LEVITAN ALMOST PERIODIC SOLUTIONS OF INFINITE-DIMENSIONAL LINEAR DIFFERENTIAL EQUATIONS

Abstract

The known Levitan’s Theorem states that the finite-dimensional linear differential equationx′=A(x+f(t)(1)with Bohr almost periodic coefficientsA(t) and f(t) admits at least one Levitan almostperiodic solution if it has a bounded solution. The main assu mption in this theoremis the separation among bounded solutions of homogeneous eq uationsx′=A(t)x .(2)In this paper we prove that infinite-dimensional linear differential equation (3) withLevitan almost periodic coefficients has a Levitan almost periodic solution if it has at least one relatively compact solution and the trivial solut ion of equation (2) is Lyapunov stable. We study the problem of existence of Bohr/Levi tan almost periodicsolutions for infinite-dimensional equation (3) in the fram ework of general nonau tonomous dynamical systems (cocycles).