Existence of solutions for neutral integro-differential equations with state-dependent delay

In this paper, we study the existence and regularity of solutions for a neutral functional integro-differential equation with state-dependent delay in Banach space. The mild solutions are investigated by Sadovskii fixed point theorem under compactness condition for the resolvent operator, and the theory of fractional power and α-norm are also used in the discussion since the nonlinear terms of the system involve spacial derivatives. The strict solutions are obtained under Hölder continuity condition. Hence this work extends and develops the existed results on this topic. In the end an example is provided to illustrate the application of the obtained results.