Compact almost automorphic solutions to integral equations with infinite delay

Given a∈L1(R)a∈L1(R) and the generator AA of an L1L1-integrable resolvent family of linear bounded operators defined on a Banach space XX, we prove the existence of compact almost automorphic solutions of the semilinear integral equation u(t)=∫−∞ta(t−s)[Au(s)+f(s,u(s))]ds for each f:R×X→Xf:R×X→X compact almost automorphic in tt, for each x∈Xx∈X, and satisfying Lipschitz and Hölder type conditions. In the scalar linear case, we prove that a∈L1(R)a∈L1(R) positive, nonincreasing and log-convex is sufficient to obtain the existence of compact almost automorphic solutions.