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).