A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions

In this paper, we discuss a class of integrodifferential equations with nonlocal conditions via a fractional derivative of the type: $$\begin{aligned}D_{t}^{q}x(t)&=Ax(t)+\int\limits_{0}^{t}B(t-s)x(s)ds+t^{n}f\left(t,x(t)\right),&&t\in [0,T],\;n\in Z^{+},\\&&&q\in(0,1],\;x(0)=g(x)+x_{0}.\end{aligned}$$ Some sufficient conditions for the existence of mild solutions for the above system are given. The main tools are the resolvent operators and fixed point theorems due to Banach's fixed point theorem, Krasnoselskii's fixed point theorem and Schaefer's fixed point theorem. At last, an example is given for demonstration.