Levitan almost periodic solutions of infinite-dimensional linear differential equations

David Cheban

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2019), pp. 56-78

Résumé

The known Levitan's Theorem states that the finite-dimensional linear differential equation \begin{equation} x'=A(t)x+f(t) \end{equation} with Bohr almost periodic coefficients $A(t)$ and $f(t)$ admits at least one Levitan almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equations \begin{equation} x'=A(t)x. \end{equation} In this paper we prove that infinite-dimensional linear differential equation (1) with Levitan almost periodic coefficients has a Levitan almost periodic solution if it has at least one relatively compact solution and the trivial solution of equation (2) is Lyapunov stable. We study the problem of existence of Bohr/Levitan almost periodic solutions for infinite-dimensional equation (1) in the framework of general nonautonomous dynamical systems (cocycles).