Distributional and entire solutions of ordinary differential and functional differential equations

Joseph Wiener,S. M. Shah

doi:10.1155/s0161171283000216

Abstract

A brief survey of recent results on distributional and entire solutions of ordinary differential equations (ODE) and functional differential equations (FDE) is given. Emphasis is made on linear equations with polynomial coefficients. Some work on generalized-function solutions of integral equations is also mentioned.

Highlights

This paper may be considered as a continuation of [I] which contains, in partfcular, a survey of recent results on entire solutions of ordinary differential equations (ODE) with polynomial coefffclents
In [49] the authors investigate the solvability of a class of functional equations, containing as a particular case differential equations of finite and of infinite order with constant coefficients, in the Banach space with weight of entire functions
It should be noted that, perhaps, the first work of this kind was [83]

Summary

INTRODUCTION

A unified approach may be used fn the study of both dlstrfb-utional and entire solutions to some classes of linear ODE and, especially, FDE with linear transformations of the argument [3]. It fs well known [4] that normal linear homogeneous systems of ODE with fnffnltely dffferentlable coefficients have no generalized-function solutions other than the classical. I. Distributional solutions to linear homogeneous FDE may be originated either by singularities of their coefficients or by deviations of argument.

DISTRIBUTIONAL AND ENTIRE SOLUTIONS OF ODE

DISTRIBUTIONAL AND ENTIRE SOLUTIONS OF FDE