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