Almost periodic and almost automorphic solutions of linear differential equations

Abstract

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