Mild solutions of a class of semilinear fractional integro-differential equations subjected to noncompact nonlocal initial conditions

Abstract

In this paper, we prove the existence of mild solutions of a class of fractional semilinear integro-differential equations of order $\beta\in(1,2]$ subjected to noncompact initial nonlocal conditions. We assume that the linear part generates an arbitrarily strongly continuous $\beta$-order fractional cosine family, while the nonlinear forcing term is of Carath\'eodory type and satisfies some fairly general growth conditions. Our approach combines the Monch fixed point theorem with some recent results regarding the measure of noncompactness of integral operators. Our conclusions improve and generalize many earlier related works. An example is provided to illustrate the main results.

Keywords

• cosine operator
• fractional integro-differential operator
• abstract differential equation
• noncompact nonlocal condition