ASYMPTOTIC-BEHAVIOR OF SOLUTIONS OF SYSTEMS OF NEUTRAL AND CONVOLUTION EQUATIONS

Suppose J = [alpha, infinity) for some alpha is an element of R or J = R and let X be a Banach space. We study asymptotic behavior of soluti ons on J of neutral system of equations with values in X. This reduces to questions concerning the behavior of solutions of convolution equa tions () H * Omega = b, where H = (H-j,H-k) is an r x r matrix, H-j,H -k is an element of D-L1', b = (b(j)) and b(j) is an element of L' (R, X), for 1 less than or equal to j, k less than or equal to r. We prov e that if Omega is a bounded uniformly continuous solution of ()with b from some translation invariant suitably closed class A, then a belo ngs to A, provided, for example, that det fi has countably many zeros on R and c(0) not subset of X. In particular, if b is (asymptotically) almost pet iodic, almost automorphic or recurrent, R is too. Our resu lts extend theorems of Bohr, Neugebauer, Bochner, Doss, Basit, and Zhi kov and also, certain theorems of Fink, Madych, Staffans, and others. Also, we investigate bounded solutions of (). This leads to an extens ion of the known classes of almost periodicity to larger classes calle d mean-classes. We explore mean-classes and prove that bounded solutio ns of () belong to mean-classes provided certain conditions hold. The se results seem new even for the simplest difference equation Omega(t + 1) - R(t) = b(r) with J = X = W and b Stepanoff almost periodic. (C) 1998 Academic Press.