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.