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International Journal of Stochastic Analysis

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Volume 14 |Article ID 265389 | https://doi.org/10.1155/S1048953301000107

A. Ya. Dorogovtsev, O. Yu. Trofimchuk, "On the stability of stationary solutions of a linear integro-differential equation", International Journal of Stochastic Analysis, vol. 14, Article ID 265389, 12 pages, 2001. https://doi.org/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, Ukraine

2Kiev University, Mechanics and Mathematics Department, Vladimirskay 64, Ukraine

Received01 Oct 1999
Revised01 Nov 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.

Copyright © 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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