Sufficient Conditions of Ergodicity of Solutions of Second Order Abstract Linear Differential Equations

This paper is devoted to second order abstract linear differential equations in a Banach space. For such equations the Cauchy problem is stated, and the behavior of its solutions as is examined. The aim of the paper is to study ergodicity and asymptotic behavior of the solutions of the strongly correct Cauchy problem. For this purpose the theory of complete second order linear differential equations in Banach spaces, developed by Fattorini, is used. As shown in the paper, for a wide class of equations the solutions are either ergodic or unbounded, depending on the initial values.