Existence of solutions for some nonautonomous partial functional differential equations with state-dependent delay

The aim of this work is to prove the existence of mild solutions for some nondensely nonautonomous partial functional differential equations with state-dependent delay in Banach spaces. We assume that the linear part is not necessarily densely defined and generates an evolution family. Our approach is based on a nonlinear alternative of Leray–Schauder type and nonautonomous evolution family with nondensely domain. We propose a reaction diffusion equation with state-dependent delay to illustrate the obtained result.