

[65] [66]

[67] [68] [69] [70]

[71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81]

385

, Behavior of solutions of functional and differential-functional equa­ tions with several transformations of the independent variable, Ukrain. Mat. Zh. 3 4 (1982), 350-356. , Probabilistic method for a class of functional-differential equations, Ukrain. Mat. Zh. 4 1 (1989), 1322-1327. , Kato problem for functional-differential equations and difference Schrodinger operators, Operator Theory: Advances and Applications 4 6 (1990), 319-321. G. A. Derfel' and S. A. Molchanov, Probabilistic and spectral methods in the theory of differential-functional equations, Uspekhi Mat. Nauk 4 2 (1987), 126. , On T. Kato's problem regarding bounded solutions of differentialfunctional equations, Functional Anal. Appl. 24 (1990), 67-69. , Spectral methods in the theory of functional-differential equations, Mat. Zametki 4 7 (1990), 42-51. G. A. Derfel' and V. N. Shevelo, Connection between the existence of summable and almost-periodic solutions of a class of differential-functional equations, Ukrain. Mat. Zh. 3 9 (1987), 437-439. P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, 1947). L. H. Erbe and B. G. Zhang, Oscillation of discrete analogues of delay equa­ tions, Differential and Integral Equations 2 (1989), 300-309. R. E s t r a d a and R. P. Kanwal, Distributional solutions of singular integral equa­ tions, J. Integral Equations 8 (1985), 41-85. , Distributional solutions of dual integral equations of Cauchy, Abel, and Titchmarsh types, J. Integral Equations 9 (1985), 277-305. , A distributional theory for asymptotic expansions, Proc. Roy. Soc. London, Ser. A 4 2 8 (1990), 399-430. W . N. Everitt and L. L. Littlejohn, Differential operators and the Legendre type polynomials, Differential and Integral Equations 1 (1988), 97-116. D. M. Fargue, Reducibilite des systemes hereditaires a des systemes dynamiques, C. R. Acad. Sci. Paris Ser. B 2 2 7 (1973), 471-473. A. Feldstein and Z. Jackiewicz, Unstable neutral functional-differential equa­ tions, Canad. M a t h . Bull. 3 3 (1990), 428-433. E. Fick and G. Hausmann, Logistic equation with memory, Phys. Rev. A (3) 4 4 (1991), 2469-2473. P. F l a m a n t , Sur une equation differentielle fonctionnelle lineaire, Rend. Circ. Mat. Palermo 4 8 (1924), 135-208. L. Fox, D. F . Mayers, J. R. Ockendon, and A. B. Tayler, On a functional differential equation, J. Inst. Maths. Applies. 8 (1971), 271-307.

386

BIBLIOGRAPHY

P. Frederickson, Dirichlet series solutions for certain functional differential equations, Lecture Notes Math. 243 (1971), 249-254. A. Friedman, Generalized Functions and Partial Differential Equations (Prentice-Hall, Englewood, New Jersey, 1963). T. Furumochi, On the delay-differential equation N(t) = N(t)(a — bN(t — 1) — cN{t - 2)), Mem. Fac. Sci. Shimane Univ. 23 (1989), 57-71. , Uniform asymptotic stability in perturbed differential equations, J. Math. Anal. Appl. 137 (1989), 207-220. T. Furumochi and A. Hayashi, Stability and oscillation in delay differential equations with piecewise constant arguments, Mem. Fac. Sci. Shimane Univ. 24 (1990), 21-30. I. M. Gel'fand and G. E. Shilov, Fourier transformation of rapidly increas­ ing functions and uniqueness questions in the solution of the Cauchy problem, Uspekhi Mat. Nauk 8 (1953), 3-54. , Generalized Functions. Vol. 1: Properties and Operations (Academic Press, New York, 1964). , Generalized Functions. Vol. 2: Spaces of Fundamental and Generalized Functions (Academic Press, New York, 1964). , Generalized Functions. Vol. 3: Theory of Differential Equations (Aca­ demic Press, New York, 1964). A. 0 . Gel'fond, Calculus of Finite Differences (Fizmatgiz, Moscow, 1959). K. Gopalsamy, Global stability in the delay-logistic equation with discrete de­ lays, Houston J. Math. 16 (1990), 347-356. K. Gopalsamy, I. Gyori, and G. Ladas, Oscillations of a class of delay equations with continuous and piecewise constant arguments, Funkcial. Ekvac. 32 (1989), 395-406. K. Gopalsamy, M. R. C. Kulenovic, and G. Ladas, Oscillations of a system of delay logistic equations, J. Math. Anal. Appl. 146 (1990), 192-202. K. Gopalsamy, M. R. S. Kulenovic, and G. Ladas, On a logistic equation with piecewise constant arguments, Differential and Integral Equations 4 (1991), 215-223. S. R. Grace, Oscillation of functional differential equations with deviating ar­ guments, J. Math. Anal. Appl. 149 (1990), 558-575. J. R. Graef, M. K. Grammatikopoulos, and P. W. Spikes, Asymptotic and os­ cillatory behavior of solutions of first order nonlinear neutral delay differential equations, J. Math. Anal. Appl. 155 (1991), 562-571. , On the asymptotic behavior of solutions of a second order nonlinear neutral delay differential equation, J. Math. Anal. Appl. 156 (1991), 23-39. , On the behavior of solutions of a first order nonlinear neutral delay differential equation, Applicable Anal. 40 (1991), 111-121.

BIBLIOGRAPHY

387

[100] B. G. Grebenshchikov, Stability of systems with variable delay that is linearly dependent on time, in Stability and Nonlinear Oscillations (Ural. Gos. Univ., Sverdlovsk, 1983) pp. 25-34. [101] , Stability with respect to the first approximation of systems with linearly time-dependent delay, Differential Equations 26 (1990), 159-162. [102] F . Gross and C. C. Yang, On meromorphic solutions of a certain class of functional-differential equations, Ann. Polon. M a t h . 2 7 (1973), 305-311. [103] E. A. Grove, I. Gyori, and G. Ladas, On the characteristic equation for equa­ tions with continuous and piecewise constant delays, Rad. M a t . 5 (1989), 2 7 1 281. [104] C. P. G u p t a , Boundary value problems for differential equations in Hilbert spaces involving reflection of the argument, J. M a t h . Anal. Appl. 1 2 8 (1987), 375-388. [105] , Existence and uniqueness theorems for boundary value problems in­ volving reflection of the argument, Nonlinear Anal. 11 (1987), 1075-1083. [106] , Two-point boundary value problems involving reflection of the argu­ ment, Internal. J. M a t h . M a t h . Sci. 10 (1987), 361-371. [107] I. Gyori, Two approximation techniques for functional-differential equations, C o m p u t . M a t h . Appl. 16 (1988), 195-214. [108] , On the approximation of the solutions of delay differential equations by using piecewise constant arguments, Internat. J. M a t h . M a t h . Sci. 1 4 (1991), 111-126. [109] I. Gyori and G. Ladas, Linearized oscillations for equations with piecewise constant argument, Differential and Integral Equations 2 (1989), 123-131. [110] , Oscillation Theory of Delay Differential Equations (Oxford University Press, 1992). [ I l l ] I. Gyori, G. Ladas, and L. Pakula, Conditions for oscillation of difference equations with applications to equations with piecewise constant arguments, SIAM J. M a t h . Anal. 22 (1991), 769-773. 112] I. Gyori, G. Ladas, and P. N. Vlahos, Global attractivity in a delay difference equation, Nonlinear Anal. 17 (1991), 473-479. 113] W . Halm, Uber die Funktional-Differentialgleichung f'(z) = f(qz) und verwandte Funktionalgleichungen, Ann. Univ. Sci. Budapest. Eotvos Sect. M a t h . 16 (1973), 3-21. 114] A. Halanay and D. Wexler, Teoria Calitativd a Sistemelor cu Impulsuri (Acad. RSR, Bucuresti, 1968). 115] J. K. Hale, Theory of Functional Differential Equations (Springer-Verlag, New York-Heidelberg-Berlin, 1977). 116] J. K. Hale, E. F. Infante, and F-S. P. Tsen, Stability in linear delay equations, J. M a t h . Anal. Appl. 1 0 5 (1985), 533-555.

388

BIBLIOGRAPHY

[117] G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities (Cambridge Univer­ sity Press, London and New York, 1952), p. 182. [118] J. Henderson and L. Lee, Continuous dependence and differentiation of solu­ tions of finite difference equations, Internat. J. Math. Math. Sci. 14 (1991), 747-756. [119] D. Henry, Small solutions of linear autonomous functional differential equa­ tions, J. Differential Equations 8 (1970), 494-501. [120] E. Hille, Ordinary Differential Equations in the Complex Domain (John Wiley & Sons, New York, 1976). [121] L. Hormander, Linear Partial Differential Operators (Springer-Verlag, New York, 1963). [122] J. Horvath, Sur I'iteration de la transformee de Hilbert d'une distribution complexe, C. R. Acad. Sci. 237 (1953), 1480-1482. [123] R. F. Hoskins, Generalised Functions (Horwood, Chichester and New York, 1979). [124] J. P. Hoyt, Remarks on a second order differential equation, Amer. Math. Monthly 74 (1967), 436. [125] Y. K. Huang, A nonlinear equation with piecewise constant argument, Appli­ cable Anal. 33 (1989), 183-190. [126] , On a system of differential equations alternately of advanced and delay type, in Differential Equations I, ed. A. R. Aftabizadeh (Ohio University Press, Athens, 1989) pp. 455-465. [127] , Oscillations and asymptotic stability of solutions of first order neutral differential equations with piecewise constant argument, J. Math. Anal. Appl. 149 (1990), 70-85. [128] G. K. Immink, Asymptotics of Analytic Difference Equations (Springer-Verlag, New York, 1981). [129] A. D. Iskenderov, A mixed boundary-value problem for loaded quasi-linear hy­ perbolic equations, Dokl. Akad. Nauk SSSR 199 (1971), 1237-1239. [130] S. Izumi, On the theory of linear functional differential equations, Tohoku Math. J. 30 (1929), 10-18. [131] K. N. Jayasree and S. G. Deo, Variation of parameters formula for the equation of Cooke and Wiener, Proc. Amer. Math. Soc. 112 (1991), 75-80. [132] D. S. Jones, Generalised Functions (McGraw-Hill, New York, 1966). [133] E. Kamke, Gewohnliche Differentialgleichungen (Akademische Verlagsgesellschaft, Geest & Portig K.-G., Leipzig, 1959). [134] R. P. Kanwal, Generalized Functions: Theory and Technique (Academic Press, New York, 1983). [135] G. Karakostas, Expression of solutions of linear differential equations of re­ tarded type, Boll. Un. Mat. Ital. 17A (1980), 428-435.

BIBLIOGRAPHY

389

[136] G. Karakostas, Y. G. Sficas, and V. A. Staikos, On the basic theory of initial value problems for delay differential equations, Boll. Un. Mat. Ital. I B (1982), 1179-1198. [137] T. Kato, Asymptotic behavior of solutions of the functional-differential equation y'(x) = ay(Xx) + by(x), in Delay and Functional-Differential Equations and Their Applications (Academic Press, New York, 1972) pp. 197-217. [138] T. Kato and J. B. McLeod, The functional differential equation y'(x) = ay(Xx) + by(x), Bull. Amer. Math. Soc. 77 (1971), 891-937. [139] A. Ya. Khinchine, Continued Fractions (Fizmatgiz, Moscow, 1961) in Russian. [140] Y. Kitamura and T. Kusano, Oscillation of first-order nonlinear differential equations with deviating argument, Proc. Amer. Math. Soc. 78 (1980), 64-68. [141] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equa­ tions (Academic Press, New York, 1986). [142] A. E. Kosulin, One-dimensional singular equations in generalized functions, Dokl. Akad. Nauk SSSR 163 (1965), 1054-1057. [143] A. Kratzer and W. Franz, Transzendente Funktionen (Akademische Verlagsgesellschaft, Leipzig, 1960). [144] S. G. Krein, Linear Differential Equations in Banach Spaces (Transl. of Math. Monogr. 29, Amer. Math. Soc, Providence, 1971). [145] V. Ja. Kreinovic, Notices Amer. Math. Soc. 26 (Apr. 1979), 191. [146] , Notices Amer. Math. Soc. 26 (Aug. 1979), 298. [147] Y. Kuang and A. Feldstein, Monotonic and oscillatory solutions of a linear neutral delay equation with infinite lag, SIAM J. Math. Anal. 23 (1990), 1633— 1641. [148] M. Kuczma, Functional Equations in a Single Variable (PWN, Warszawa, 1968). [149] G. K. Kulev and D. D. Bainov, Stability of the solutions of impulsive integrodifferential equations in terms of two measures, Internat. J. Systems Sci. 21 (1990), 2225-2239. [150] R. G. Kuller, On the differential equation f — f a g, where g o g = / , Math. Mag. 42 (1969), 195-200. [151] V. G. Kurbatov, Linear Differential-Difference Equations (Voronezh. Gos. Univ., Voronezh, 1990). [152] Kh. Yu. Kurdanov, The influence of an argument deviation on the behavior of solutions of differential equations, Differentsial'nye Uravneniya 15 (1979), 994. [153] T. Kusano and H. Onose, Oscillations of functional differential equations with retarded argument, J. Differential Equations 15 (1974), 269-277. [154] G. Ladas, Sharp conditions for oscillations caused by delays, Applicable Anal. 9 (1979), 93-98. [155] , Oscillations of equations with piecewise constant mixed arguments,

390

BIBLIOGRAPHY

in Differential Equations and Applications II, ed. A. R. Aftabizadeh (Ohio University Press, Athens, 1988) pp. 64-69. [156] , Explicit conditions for the oscillation of difference equations, J. Math. Anal. Appl. 153 (1990), 276-287. [157] G. Ladas, E. C. Partheniadis, and J. Schinas, Existence theorems for second order differential equations with piecewise constant arguments, in Lecture Notes in Pure and Appl. Math. 118 (Dekker, New York, 1989) pp. 389-395. [158] G. Ladas, C. G. Philos, and Y. G. Sficas, Existence of positive solutions for certain difference equations, Utilitas Math. 38 (1990), 119-128. [159] G. Ladas, C. Qian, P. N. Vlahos, and J. Yan, Stability of solutions of linear nonautonomous difference equations, Applicable Anal. 41 (1991), 183-191. [160] G. Ladas and I. P. Stavroulakis, Oscillations caused by several retarded and advanced arguments, J. Differential Equations 44 (1982), 134-152. [161] G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala, Monotone Iterative Tech­ niques for Nonlinear Differential Equations (Pitman, London, 1985). [162] V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations (World Scientific Publishing Co., London, 1989). [163] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, 2 vols. (Academic Press, New York, 1969). [164] B. S. Lalli, Sufficient conditions for oscillations of certain neutral delay differ­ ential equations, Math. Student 57 (1989), 84-92. [165] B. S. Lalli, B. G. Zhang, and J. Z. Li, On the oscillation of solutions and existence of positive solutions of neutral difference equations, J. Math. Anal. Appl. 158 (1990), 213-233. [166] J. P. LaSalle, The Stability and Control of Discrete Processes (Springer-Verlag, New York, 1986). [167] W. Layton, Periodic solutions of nonlinear delay equations, J. Math. Anal. Appl. 77 (1980), 198-204. [168] , Existence of almost periodic solutions to delay differential equations with Lipschitz nonlinearities, J. Differential Equations 55 (1984), 151-164. [169] N. N. Lebedev, Special Functions and Their Applications (Dover, New York, 1972). [170] W. Leighton, On the differential equation xy" — (x + n)y' + ny = 0, Amer. Math. Monthly 76 (1969), 52-53. [171] T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992. [172] Z.-H. Li, The asymptotic estimates of solutions of difference equations, J. Math. Anal. Appl. 94 (1983), 181-192. [173] M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions (Cambridge University Press, London, 1958).

BIBLIOGRAPHY

391

[174] E.-B. Lirn, Asymptotic behavior of solutions of the functional differential equa­ tion x'{t) = Ax{Xt) + Bx(t),X> 0, J. Math. Anal. Appl. 55 (1976), 794-806. [175] , Asymptotic bounds of solutions of the functional differential equation x'(t) = ax(Xt) + bx(t) + / ( t ) , 0 < A < 1, SIAM J. Math. Anal. 9 (1978), 915-920. [176] L. L. Littlejohn, On the classification of differential equations, having orthog­ onal polynomial solutions, Ann. Mat. Pura Appl. 138 (1984), 35-53. [177] L. L. Littlejohn and R. P. Kanwal, Distributional solutions of the hypergeometric differential equation, J. Math. Anal. Appl. 122 (1987), 325-345. [178] R. Lucic, On a functional differential equation, Publ. Electrotechn. Fac. Univ. Belgrade, Ser. Math. Phys. 338-352 (1971), 55-56. [179] , On a class of functional differential equations, Publ. Electrotechn. Fac. Univ. Belgrade, Ser. Math. Phys. 461-497 (1974), 31-32. [180] , Functional differential equations whose arguments form a finite group, Publ. Electrotechn. Fac. Univ. Belgrade, Ser. Math. Phys. 498-541 (1975), 133-135. [181] S. M. V. Lunel, A sharp version of Henry's theorem on small solutions, J. Differential Equations 62 (1986), 266-274. [182] , Series expansions and small solutions for Volterra equations of convo­ lution type, J. Differential Equations 85 (1990), 17-53. [183] , Small solutions and completeness for linear functional differential equations, in Contemporary Mathematics 129, eds. J. R. Graef and J. K. Hale (Amer. Math. Soc, Providence, 1992) pp. 127-152. [184] G. I. Marchuk, Mathematical Models in Immunology (Nauka, Moscow, 1991). [185] A. A. Martynyuk, V. Lakshmikantham, and S. Leela, Stability of Motion: The Method of Integral Inequalities (Naukova Dumka, Kiev, 1989). [186] M. T. Matraimov, Existence of bounded solutions of differential equations with reflection of the argument, in Studies in Integro-Differential Equations 22 (Ilim, Frunze, 1989) pp. 86-88. [187] B. Mazbic-Kulma, On an equation with reflection of order n, Studia Math. 35 (1970), 69-76. [188] N. McDonald, Time lags in biological models, in Lecture Notes in Biomathematics 27 (Springer-Verlag, Berlin, 1978). [189] J. B. McLeod, The functional differential equation y'(x) = ay(\x) + by(x) and generalizations, in Lecture Notes Math. 280 (1972), 308-313. [190] R. E. Mickens, Difference Equations (Van Nostrand, New York, 1990). [191] S. G. Mikhlin, Linear Equations in Partial Derivatives (Fizmatgiz, Moscow, 1977). [192] N. N. Milev and D. D. Bainov, Stability of linear impulse differential equations, Internat. J. Systems Sci. 21 (1990), 2217-2224.

392

BIBLIOGRAPHY

, Ordinary dichotomy and perturbations of the coefficient matrix of the linear impulsive differential equation, Internat. J. Math. Math. Sci. 14 (1991), 763-768. A. A. Mirolyubov and M. A. Soldatov, Linear Homogeneous Difference Equa­ tions (Nauka, Moscow, 1981). , Linear Nonhomogeneous Difference Equations (Nauka, Moscow, 1986). 0 . P. Misra and J. L. Lavoine, Transform Analysis of Generalized Functions (North-Holland, Amsterdam, 1986). B. S. Mityagin, On an infinitely differentiate function with the given values of the derivatives at a point, Dokl. Akad. Nauk SSSR 138 (1961), 289-292. S. Mizohata, The Theory of Partial Differential Equations (Cambridge Univer­ sity Press, 1973). A. Z. Mohon'ko, Differential and functional differential equations with factors, Differentsial'nye Uravneniya 15 (1979), 1713-1715. V. S. Mokeichev, Differential Equations with Deviating Argument (Kazan' Uni­ versity Press, Kazan', 1985). V. Mure§an, The existence of solution of some functional-differential equations, in Seminar on Differential Equations, 9-16, Preprint, 88-8 (Univ. Babe§-Bolyai, Cluj-Napoca, 1988) A. N. Murovtsev, Analytic solutions of functional-differential equations, Ukrainian Math. J. 42 (1990), 952-959. A. D. Myshkis, On certain problems in the theory of differential equations with deviating argument, Uspekhi Mat. Nauk 32 (1977), 173-202. M. A. Naimark, Linear Differential Operators (Fizmatgiz, Moscow, 1954). A. M. Nakhushev and V. N. Borisov, Boundary-value problems for loaded parabolic equations and their applications to the prediction of a groundwave level, Differentsial'nye Uravneniya 13 (1977), 105-110. D. Nemzer, Operational calculus and differential equations with infinitely smooth coefficients, Internat. J. Math. Math. Sci. 13 (1990), 405-410. T. A. Newton, On using a differential equation to generate polynomials, Amer. Math. Monthly 81 (1974), 592-601. R. Paley and N. Wiener, Fourier Transforms in the Complex Domain (Amer. Math. Soc. Colloq. Publ. 19, Providence, 1934). L. Pandolfi, Some observations on the asymptotic behavior of the solutions of the equation x(t) = A(t)x(\t) + B(t)x(t),\ > 0, J. Math. Anal. Appl. 67 (1979), 483-489. A. A. Pankov, Bounded and Almost Periodic Solutions of Nonlinear Differential-Operator Equations (Naukova Dumka, Kiev, 1985).

BIBLIOGRAPHY

393

[211] G. Papaschinopoulos, On a class of third order neutral delay differential equa­ tions with piecewise constant argument, Internat. J. Math. Math. Sci., to ap­ pear. [212] M. Parodi, La Localisation des Valeurs Caracteristiques des Matrices et ses Applications (Gauthier-Villars, Paris, 1959). [213] E. C. Partheniadis, Stability and oscillation of neutral delay differential equa­ tions with piecewise constant argument, Differential and Integral Equations 1 (1988), 459-472. [214] G. P. Pelyukh, Existence and uniqueness of C + 1 -solutions to nonlinear differential-functional equations of neutral type, in Differential-Functional and Difference Equations (Kiev, 1981) pp. 57-64. [215] , Representation of asxjmptotically periodic solutions of nonlinear dif­ ference equations, Ukrain. Mat. Zh. 42 (1990), 833-837. [216] G. P. Pelyukh and A. N. Sharkovskii, Introduction to the Theory of Functional Equations (Naukova Dumka, Kiev, 1974). [217] A. Peterson and J. Ridenhour, Oscillation of second order linear matrix differ­ ence equations, J. Differential Equations 89 (1991), 69-88. [218] , A disconjugacy criterion of W. T. Reid for difference equations, Proc. Amer. Math. Soc. 114 (1992), 459-468. [219] P. Pfluger, Matrizenkettenbruche (Juris Druck Verlag, Zurich, 1966). [220] V. M. Polishchuk and A. N. Sharkovskii, General solution of linear differentialdifference equations of neutral type, in Differential-Difference Equations (Naukova Dumka, Kiev, 1971) pp. 126-139. [221] , Representation of the solutions of linear differential-difference equa­ tions of neutral type, Differentsial'nye Uravneniya 9 (1973), 1627-1645. [222] H. Poorkarimi and J. Wiener, Bounded solutions of nonlinear hyperbolic equa­ tions with delay, in Nonlinear Analysis and Applications, ed. V. Lakshmikantham (Marcel Dekker, New York, 1987) pp. 471-478. [223] D. Przeworska-Rolewicz, Equations with Transformed Argument. An Algebraic Approach (Elsevier-PWN, Amsterdam-Warszawa, 1973). [224] B. S. Razumikhin, Stability of Hereditary Systems (Nauka, Moscow, 1988). [225] I. M. Repin, Quadratic Liapunov functional for systems with delays, Prikl. Mat. Mekh. 29 (1965), 564-566. [226] G. F. Roach, Green's Functions (Cambridge University Press, 1982). [227] L. B. Robinson, Complemente a une etude sur I 'equation fonctionelle d'Izumi, Bull. Soc. Math. France 64 (1936), 213-215. [228] , Une pseudo-fonction et I'equation d'Izumi, Bull. Soc. Math. France 64 (1936), 66-70. [229] V. S. Rogozhin, A general theory of solving boundary problems in the space of generalized functions, Dokl. Akad. Nauk SSSR 164 (1965), 277-280.

394

BIBLIOGRAPHY

[230] E. Yu. Romanenko, Asymptotic behavior of the solutions of a class of functional-differential equations, Ukrainian Math. J. 41 (1989), 1314-1319. [231] , Representation of a local general solution of a class of functionaldifferential equations, Ukrainian Math. J. 42 (1990), 182-186. [232] E. Yu. Romanenko and A. N. Sharkovskii, Asymptotic behavior of the solutions of linear functional-differential equations, in Asymptotic Behavior of Solutions of Functional-Differential Equations (Akad. Nauk Ukr. SSR, Inst. Mat., Kiev, 1978) pp. 5-39. [233] E. E. Rosinger, Nonlinear Partial Differential Equations, Sequential and Weak Solutions (North-Holland, New York, 1980). [234] V. A. Rvachev, Compactly-supported solutions of functional-differential equa­ tions and their applications, Uspekhi Mat. Nauk 45 (1990), 77-103. [235] T. L. Saaty, Modern Nonlinear Equations (Dover, New York, 1981). [236] A. M. Samoilenko and M. Ilolov, On the theory of evlolution equations with impulse action, Dokl. Akad. Nauk SSSR 316 (1991), 822-825. [237] A. M. Samoilenko and Kh. Z. Mustafaev, On the averaging principle for a class of systems of differential equations with deviating argument, Ukrain. Mat. Zh. 42 (1990), 1363-1369. [238] A. M. Samoilenko and N. A. Perestyuk, Stability of the solutions of differential equations with impulse effect, Differentsial'nye Uravneniya 13 (1977), 1981— 1992. [239] L. Schwartz, Theorie des Distributions, 2 vols. (Hermann, Paris, 1950-51). [240] , Methodes Mathematiques pour les Sciences Physiques (Hermann &: Cie, Paris, 1961). [241] G. Seifert, On an interval map associated with a delay logistic equation with discontinuous delays, in Delay Differential Equations and Dynamical Systems, eds. S. Busenberg and M. Martelli (Lecture Notes in Math. 1475, SpringerVerlag, Berlin-Heidelberg-New York, 1991) pp. 243-249. [242] S. M. Shah, Univalence of a function f and its successive derivatives when f satisfies a differential equation, J. Math. Anal. Appl. 133 (1988), 79-92. [243] S. M. Shah, H. Poorkarimi, and J. Wiener, Bounded solutions of retarded non­ linear hyperbolic equations, Bull. Alld. Math. Soc. 1 (1986), 1-14. [244] S. M. Shah and J. Wiener, Advanced differential equations with piecewise con­ stant argument deviations, Internat. J. Math. Math. Sci. 6 (1983), 671-703. [245] , Distributional and entire solutions of ordinary differential and func­ tional differential equations, Internat. J. Math. Math. Sci. 6 (1983), 243-270. [246] , Reducible functional differential equations, Internat. J. Math. Math. Sci. 8 (1985), 1-27. [247] , Existence theorems for analytic solutions of differential and functional differential equations, J. Ramanujan Math. Soc. 1 (1986), 85-97.

BIBLIOGRAPHY

395

[248] A. N. Sharkovskii, Functional-differential equations with a finite group of ar­ gument transformations, in Asymptotic Behavior of Solutions of FunctionalDifferential Equations (Akad. Nauk Ukrain., Inst. Mat., Kiev, 1978) pp. 118142. [249] A. N. Sharkovskii, Yu. L. Maistrenko, and E. Yu. Romanenko, Difference Equa­ tions and Their Applications (Naukova Dumka, Kiev, 1986). [250] G. E. Shilov, Generalized Functions and Partial Differential Equations (Gordon k Breach, New York, 1968). [251] 0 . Shisha and C. B. Mehr, On involutions, J. Nat. Bur. Stand. 71B (1967), 19-20. [252] L. Silberstein, Solution of the equation f'(x) = f(l/x), Philos. Magazine 30 (1940), 185-186. [253] R. Silkowski, A star shaped condition for stability of linear retarded functional differential equations, Proc. Roy. Soc. Edinburgh, Sect. A 83 (1979), 189-198. [254] P. S. Simeonov and D. D. Bainov, Asymptotic equivalence of linear and non­ linear systems with impulse effect, Riv. Mat. Univ. Parma 4 (1986), 1-8. [255] S. L. Sobolev, Methode nouvelle a resoudre le probleme de Cauchy pour les equations lineaires hyperboliques normales, Mat. Sb. (N.S.) 1 (1936), 39-72. [256] V. A. Staikos and P. Ch. Tsamatos, On linear differential equations with re­ tarded arguments, Math. Nachr. 115 (1984), 167-188. [257] G. Stepan, Retarded Dynamical Systems: Stability and Characteristic Func­ tions (Pitman Research Notes in Mathematics Series, Harlow-New York, 1989). [258] S. Strelitz, Asymptotic Properties of Analytic Solutions of Differential Equa­ tions (Mintis, Vilnius, 1972). [259] G. Szego, Orthogonal Polynomials, ^th edition (AMS Colloquium Publ. XXIII, Providence, 1978). [260] A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Pergamon Press, New York, 1963). [261] F. Treves, Basic Linear Partial Differential Equations (Academic Press, New York, 1975). [262] R. A. Usmani, A uniqueness theorem for a boundary value problem, Proc. Amer. Math. Soc. 77 (1979), 329-335. [263] K. G. Valeev, Linear differential equations with delay linearly depending on the argument, Sibirsk. Mat. Zh. 5 (1964), 290-309. [264] , On solutions of some functional equations, Issled. Integro-Diff. Uravn. v Kirgizii 5 (1968), 85-89. [265] B. van der Pol and H. Bremmer, Operational Calculus based on the Two-Sided Laplace Integral (Cambridge University Press, 1960). [266] Yu. A. Ved' and M. T. Matraimov, Problems of well-posedness, boundedness, and stabilization of solutions of the Cauchy problem for differential equations

396

[267] [268] [269] [270] [271] [272] [273] [274] [275] [276] [277] [278] [279] [280] [281] [282] [283]

[284] [285]

BIBLIOGRAPHY

with reflection of the argument, in Studies in Integro-Differential Equations 22 (Ilim, Frunze, 1989) pp. 73-85. V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1976). , Generalized Funcitons in Mathematical Physics (Nauka, Moscow, 1976). P. N. Vlahos, Oscillation and stability of an equation with piecewise constant quadratic nonlinearity, Dynamic Systems and Applications, to appear. F. Vogl, Uber ein System linearer Funktional-Differentialgleichungen, Z. Angew. Math. Mech. 60 (1980), 7-17. W. Walter, Differential and Integral Inequalities (Springer-Verlag, New York, 1970). W. Wasow, Asymptotic Expansions for Ordinary Differential Equations (Wiley, New York, 1965). J. Wiener, On Silberstein's functional equation, Ucen. Zap. Ryazan. Ped. Inst. 41 (1966), 5-8. , Differential equations with involutions, Differentsial'nye Uravneniya 5 (1969), 1131-1137. , Differential equations in partial derivatives with involutions, Differ­ entsial'nye Uravneniya 6 (1970), 1320-1322. , Differential equations with periodic transformations of the argument, Izv. Vyss Ucebn. Zaved. Radiofiz. 16 (1973), 481-484. , Investigaiton of some functional differential equations with a regular singular point, Differentsial'nye Uravneniya 10 (1974), 1891-1894. , Periodic maps in the study of functional differential equations, Differ­ entsial'nye Uravneniya 3 (1974), 34-45, Ryazan'. , Solutions of linear systems in generalized functions, Differentsial'nye Uravneniya 11 (1975), 1128-1130. , Existence of solutions of differential equations with deviating argument in the space of generalized functions, Sibirsk. Mat. Zh. 17 (1976), 1403-1405. , A retarded type system with infinitely differentiable coefficients has solutions in generalized functions, Uspekhi Mat. Nauk 31 (1976), 227-228. , On the entire solutions of a linear differential-functional system, Dif­ ferentsial'nye Uravneniya 13 (1977), 552-555. , Generalized-function solutions of differential equations with a count­ able number of argument deviations, Differentsial'nye Uravneniya 14 (1978), 355-358. , Generalized-function solutions of linear systems, J. Differential Equa­ tions 38 (1980), 301-315. , Distributional and entire solutions of linear functional differential equations, Internat. J. Math. Math. Sci. 5 (1982), 729-736.

BIBLIOGRAPHY

[286]

397

, Generalized-function and analytic solutions of functional differential equations, in Dynamical Systems II, eds. A. R. Bednarek and L. Cesari (Aca­ demic Press, New York, 1982) pp. 655-661. [287] , Generalized-function solutions of differential and functional differen­ tial equations, J. Math. Anal. Appl. 88 (1982), 170-182. [288] , Differential equations with piecewise constant delays, in Trends in the Theory and Practice of Nonlinear Differential Equations, ed. V. Lakshmikantham (Marcel Dekker, New York, 1983) pp. 547-552. [289] , Pointwise initial-value problems for functional differential equations, in Differential Equations, eds. I. W. Knowles and R. T. Lewis (North-Holland, New York, 1984) pp. 571-580. [290] , Boundary-value problems for partial differential equations with piecewise constant delay, Internat. J. Math. Math. Sci. 14 (1991), 301-321. [291] J. Wiener and A. R. Aftabizadeh, Boundary value problems for differential equations with reflection of the argument, Internat. J. Math. Math. Sci. 8 (1985), 151-163. [292] , Differential equations alternately of retarded and advanced type, J. Math. Anal. Appl. 129 (1988), 243-255. [293] J. Wiener and K. L. Cooke, Oscillations in systems of differential equations with piecewise constant argument, J. Math. Anal. Appl. 137 (1989), 221-239. [294] , Coexistence of analytic and distributional solutions for linear differ­ ential equations I, J. Math. Anal. Appl. 148 (1990), 390-421. [295] J. Wiener, K. L. Cooke, and S. M. Shah, Coexistence of analytic and distribu­ tional solutions for linear differential equations II, J. Math. Anal. Appl. 159 (1991), 271-289. [296] J. Wiener and L. Debnath, Partial differential equations with piecewise constant delay, Internat. J. Math. Math. Sci. 14 (1991), 485-496. [297] , The Fourier method for partial differential equations with piecewise continuous delay, in Contemporary Mathematics 129, eds. J. R. Graef and J. K. Hale (Amer. Math. Soc, Providence, 1992) pp. 241-263. [298] , A parabolic differential equation with unbounded piecewise constant delay, Internat. J. Math. Math. Sci. 15 (1992), 339-346. [299] , A wave equation with discontinuous time delay, Internat. J. Math. Math. Sci. 15 (1992), 781-792. [300] J. Wiener, L. Debnath, and S. M. Shah, Analytic solutions of nonlinear neutral and advanced differential equations, Internat. J. Math. Math. Sci. 9 (1986), 365-372. [301] J. Wiener and S. M. Shah, Continued fractions arising in a class of functional differential equations, J. Math. Phys. Sci. 21 (1987), 527-543.

398

BIBLIOGRAPHY

[302] H. Wittich, Neuere Untersuchungen uber Eindeutige Analytische Funktionen (Springer-Verlag, Berlin, 1955). [303] K. Yosida, Operational Calculus (Springer-Verlag, New York, 1984). [304] J. S. Yu, First-order nonlinear differential inequalities with deviating argu­ ments, Acta Math. Sinica 33 (1990), 152-159. [305] D. Zeitlin, On a class of ordinary linear differential equations having Y^kLo c * 1 * and E L o ckXk as solutions, Amer. Math. Monthly 84 (1977), 716-720. [306] A. H. Zemanian, Distribution Theory and Transform Analysis (McGraw-Hill, New York, 1965). [307] , Generalized Integral Transformations (Interscience, New York, 1969). [308] B. G. Zhang and K. Gopalsamy, Global attractivity and oscillations in a peri­ odic delay-logistic equation, J. Math. Anal. Appl. 150 (1990), 274-283. [309] M. Zima, On the existence and uniqueness of solution of a certain initial value problem, Zeszyty Nauk. Politech. Rzeszowskiej. Mat. Fiz. 7 (1988), 113-118.

Author Index Abel, 385 Aczel, 381 Aftabizadeh, 81, 91, 107, 114, 124, 235, 249, 381,388, 390,397 Agarwal, 381 Aleksidze, 381 Aliev, 306, 381,382 Antonevich, 320, 382 Arino, 81, 382 Ascoli, 262, 264, 265

Cushing, 384 Datko, 69, 235, 384 Debnath, 178, 190, 272, 397 Deo, 114,388 Derfel', 320, 384, 385 Dirac, xii, 179, 292-294, 326, 385 Dyson, 320, 383 Erbe, 385 Estrada, 378, 385 Euler, 3, 215, 225, 228, 247, 347, 381, 382 Everitt, 359, 385

Bainov, 78, 81, 382, 389-392, 395 Banach, 19, 25, 52, 261, 262, 389 Bebernes, 382 Belair, 382 Bellman, 382 Bessel, 302, 327, 336, 374 Bodnarchuk, 382 Bogdanov, 382 Bolkovoi, 320, 382 Borisov, 392 Borok, 382 Bremermann, 382 Bremmer, 295, 395 Bruwier, 244, 320, 383 Busenberg, 270, 383, 384, 394 Bykova, 320, 383

Fargue, 270, 385 Feldstein, 321, 385, 389 Ferreira, 383 Fick, 385 Flamant, 321, 385 Fourier, 170, 172, 174, 176, 181, 182, 184,189, 294, 307, 310, 321, 322, 360, 382, 386 Fox, 321, 385 Franz, 389 Frederickson, 321, 386 Friedman, 386 Furumochi, 157, 386 Gaines, 382 Gegenbauer, 338, 339 Gel'fand, 298, 386 Gel'fond, 386

Cao, 378, 383 Carr, 320 Carvalho, 3, 157, 383 Castelan, 231, 383 Cauchy, 20, 22-25, 28, 51-53, 176, 183, 184, 189, 251, 252, 254, 281, 294, 380, 382, 383, 385, 386, 395 Cesari, 397 Chambers, 320, 383 Chaus, 383 Chebotarev, 383 Chebyshev, 335-337, 339 Chuanxi, 383 Coddington, 383 Cooke, 1, 3, 4, 68, 79, 114, 115, 137, 157, 279, 325, 378, 382-384, 388, 397 Corduneanu, 384

Gopalsamy, 157, 386, 398 Grace, 81, 386 Graef, 81, 386, 391, 397 Grammatikopoulos, 81, 386 Grebenshchikov, 321, 387 Green, 250, 393 Gronwall, 78 Gross, 321, 387

Grove, 157, 387 Gupta, 249, 387 Gyori, 79, 81, 156, 157, 382, 383, 386, 387 Hahn, 321, 387 Halanay, 78, 387

399

400

AUTHOR INDEX

Hale, 1, 69,387, 391,397 Hardy, 388 Hausmann, 385 Hayashi, 157, 386 Heaviside, 293, 294, 298, 361 Helmholtz, 294 Henderson, 388 Henry, 378, 388, 391 Hermite, 333, 334 Hilbert, x, 176, 178, 387, 388 Hille, 388 Hormander, 388 Horvath, 378, 388 Hoskins, 295, 388 Hoyt, 343, 388 Huang, 114, 157, 235, 381, 388 Ilolov, 78, 394 Immink, 388 Infante, 69, 231, 383, 387 Iskenderov, 388 Izumi, 279, 321, 388,393 Jackiewicz, 321, 385 Jacobi, 355, 356 Jayasree, 114, 388 Jones, 388 Jordan, 141, 152 Kamke, 388 Kanwal, 298, 325, 326, 378, 385, 388, 391 Karakostas, 321, 388, 389 Kato, 82, 157, 320, 321, 385, 389 Khinchine, 389 Kirchoff, 294 Kitamura, 81, 389 Knowles, 384, 397 Kolmanovskii, 389 Kosulin, 378, 389 Kovalevsky, 380 Kratzer, 389 Krein, 389 Kreinovic, 321, 389 Kronecker, 232 Kuang, 321, 389 Kuczma, 389 Kulenovic, 157, 386 Kulev, 78, 389

Kuller, 229, 389 Kurbatov, 389 Kurdanov, 389 Kuruklis, 383 Kusano, 81, 389 Ladas, 2, 81, 96, 114, 156, 157, 382, 383, 386, 387, 389, 390 Ladde, 390 Laguerre, 340-342 Lakshmikantham, 78, 384, 390, 391, 393, 397 Lalli, 81, 390 Laplace, 186,187, 292, 301, 3 2 0 , 3 2 2 - 3 2 4 , 3 2 7 332, 334, 336, 337, 339, 341, 345, 347, 349, 350, 352, 354, 355,358-360, 362-366, 369-372, 374-377 LaSalle, 390 Lavoine, 392 Lay ton, 390 Lebedev, 390 Lebesgue, 296 Lee, 388 Leela, 390, 391 Legendre, 331, 332, 339, 359, 385 Leighton, 343, 390 Levinson, 383 Lewis, 384, 397 Li, J. Z., 390 Li, T.-Y., 390 Li, Z.-H., 390 Liapunov, xiii, 231, 235, 379, 383, 393 Lighthill, 390 Lim, 321, 391 Lipschitz, 205, 206, 208, 212, 262, 274, 278, 390 Littlejohn, 325, 326, 359, 360, 385, 391 Littlewood, 388 Lucic, 391 Lunel, 378, 391 Maistrenko, 395 Marchuk, 391 Martelli, 383, 384, 394 Martynyuk, 391 M a t r a i m o v , 3 9 1 , 395 Mayers, 321, 385 Mazbic-Kulma, 244, 391 McDonald, 391

AUTHOR INDEX

McLeod, 82, 157, 320, 321, 389, 391 Mehr, 216, 395 Mickens, 391 Mikhlin, 391 Milev, 78, 391, 392 Mirolyubov, 392 Misra, 392 Mityagin, 392 Mizohata, 392 Mohon'ko, 321, 392 Mokeichev, 392 Molchanov, 320, 385 Mures.an, 321, 392 Murovtsev, 321, 392 Mustafaev, 321, 394 Myshkis, 1,81,382,392 Naimark, 392 Nakhushev, 392 Nemzer, 392 Newton, 343, 392 Nosov, 389 Ockendon, 321, 385 Onose, 81, 389 Pakula, 156, 387 Paley, 392 Pandolfi, 321, 392 Pankov, 392 Papaschinopoulos, 114, 157, 393 Parodi, 393 Partheniadis, 114, 157, 390, 393 Pelyukh, 321,393 Perestyuk, 78, 394 Peterson, 393 Pfluger, 393 Philos, 390 Planck, 321 Poisson, 294 van der Pol, 295, 395 Polishchuk, 321,393 Polya, 388 Poorkarimi, 201, 384, 393, 394 Przeworska-Rolewicz, 213, 393 Qian, 390 Razumikhin, 393

Repin, 235, 393 Ridenhour, 393 Roach, 393 Robinson, 321, 393 Rodrigues, 332 Rogozhin, 378, 393 Romanenko, 321, 394, 395 Rosinger, 394 Rvachev, 321, 394 Saaty, 394 Samarskii, 395 Samoilenko, 78, 321, 394 Schauder, 262, 264, 265 Schinas, 114, 157, 390 Schrodinger, 175, 385 Schwartz, 292, 295, 298 Schwarz, 251, 252, 254 Seifert, 157, 394 Sficas, 81, 382, 389, 390 Shah, 1, 33, 201, 272, 325, 394, 397 Sharkovskii, 221, 321, 393-395 Shevelo, 320, 385 Shilov, 189, 298, 386, 395 Shisha, 216, 395 Silberstein, 213, 228, 395 Silkowski, 69, 395 Simeonov, 78, 390, 395 Skorobogat'ko, 382 Sobolev, 292, 395 Soldatov, 392 Spikes, 81, 386 Staikos, 321,389, 395 Stavroulakis, 81, 96, 390 Stepan, 395 Stirling, 319, 375, 376 Strelitz, 395 Szego, 395 Tayler, 321, 385 Taylor, 259, 280, 299, 306, 335, 341 Tikhonov, 395 Titchmarsh, 385 Travis, 270, 383 Treves, 395 Tsamatos, 321, 395 Tsen, 69, 387 Turi, 384

401

402

AUTHOR INDEX

Turner, 384 Usmani, 395 Valeev, 244, 321, 395 Vandermonde, 17, 30, 42 Vatsala, 390 Ved', 395 Vladimirov, 396 Vlahos, 157, 387, 390, 396 Vogl, 321, 396 Volterra, 201, 203, 391 Walter, 396 Wasow, 396 Wexler, 78, 387 Wiener, 1,4,33,68,81,91, 107, 114,115, 124, 137, 178, 190, 201, 235, 249, 272, 279, 325, 381, 384, 388, 393, 394, 396 Wiener, N., 392 Wittich, 398 Xu, 107, 381 Yan, 390 Yang, 321, 387 Yorke, 390 Yosida, 398 Yu, 398 Zahariev, 81,382 Zeitlin, 344, 398 Zemanian, 298, 398 Zhang, 385, 390, 398 Zhitomirskii, 320, 382 Zima, 321, 398

Subject Index absolute value, 12, 13 absolutely integrable, 138 advanced differential equation, 104 advanced differential inequality, 94, 95, 101, 103, 104 algorithm, 57 almost periodic, 239-241 approximation, 3 uniformly good, 79 arbitrarily large zeros, 92, 150 argument, xi advanced, 29 continuous, ix deviation of, x, xii discontinuous, ix discrete, x lagging piecewise constant, 1 piecewise continuous, 1 linearly transformed, xi, xii piecewise constant, 157, 163, 191, 201 proportional to (, xiii, 371 retarded, 29 argument deviation, 81, 96, 115, 124, 137, 266, 309, 315, 317, 364, 371 Ascoli's theorem, 262, 264, 265 asymptotic behavior, 3, 51, 149 asymptotic properties, xi, 200 asymptotic series, 373 asymptotic stability, 14, 40, 70, 120, 199 autonomous system, 152

bounded inverse, 20 boundedness of solutions, 212 branching point, 367 Cauchy inequality, 281 Cauchy problem, 20, 22-25, 51, 52, 183, 184, 189 abstract, 176, 189 correct, 22, 23, 25 generalized, 52 uniformly correct, 25, 28, 53 weakened, 23 Cauchy-Schwarz inequality, 251, 252, 254 chaotic behavior, 2, 3 characteristic data, 201 characteristic roots, 268 characteristic variety, 182 Chebyshev polynomial, 335-337 Chebyshev's equation, 339 coefficient analytic, xii constant, 4 holomorphic, 372 infinitely smooth, xii polynomial, xii, xiii, 142, 309, 326, 328, 347, 365, 371, 372, 374 compact interval, 79 complete orthonormal set, 167 complicated dynamics, 3 constant lag, 70 continued fraction, 9-11, 13 branching, 19 infinite, 10 matrix, 19 continued-fraction expansion, 58 continuous linear functional, xii continuous vector field, 2, 6 control theory, vii, 3 convolution, 179 countable set, 170

backward continuation, 7, 8, 20, 22, 30, 33, 46, 48, 117, 129, 138, 140, 141 Banach space, 19, 25, 52, 261, 262 Bessel's equation, 302, 327 generalized, 374 of imaginary argument, 336 "best possible" condition, 92, 107, 123, 124, 134 binomial expansion, 185 boundary condition, 164, 170, 172, 174, 190, 196, 249, 252-254, 256, 263, 265 boundary-value problem, x, 164, 170,176, 184, 221,249-251,262,264

data on characteristics, 203 decay, xii delay, 2, 4, 71, 72 continuous, 201

403

404

S U B J E C T INDEX

delay, (cont'd) discontinuous, 164 discontinuous time, 190 discrete, 78, 201 piecewise constant, 78, 157, 180, 201 piecewise continuous, x, 163 unbounded, 82 delay differential inequality, 92, 93, 96, 99, 100 delayed action, 158 delta function, xii, 4, 292-295, 325 delta functional, 297 derivative distributional, 4, 297, 298 one-sided, 4, 19, 25, 49, 72, 84, 88, 115, 125, 137, 158, 165, 177, 205 strong, 19, 25 determinant, 9, 17, 30, 42 deviating argument, 213, 265 deviation proportional to the argument, 272 diagonal elements, 141 differentiable m a p , 239 differential equation, 1 advanced, 95 delay, 87, 93, 99, 100 functional, 29, 157, 213, 221, 231, 235, 238, 253, 256, 265, 269, 271, 306, 309, 320, 364, 370, 377, 378 impulsive, 3, 78 linear delay, 87, 89 linear functional, 91 loaded partial, 181 logistic, 3 neutral type, x partial, x, 163, 178 hyperbolic, 201 loaded, 184 with involutions, 265 reducible functional, 270 retarded, xiii, 82, 231 with involutions, 213, 217, 249 with lagging arguments, 1 with parameters, 31, 73, 82, 125 with piecewise constant arguments, 1 with piecewise constant delay, 178 with reflection of the argument, 222, 235, 241, 242, 244, 249 with rotation, 241 differential inequality, 81, 82, 84

diffusion, 163 Dirac delta function, xii, 292, 293, 326 Dirac delta functional, 179 direct product, 54, 232 discrete delay, x discrete spectrum, 177, 178 discrete times, 163, 164 disk, 272, 273, 276, 281, 373 distribution, xi, 295, 297, 301, 307, 308, 315, 353-355, 361, 363-365, 368, 376 finite-order, xii, 360, 364 infinite-order, 325 of slow growth, 361 order of, 326 regular, xii, 297 singular, xii, 297 slow growth, 322 tempered, 322 distribution theory, xi, 271, 292, 321, 325 domain, 19, 23-25, 171 constant, 53 dense, 23, 24, 53, 176 dynamical system, x continuous, 2 discrete, 2, 30 hybrid of, 2 eigenfunctions, 166, 167, 169, 176, 177 matrix, 192 orthogonal, 170 orthonormal, 170 eigenvalue, 137, 142, 144, 145, 152-154, 166, 167, 170, 177, 191, 195, 198, 199, 233, 289, 290 complex conjugate, 149 negative, 142, 143, 149 negative integer, 308, 311, 369 positive, 144, 146, 149 positive distinct, 147 real distinct, 147 with equal moduli, 149 eigenvector, 152-154, 233 energy space, 176 EPCA, x, xi, 1, 3, 78, 81, 163, 164, 181 advanced, xi alternately advanced and retarded, 157 approximation, x asymptotically stable, 157

S U B J E C T INDEX

E P C A , (cont'd) delay, 107 linear retarded, 4 linear with variable coefficients, 51, 59 mixed, xi neutral, xi, 157 nonlinear, 157 of advanced type, 28 of mixed type, 28 of neutral type, 28 oscillation of, 157 oscillatory, 157 retarded, xi stability of, 157 systems of, 82 with variable coefficients, xi equation, 2 absolutely nondegenerate, 182, 183 alternately advanced and delay, 157 alternately advanced and retarded, 82, 114, 115, 137 asymptotically stable, 70, 71, 157 autonomous difference, 30 characteristic, 19, 40, 68, 150, 157, 182, 215, 226, 227 confluent hypergeometric, 303, 325, 326, 350, 359, 372 degenerate, 182, 183 delay, xii, 44, 378 delay differential, 1, 78 determinantal, 233 difference, x, xi, 2, 3, 5, 11, 32, 56, 68, 79, 159 differential-difference, 27, 70, 71, 77, 81, 208, 370, 371 stability of, 69 diffusion-convection, 163 exponentially stable, 79 functional, x, 228 functional differential, ix-xi, 1, 2, 6 impulsive, x, xi reducible, xiii retarded, 2 with delay, ix, 1 homogeneous, 25, 44 hyperbolic, 201 hypergeometric, 326, 347, 355-358 impulsive, ix, 4, 72

405

equation, (cont'd) indicial, 224 integral, 273, 321 linear differential-difference, 96 linear functional differential, 63 linear neutral, 279 loaded, ix, x, 3, 4, 181 matrix, 196, 244, 307 neutral, 157 nondegenerate, 182 nonhomogeneous, 20 nonhomogeneous difference, 41 nonlinear advanced, 275 nonlinear hyperbolic, 201, 212 nonlinear neutral, 272 of advanced type, 2, 44, 115, 124, 219, 275 of delay type, 2 of mixed type, 2, 28, 55 of neutral type, 2, 44, 58 of retarded type, 2, 115, 124, 219 oscillatory, 157 parabolic, 193 partial differential, xiii reducible, 243, 244, 247 retarded nonlinear hyperbolic, 201 scalar, 57, 59, 62, 78 singular integral, 378 truncated, 344 wave, 190 with deviating arguments, 81 with discontinuous arguments, 1 with involutions, 213, 265 with linear argument transformations, 213 with linearly transformed arguments, xiii, 81, 320 with piecewise constant argument, 191 with rotation of the argument, 213, 241 with unbounded delay, 21 zero solution, 79 asymptotically stable, 68 estimate, 53, 210 Euler scheme, 3 Euler's equation, 215, 225, 226, 228, 247, 347 expansion, 172 exponential estimate, 27 exponential growth, 20, 369

406

SUBJECT INDEX

feedback control, 68 feedback delay, 3 finite group, 221 fixed point, xi, 217, 219, 220, 222, 262, 264266, 269, 272 Fourier coefficient, 172 Fourier expansion, 174 Fourier integral, 294 Fourier method, 176 Fourier series, 170 Fourier transform, 181, 182, 184, 321 Fourier transformation, 189, 307, 310, 322, 360 fractional part, 5, 8, 33 function, 295 abstract, 176, 177, 189 analytic, 148, 366 bounded, 20, 51, 64, 66 bounded uniformly, 27 discontinuous, 2 distribution, 296 entire, xi, 272, 283, 285, 291, 325, 365, 366, 373, 374, 378 entire of zero order, 286, 320 generalized, xii, 271, 295-297, 325 holomorphic, 273, 276, 330, 331 increasing, 71 infinitely smooth, 189, 295, 297 initial, viii, 196 iteration of, 83 locally integrable, xii, 296, 297, 363, 364 meromorphic, 182 monotinically increasing, 66 of rapid decay, 362 of zero order, 291 oscillatory, 174, 194, 269 positive, 66 rational, xii, 326, 329 real-valued, 169 unbounded, 194 functional, 296, 310, 315, 362 continuous linear, 295, 296, 361 linear, 310, 322 symmetric bilinear, 176 functional differential inequality, 121, 122 fundamental system of solutions, 225, 226, 231 fusion of solutions, 8, 75

gamma function, 338, 340, 373 Gegenbauer polynomial, 338, 339 general integral, 31, 73 generalized eigenmatrix pair, 234 greatest-integer function, xi, 1, 3, 4, 28, 74, 82, 91, 115, 124, 157, 163, 164, 178, 190, 201 Green's function, 250 Gronwall integral inequality, 78 half-integer points, 137 heat flow, 163 Heaviside function, 298, 361 Hermite polynomial, 333, 334 Hilbert space, x, 176, 178 hybrid control system, 3 hybrid system, x with time delay, 68 hypergeometric series, 360 identity matrix, 244 impulse, 292 increasing sequence, 72 index of parabolicity, 189 infinitely differentiable functions, 361 infinitely smooth functions of rapid decay, 361 initial condition, 4, 16, 20, 21, 23, 30, 32, 44, 46, 78, 94, 164, 177, 179, 180, 185, 189, 190, 197, 218, 219, 222, 227, 232, 233, 246, 266, 286 pointwise, 21 initial data, ix, 2, 22 initial function, 2, 166, 168, 170, 175, 190 initial interval, 2 initial value, 265 initial vector, 143, 147, 148, 153, 154 initial-value problem, x, xi, 2, 4, 8, 25-28, 31, 33, 44, 49, 55, 72, 82, 137, 164, 188, 233, 249, 272, 276 homogeneous degenerate, 184 inner product, 170, 176 instability nonoscillatory, 71 oscillatory, 71 integral endpoints, 19, 35 integral transformation, xi, 184, 272, 319, 325, 371

S U B J E C T INDEX

integral values, 21 intervals of constancy, xi, 1 inverse problem, 181 involution, 215, 216, 218, 219, 222, 229, 265, 266 hyperbolic, 222, 269 strong, 216-219 iteration, 126, 220 iteration of a function, 215 iteration process, 63, 65, 207, 314 Jacobi equation, 356 Jacobi polynomial, 355, 356 Jordan matrix, 141, 152 Kronecker product, 232 lagging times, 159 Laguerre polynomial, 340, 341 Laguerre's equation, 340-342 Laplace transform, 292, 323, 324, 327-329, 345, 347, 362-366, 369-371, 374 bilateral, 362 formal, 363, 364 inverse, 187 two-sided, 186, 187 Laplace transformation, 301, 320, 322, 327, 329, 331, 332, 334, 336, 339, 341, 345, 347, 349, 350, 358, 360 Legendre polynomial, 331, 332, 359 Legendre's equation, 339 Liapunov function, 231, 379 Liapunov functional, xiii, 231, 235 linear argument transformation, 272 linear combination, 195 linear problem, 75 Lipschitz condition, 205, 206, 208, 212, 262, 274, 278 locally integrable, 140, 141 mapping, 89, 215, 261 contraction, 239, 263 hyperbolic involutory, 217 matrix, 137, 142, 233, 308, 311, 368, 369 commutative, 229 diagonal, 141, 144, 146, 151-153 diagonalizable, 152 entry of, 151, 195 identity, 192, 310

407

matrix, (cont'd) inverse, 281, 283, 313 negative definite, 144 nonsingular, 47, 56, 58,140, 192, 316, 317 norm of, 63, 279, 310 uniformly bounded, 63, 65 positive definite, 231 maximal interval, 69 method of steps, 166, 175, 208 method of successive approximations, 205, 209 mixed problem, 265 moduli, 13 modulus, 308 monotone iterative technique, 82, 87 monotone sequence, 87, 89 neighborhood, 294, 298, 306, 308, 315-318, 330, 331, 345, 364 non-negative integer root, 300, 301 non-negative integer zero, 302 nonpositive integer roots, 308 nonzero vector, 30 norm, 176, 369 numerical approximation, 3 open unit disk, 19 operator, 20, 23, 24, 27, 28, 52, 53, 169, 172, 176, 222, 225, 238, 242, 243, 246, 248, 273 abstract elliptic, 24 bounded inverse, 52 bounded linear, 20, 51 closed, 23, 53 differential, 347 evolution, 52 identity, 243 infinitely smooth semigroup, 189 linear, 176, 178 linear constant, 19, 23, 25 linear differential, 180, 231, 243 positive, 176 positive definite, x, 176-178 resolvent of, 24 semigroup, 26 semigroup of, 23 semigroup of multiplication, 189 strongly continuous, 23, 51, 53, 54 symmetric, 176, 177

408

S U B J E C T INDEX

operator, (cont'd) unbounded, 53 uniformly bounded, 54 order of distribution, 292 order of growth, 282, 285, 291, 365, 366, 370 orthogonal polynomial, 330, 331, 358, 360 oscillation, x, xi, 81, 82, 123 oscillation in systems, 137 oscillatory component, 143, 149, 200 oscillatory properties, 81, 91, 105, 156, 190 parameter space, 70 partial sum, 343 period, 38, 39, 114, 124, 152-154 period three, 3 periodicity, x, xi piecewise constant analogue, 157 piecewise constant time, 163, 190 Planck's law, 321 pole, 326, 349, 363, 367 polynomial coefficient, 282, 306 quantum mechanics, 19 recursion formula, 167 recursion relation, ix, 2, 5, 7, 18, 24, 167, 181, 205, 226, 283, 372 regular point, 23 regular singular point, 289, 292 residue, 349, 363 Rodrigues' formula, 332 root, 8, 12, 15, 1 7 , 3 1 , 4 0 , 4 2 , 6 9 , 70 modulus of, 68 non-negative integer, 330, 331, 358, 359 nonzero integer, 355 of unity, 153, 171 positive, 157 scalar problem, 20, 65 scalar product, 144 Schauder's theorem, 262, 264, 265 Shrodinger equation, 175 self-adjoint, 170 semigroup, 28 separation of variables, xiii, 165, 168, 172, 174, 175, 190, 196,265, 267 sequence complete, 166, 192 orthonormal, 166, 192

sharp condition, 93, 95, 123, 124, 134 Silberstein's equation, 228 singular point, 339, 342 singularity, xii, 271, 289, 309, 364-367, 371 small oscillations, 19 solution, 13, 14, 17, 19, 30 analytic, xi, xiii, 44, 271, 272, 276, 279, 325, 340 asymptotic, 81 asymptotically stable, 13, 14, 17, 19, 31, 33-35, 39, 41, 49, 69, 112, 118, 129, 130, 141, 195, 198, 199,201 bounded, 43, 124, 135, 199, 201, 207, 208, 235, 237, 239, 369 classical, 271, 326 component of, 144, 147 nonoscillatory, 147 oscillatory, 147, 149 decreasing, 161 distributional, xi-xiii, 63, 271, 272, SOSSOS, 309,318,319,325-330,332-334, 336,338-348,351, 352, 356-358,360, 365, 366, 369, 370, 374-377 finite order, 292 infinite order, xii, 309 entire, xi-xiii, 272, 282, 325, 371, 377 of zero order, xii transcendental, xii entire transcendental, 271, 371 entry of, 195 eventually negative, 93, 95, 99, 100, 104, 122, 123 eventually positive, 92-95, 97, 99-101, 104,121-123 exponentially asymptotically stable, 66 finite-order, 292, 303, 304, 349, 350, 367 finite-order distributional, 306, 325, 329, 340, 342, 347, 349, 354, 366, 367 formal, 166, 167, 372, 375 formal distributional, 344, 360, 361, 364, 372-374 formal infinite-order distributional, 369 fundamental, 179, 187 general, 5-7, 16-18, 30, 4 1 , 44, 45, 75, 116, 128,224, 2 3 0 , 2 4 7 , 2 4 8 generalized, ix generalized-function, xi, 271, 306 growth of, 54, 59

SUBJECT INDEX

solution, (cont'd) holomorphic, 272, 275, 276, 283, 285, 286, 319 increasing, 160 infinite-order, 311, 366, 367, 370, 371 infinite-order distributional, 364, 374, 378 infinitely many, 38 linearly independent, 171, 231, 233, 234 lower, 82, 84, 86, 89, 90 matrix, 289 maximal, 87, 89, 91 minimal, 87, 89, 91 monotonic, 159 negative, 101, 105 non-negative integer, 354 nonoscillating, 230 nonoscillatory, 93, 94, 96, 100, 101, 104, 105, 144 nonoscillatory component of, 146 nontrivial, 18, 30, 120, 159, 161, 184, 299, 301,307, 369 nontrivial oscillatory, 161 nonzero, 18, 129, 130, 135, 300, 302, 348 null stability of, 68 of zero order, 272, 372 oscillating, 3, 230 oscillatory, 81, 91-93, 95, 96, 99, 104, 106, 107,112-114,122,123,132,134,135, 137, 142, 143, 151, 153, 157,200 particular, 17 periodic, 3, 38, 39, 81, 107, 112-114, 132, 134-137, 152-154, 157 polynomial, xii, 325-329, 331, 339, 342, 343, 345, 349, 351, 352, 354, 355, 358, 359 positive, 105, 160 rational, 325, 326, 328-330, 333, 334, 336, 337, 341, 342, 347, 348, 357, 358, 366 regular, 280 singular distributional, xii, 352 small, xii, 378 stable, 33-35 strongly continuously differentiable, 177 two-sided, xi, 2 unbounded, 37-39, 159, 160, 199, 200, 267, 269 unique, 94

409

solution, (cont'd) unstable, 119, 197, 198, 200 upper, 82, 84, 86, 87, 89 weak, 176-178 weakened, 23 with precisely one zero, 35, 37-39 zero, 69 space, 273, 295, 314 bounded operators, 52 contraction of, 275, 279 finite-order distributions, xii, 298 generalized functions, xi, 312, 316, 364, 374, 378 generalized-function, 309 infinite-order distributions, xii infinitely smooth functions, xii metric, 274, 278 tempered distributions, 321 vector, 231 spectrum, 20, 321 stability, x, xi, 59, 68, 269 stability region, 70-72 stabilization, 3 step function, 72, 75 Stirling's formula, 319, 375, 376 substitution, 156, 172, 195, 196, 202 support, 300, 308, 310, 311, 322, 342, 364 bounded, 296 compact, 184, 295 of distribution, 322 of test function, 322 system parabolic by Shilov, 189 Taylor coefficient, 280 Taylor expansion, 299, 306 Taylor sum, 335, 341 Taylor's formula, 259 test function, 295, 297, 310, 322, 365 rapid descent, 322 theory of distributions, 292 uniformly bounded inverse, 59 uniqueness class, 183 unit circle, 70 unit impulse, 293 unit step function, 293 Vandermonde's determinant, 17, 30, 42

410

variable monotonically decreasing, 64 variation of parameters,! 42, 78, 231 vector locally integrable, 41 vector-function locally integrable, 138 Volterra integral equation, 201, 203 weak convergence, 296 weight distributions, 360

SUBJECT INDEX