Generalized Solutions of Functional Differential Equations

425 Pages·1993·6.85 MB·English

by  Joseph Wiener

Generalized Solutions of
Functional Differential Equations
This page is intentionally left blank
GENERALIZED
SOLUTIONS OF
FUNCTI ONAL
DIFFERENTIAL
UATIONS
JOSEPH WIENER
The University of Texas-Pan American
USA
'
World Scientific
I
Singapore New Jersey• London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P0 Box 128, Farrer Road, Singapore 9128
USA office: Suite 1B, 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 This book, or parts thereof may not be repmducedin any form
orby any means, electronic or mechanical, including photocopying, recordingorany
information storage and retrieval system now known or to be invented, without
written permission from the Publisher.
ISBN 981-02-1207-0
Fx 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
Chapter Differential Equations with Piecewise
1. Continuous Arguments
1.1 Linear Retarded EPCA with Constant Coefficients 4
1.2 Some Generalizations 16
1.3 EPCA of Advanced, Mixed, and Neutral Types 28
1.4 Asymptotic Behavior of Linear EPCA with
Variable Coefficients 51
1.5 Stability as a Function of Delay 68
1.6 EPCA and Impulsive Equations 72
Chapter Oscillatory and Periodic Solutions of
Differential Equations with Piecewise
Continuous Arguments 81
2.1 Differential Inequalities with Piecewise
Continuous Arguments 82
2.2 Oscillatory Properties of First-Order Linear
Functional Differential Equations 91
V
vi CONTENTS
2.3 Oscillatory and Periodic Solutions of Delay EPCA 107
2.4 Differential Equations Alternately of Retarded
and Advanced Type 114
2.5 Oscillations in Systems of Differential Equations
with Piecewise Continuous Arguments 137
2.6 A Piecewise Constant Analogue of a Famous FDE 157
Chapter Partial Differential Equations with
.1 Piecewise Continuous Delay 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 A Wave Equation with Discontinuous Time Delay 190
3.4 Bounded Solutions of Retarded Nonlinear Hyper-
bolic Equations 201
Chapter A Reducible Functional Differential
9 Equations
213
4.1 Differential Equations with Involutions 213
4.2 Linear Equations 222
4.3 Bounded Solutions for Differential Equations with
Reflection of the Argument 235
4.4 Equations with Rotation of the Argument 241
4.5 Boundary-Value Problems for Differential
Equations with Reflection of the Argument 249
4.6 Partial Differential Equations with Involutions 265
CONTENTS vii
Chapter Analytic and Distributional Solutions
of Functional Differential Equations 271
5.1 Holomorphic Solutions of Nonlinear Neutral
Equations 272
5.2 Holomorphic Solutions of Nonlinear Advanced
Equations 275
5.3 Analytic and Entire Solutions of Linear Systems 279
5.4 Finite-Order Distributional Solutions 292
5.5 Infinite-Order Distributional Solutions 309
5.6 An Integral Equation in the Space of Tempered
Distributions 321
Chapter Coexistence of Analytic and
U Distributional Solutions for Linear
Differential Equations 325
6.1 Distributional, Rational, and Polynomial
Solutions of Linear ODE 326
6.2 Application to Orthogonal Polynomials 330
6.3 Interesting Properties of Laguerre's Equation 340
6.4 The Hypergeometric and Other Equations 347
6.5 The Confluent Hypergeometric Equation 359
6.6 Infinite-Order Distributional Solutions Revisited 364
Open Problems 379
Bibliography 381
Author Index 399
Subject Index 403
This page is intentionally left blank
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 mi-
ix