A Note on a Semilinear Fractional Differential Equation of Neutral Type with Infinite Delay

We deal in this paper with the mild solution for the semilinear fractional differential equation of neutral type with infinite delay: Dαx(t)+Ax(t)=f(t,xt), t∈[0,T], x(t)= (t), t∈] ∞,0], with T>0 and 0<α<1. We prove the existence (and uniqueness) of solutions, assuming that A is a linear closed operator which generates an analytic semigroup (T(t))t≥0 on a Banach space by means of the Banach's fixed point theorem. This generalizes some recent results.