Existence Results for Abstract Functional Differential Equations with Infinite State-Dependent Delay and Applications

This paper is devoted to the investigation of the existence of mild solutions of abstract retarded functional differential equations with infinite state-dependent delay. We obtain the result concerning the existence of mild solutions for the equations with state-dependent delays as a fixed point of the solution operator of an associated abstract retarded functional differential equation with time-varying delays. We are concerned with equations that present a phenomenon of lacunary memory. We apply our results to study the existence of solutions of a state-dependent partial differential equation with infinite state-dependent delay. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023.