## A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions.(English)Zbl 1235.34210

Summary: We discuss a class of integrodifferential equations with nonlocal conditions via a fractional derivative of the type: $\begin{split} D^q_tx(t)= Ax(t)+\int^t_0B(t-s)x(s)ds+t^nf(t,x(t)),\quad t\in [0,T],\;n\in Z^+,\\ q\in(0,1],\;x(0)=g(x)+x_0.\end{split}$ Some sufficient conditions for the existence of a mild solution are given. The main tools are the resolvent operator and fixed point theorems due to Banach’s fixed point theorem, Krasnoselskii’s fixed point theorem and Schaefer’s fixed point theorem. Finally, an example is given.

### MSC:

 34K37 Functional-differential equations with fractional derivatives 34K30 Functional-differential equations in abstract spaces 45J99 Integro-ordinary differential equations
Full Text: