Existence of S-asymptotically ω-periodic solutions to abstract integro-differential equations

•We study the existence of solutions for a class of abstract equations.•We construct an exponentially stable resolvent operator.•Solutions S-asymptotically ω-periodic are obtained.•We obtained existence results with less restrictions on the constants.•We provide a concrete example for the abstract results.

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 formddt[x(t)+∫0tN(t-s)x(s)ds]=Ax(t)+∫0tB(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 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.