On Massera and Bohr-Neugebauer type theorems for some almost automorphic differential equations

Abstract

This paper is a continuation of the work in the literature [21] related to the differential equation x′=A(t)x+f(t). We give new Massera and Bohr-Neugebauer type theorems when the coefficients are almost automorphic. For the Massera theorem, unlike in the almost periodic case (see [21]), we show that Favard's condition is only needed to hold for A(t) (not necessarily for all its hull) in order to obtain a compactly almost automorphic solution when the coefficients are almost automorphic. Moreover, we give many examples of matrices A(t) satisfying Favard's condition. We conclude the work by some new results of Bohr-Neugebauer type (all bounded solutions are almost automorphic).