Generalized Solutions of Functional Differential Equations

✍ Scribed by Joseph Wiener

• Publisher: World Scientific
• Year: 1993
• Tongue: English
• Leaves: 425
• Category: Library

✦ Synopsis

This book addresses the nedd for investigation of functional differential equations with discontinuous delays. Such equations provide a mathematical model for a physical or biological system in which the rate of change of the system depends upon its past history. Work is described that has been done by the author and others over thelast few years on differential equations with piecewise continuous arguments (EPCA). They include, as particular cases, impulse and loaded equations of control theory and are similar in structure to those found in certain sequential-continuous models of disease dynamics. Considerable attention is given to the study of initial and boundary-value problems for partial differential equations of mathematical physic with discontinuous time delays. A large part of the book is devoted to the exploration of differential and functional differential equations in spaces of generalized functions (distributions) and contains a wealth of new information in this area. In each of the directions discussed in this book, there appears to be ample opportunity for extending the known results.

Contents

Preface

CHAPTER 1 Differential Equations with Piecewise Continuous Arguments

 1. Linear Retarded EPCA with Constant Coefficients

 2. Some Generalizations

 3. EPCA of Advanced, Mixed, and Neutral Types

 4. Asymptotic Behavior of Linear EPCA with Variable Coefficients

 5. Stability as a Function of Delay

 6. EPCA and Impulsive Equations

CHAPTER 2 Oscillatory and Periodic Solutions of Differential Equations with Piecewise Continuous Arguments

 1. Differential Inequalities with Piecewise  continuous Arguments

 2. Oscillatory Properties of First-Order Linear Functional Differential Equations

 3. Oscillatory and Periodic Solutions of Delay EPCA

 4. Differential Equations Alternately of Retarded and Advanced Type

 5. Oscillations in Systems of Differential Equations with Piecewise Continuous Arguments

 6. A Piecewise Constant Analogue of a Famous FDE

CHAPTER 3 Partial Differential Equations with Piecewise Continuous Delay

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

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

 3. A Wave Equation with Discontinuous Time Delay

 4. Bounded Solutions of Retarded Nonlinear Hyperbolic Equations

CHAPTER 4 Reducible Functional Differential Equations

 1. Differential Equations with Involutions

 2. Linear Equations

 3. Bounded Solutions for Differential Equations with Reflection of the Argument

 4. Equations with Rotation of the Argument

 5. Boundary-Value Problems for Differential Equations with Relfection of the Argument

      5.1. Preliminary Results. F

      5.2. Main Results.

 6. Partial Differential Equations with Involutions

CHAPTER 5 Analytic and Distributional Solutions of Functional Differential Equations

 1. Holomorphic Solutions of Nonlinear Neutral Equations

 2. Holomorphic Solutions of Nonlinear Advanced Equations

 3. Analytic and Entire Solutions of Linear Systems

 4. Finite-Order Distributional Solutions

 5. Infinite-Order Distributional Solutions

 6. An Integral Equation in the Space of Tempered Distributions

CHAPTER 6 Coexistence of Analytic and Distributional Solutions for Linear Differential Equations

 1. Distributional, Rational, and Polynomial Solutions of Linear ODE

 2. Application to Orthogonal Polynomials

 3. Interesting Properties of Laguerre's Equation

 4. The Hypergeometric and Other Equations

 5. The Confluent Hypergeometric Equation

 6. Infinite-Order Distributional Solutions Revisited

Open Problems

Bibliography

Author Index

Subject Index