Existence of solutions to a class of partial functional integrodifferential equations with nonlocal conditions (Q2910736)

The aim of this work is to prove the existence of mild solutions for some integrodifferential equations with nonlocal conditions. The linear part is assumed to have a resolvent operator in the sense provided by Grimmer, the nonlinear part is assumed to be continuous or Lipschitz. The resolvent operator plays a crucial role in that paper, since it helps to get a variation of constant formula for the mild solutions. The author gives enough conditions ensuring the existence of the resolvent operators and then uses this formula and many fixed point theorems to show the existence of the mild solution, like Banach's fixed point theorem or the Leary-Schauder alternative. Banach's fixed point theorem is applied when a Lipschitz condition is assumed. The compactness of the resolvent operator is used when the Leary-Schauder alternative is applied.