Compact almost automorphic solutions to integral equations with infinite delay - Henríquez, Hernán R.; Lizama, Carlos (Nonlinear analysis, 2009)
Given a ∈ L 1 ( R ) and the generator A of an L 1 -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 ) = ∫ − ∞ 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 ∈ L 1 ( R ) positive, nonincreasing and log-convex is sufficient to obtain the existence of compact almost automorphic solutions.