Abstract

Given a?L1(R) and A the generator of an L1-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=?-8 ta(t-s)[Au(s)+f(s,u(s))]ds for each f:R×X?X almost automorphic in t, uniformly in x?X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a?L1(R) positive, nonincreasing and log-convex is already sufficient. © Springer Science+Business Media, LLC 2009.