Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay

We deal in this paper with the mild solution for fractional semilinear differential equations with infinite delay: Dαx(t)=Ax(t)+f(t,xt,Bx(t)),t∈[0,T],x(t)=ϕ(t),t∈]-∞,0] with T>0T>0 and 0<α<10<α<1. We prove the existence (and uniqueness) of solutions, assuming that A   generates an αα-resolvent family (Sα(t))t⩾0(Sα(t))t⩾0 on a complex Banach space XX by means of classical fixed points methods.