ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

We analyze the existence of almost periodic (respectively, almost automorphic, recurrent) solutions of a linear non-homogeneous differential (or difference) equation in a Banach space, with almost periodic (respectively, almost automorphic, recurrent) coeffcients. Under some conditions we prove that one of the following alternatives is fulfilled: (i) There exists a complete trajectory of the corresponding homogeneous equation with constant positive norm; (ii) The trivial solution of the homogeneous equation is uniformly asymptotically stable. If the second alternative holds, then the non-homogeneous equation with almost periodic (respectively, almost automorphic, recurrent) coefficients possesses a unique almost periodic (respectively, almost automorphic, recurrent) solution. We investigate this problem within the framework of general linear nonautonomous dynamical systems. We apply our general results also to the cases of functional-differential equations and difference equations.