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
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
Copyright © 2001 A. Ya. Dorogovtsev and O. Yu. Trofimchuk. 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.
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.