Existence for Nonautonomous Fractional Integrodifferential Equations with Nonlocal Conditions

Abstract

We study the existence of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with nonlocal conditions in a separable Banach space $X$. Combining the techniques of operator semigroup, noncompactness measures, and the fixed point theory, we obtain new existence of mild solution without the assumptions that the nonlinearity $f$ satisfies a Lipschitz type condition and the semigroup $\{\text{exp}(-tA(s))\}$ generated by ${\{-A(s)\}}_{s\in [0,T]}$ is compact. An application of the abstract result is also given.