Existence of Some Semilinear Nonlocal Functional Differential Equations of Neutral Type

This paper is concerned with the existence of mild and strong solutions onthe interval for some neutral partial differentialequations with nonlocal conditions. The linear part of theequations is assumed to generate a compact analytic semigroup ofbounded linear operators, whereas the nonlinear part satisfies theCarathëodory condition and is bounded by some suitablefunctions. We first employ the Schauder fixed-point theorem to provethe existence of solution on the interval for that is small enough, and, then, by letting and using a diagonal argument, we have the existence results onthe interval . This approach allows one to drop thecompactness assumption on a nonlocal condition, which generalizesrecent conclusions on this topic. The obtained results will beapplied to a class of functional partial differential equations with nonlocal conditions