Asymptotic Behavior of Solutions of Systems of Neutral and Convolution Equations

Suppose =[α, ∞) for someα or = and letXbe a Banach space. We study asymptotic behavior of solutions on of neutral system of equations with values inX. This reduces to questions concerning the behavior of solutions of convolution equations (*)H*Ω=b, whereH=(Hj, k) is anr×rmatrix,Hj, k ′L1,b=(bj) andbj ′( , X), for 1 j, k r. We prove that ifΩis a bounded uniformly continuous solution of (*) withbfrom some translation invariant suitably closed class , thenΩbelongs to , provided, for example, that det Hhas countably many zeros on andc0 X. In particular, ifbis (asymptotically) almost periodic, almost automorphic or recurrent,Ωis too. Our results extend theorems of Bohr, Neugebauer, Bochner, Doss, Basit, and Zhikov and also, certain theorems of Fink, Madych, Staffans, and others. Also, we investigate bounded solutions of (*). This leads to an extension of the known classes of almost periodicity to larger classes called mean-classes. We explore mean-classes and prove that bounded solutions of (*) belong to mean-classes provided certain conditions hold. These results seem new even for the simplest difference equationΩ(t+1)−Ω(t)=b(t) with =X= andbStepanoff almost periodic