On the stability of stationary solutions of a linear integro-differential equation

A Ya Dorogovtsev,O Yu Trofimchuk

doi:10.1155/s1048953301000107

Abstract

In this paper the following two connected problems are discussed. The problem of the existence of a stationary solution for the abstract equation ϵx″(t)+x′(t)=Ax(t)+∫−∞tE(t−s)x(s)ds+ξ(t),t∈R containing a small parameter ϵ in Banach space B is considered. Here A∈ℒ(B) is a fixed operator, E∈C([0,+∞),ℒ(B)) and ξ is a stationary process. The asymptotic expansion of the stationary solution for equation (1) in the series on degrees of e is given.We have proved also the existence of a stationary with respect to time solution of the boundary value problem in B for a telegraph equation (6) containing the small parameter ϵ. The asymptotic expansion of this solution is also obtained.

Highlights

II" Let (B ]]) be a complex Banach space, 0 the zero element in B, and (B) the Banach space of bounded linear operators on B with the operator norm, denoted by the symbol I1" I1" For a B-valued function, continuity and differentiability refer to continuity and differentiability in the B-norm
For an (B)-valued function, continuity is the continuity in the operator norm
Proof: We return to the proof of Lemma 3 where the stationary solution x for equation (3) was given

Summary

Kiev University

Mechanics and Mathematics Department Vladimirskay 6, 01033 Kiev-33 Ukraine (Received October, 1999; Revised November, 2000). In this paper the following two connected problems are discussed. The problem of the existence of a stationary solution for the abstract equation. Containing a small parameter e in Banach space B is considered. A E (B) is a fixed operator, E C([0, -t-c),(B)) and is a stationary process. The asymptotic expansion of the stationary solution for equation (1). In the series on degrees of is given. We have proved the existence of a stationary with respect to time. Containing the small parameter The asymptotic expansion of this solution is obtained. AMS subject classifications: 34G10, 60G20, 60H15, 60H99

Introduction

It is clear that and for every

It o

Let Yl be a unique stationary solution for equation