Journal of Applied Mathematics and Stochastic Analysis 
Volume 14 (2001), Issue 2, Pages 139-150
doi:10.1155/S1048953301000107

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

A. Ya. Dorogovtsev1 and O. Yu. Trofimchuk2

 1Kiev Institute of Business and Technology, Blvd. T. Shevchenko, 4, 311, Kiev-33 01033, Ukraine
2Kiev University, Mechanics and Mathematics Department, Vladimirskay 64, Kiev-33 01033, Ukraine
Received 1 October 1999; Revised 1 November 2000

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.