Mild solutions for some partial functional integrodifferential equations with finite delay in Fréchet spaces

In this work, we study the existence of mild solutions for some partial functional integrodifferential equations with finite delay in a Frechet spaces. We assume that the linear part has a resolvent operator in the sense given by Grimmer (Trans Am Math Soc 273: 333–349, 1982). The nonlinear part is a sum of a Lipschitzian function and another satisfies the Caratheodory’s conditions. Our approach is based on a nonlinear alternative of Avramescu type and the resolvent operators theory. An application is provided to a reaction-diffusion equation with delay.