Weighted pseudo antiperiodic solutions for fractional integro-differential equations in Banach spaces

In this paper we prove the existence of weighted pseudo antiperiodic mild solutions for fractional integro-differential equations in the formDαu(t)=Au(t)+∫-∞ta(t-s)Au(s)ds+f(t,u(t)),t∈R,where f(·,u(·))f(·,u(·)) is Stepanov-like weighted pseudo antiperiodic and A   generates a resolvent family of bounded and linear operators on a Banach space X,a∈Lloc1(R+) and α>0α>0. Here the fractional derivative is considered in the sense of Weyl. Also, we give a short proof to show that the vector-valued space of Stepanov-like weighted pseudo antiperiodic functions is a Banach space.