In this paper the following two connected problems are discussed. The problem of the existence of a stationary solution for the abstract equation?x″(t)+x′(t)=Ax(t)+∫?∞tE(t?s)x(s)ds+ξ(t),t∈Rcontaining a small parameter?in Banach spaceBis considered. HereA∈?(B)is a fixed operator,E∈C([0,+∞),?(B))andξis 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 inBfor a telegraph equation (6) containing the small parameter?. The asymptotic expansion of this solution is also obtained.
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