Abstract

Given a ? L1 (R) and the generator A of an L1-integrable resolvent family of linear bounded operators defined on a Banach space X, we prove the existence of compact almost automorphic solutions of the semilinear integral equation u (t) = ?- 8 t a (t - s) [A u (s) + f (s, u (s))] d s for each f : R × X ? X compact almost automorphic in t, for each x ? X, and satisfying Lipschitz and Hölder type conditions. In the scalar linear case, we prove that a ? L1 (R) positive, nonincreasing and log-convex is sufficient to obtain the existence of compact almost automorphic solutions. © 2009 Elsevier Ltd. All rights reserved.