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

[摘要] In this paper the following two connected problems are discussed. The problem of the existence of a stationary solution for the abstract equationϵx″(t)+x′(t)=Ax(t)+∫−∞tE(t−s)x(s)ds+ξ(t),t∈Rcontaining a small parameterϵin Banach spaceBis considered. HereA∈ℒ(B)is a fixed operator,E∈C([0,+∞),ℒ(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 parameterϵ. The asymptotic expansion of this solution is also obtained.

[学科分类] 应用数学