Almost automorphic solutions to abstract Volterra equations on the line

Given a∈L1(R)a∈L1(R) and AA the generator of an L1L1-integrable family of bounded and linear operators defined on a Banach space XX, 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:R×X→Xf:R×X→X SpSp-almost automorphic in tt, uniformly in x∈Xx∈X, and satisfying diverse Lipschitz type conditions. For the scalar linear case, we prove that a∈L1(R)a∈L1(R) completely monotonic is already sufficient.