Almost Automorphic Solutions to Abstract Volterra Equations on the Line

Given a?L1(?) and A the generator of an L 1-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of an almost automorphic mild solution to the semilinear integral equation u(t)=?-? ta(t-s)[Au(s)+f(s,u(s))]ds for each f:?×X?X S p-almost automorphic in t, uniformly in x?X, and satisfying diverse Lipschitz type conditions. For the scalar linear case, we prove that a?L1(R) completely monotonic is already sufficient. © 2011 Elsevier Ltd. All rights reserved.