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

Dorogovtsev, A. Ya.; Trofimchuk, O. Yu.

doi:https://doi.org/10.1155/S1048953301000107

International Journal of Stochastic Analysis

/ / Article

On this page

AbstractCopyrightRelated articles

Open Access

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

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.

No related content is available yet for this article.

PDF

Download Citation

Citation

Order printed copiesOrder

Downloads757

Citations