Existence of S-Asymptotically ?-Periodic Solutions to Abstract Integro-Differential Equations

The main aim of this work is to study the existence of S-asymptotically ω-periodic solutions for a class of abstract integro-differential equations modeled in the following form d/dt[x(t) + ∫0t N(t-s)x(s)ds] = Ax(t) + ∫0t B(t-s)x(s)ds + f(t,x(t)), t ≥ 0, x(0) = x0 ∈ X, where A, B(t) for t ≥ 0 are closed linear operators defined on a common domain D(A) which is dense in X,N(t) for t ≥ 0 are bounded linear operators on X, and f : [0, ∞) x X → X is an appropriate function. The existence results are obtained by applying the theory of exponentially stable resolvent operators. We also discuss an application of these results. © 2015 Elsevier Inc. All rights reserved.