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

Abstract

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 \[t\rightarrow +\infty\] 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. For the solutions to be ergodic, conditions on the linear operators-coefficients of the differential equation and the initial values of the Cauchy problem are obtained. In case of ergodic solutions, exact values of ergodic limits are given. In case of unbounded solutions, asymptotic behavior of solutions is described. Results obtained in this paper are a generalization of the previously known results concerning ergodic properties of the solutions for the Cauchy problem for the incomplete second order equations.