Almost periodic and almost automorphic solutions of linear differential equations (Q1948776)

In this work the authors generalize the results of \textit{P. Cieutat} and \textit{A. Haraux} [Port. Math. (N.S.) 59, No. 2, 141--159 (2002; Zbl 1018.34049)] to the case of general linear nonautonomous dynamical systems (cocycles). More precisely, they describe the asymptotic behavior of nonautonomous linear contractive systems with minimal base and prove that, under some contracting and compactness assumptions, the associated homogeneous system either has a complete trajectory with constant positive norm, or the trivial solution is uniformly asymptotically stable. Moreover, when the second alternative holds, the nonautonomous dynamical system possesses a unique solution of the same kind as the coefficients; that is, almost periodic (resp. almost automorphic, recurrent) coefficients give rise to an almost periodic (resp. almost automorphic, recurrent) solution.