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

In this paper the following two connected problems are discussed. The problem of the existence of a stationary solution for the abstract equation ϵ x ″ ( t ) + x ′ ( t ) = A x ( t ) + ∫ − ∞ t E ( t − s ) x ( s ) d s + 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.