Existence of Some Semilinear Nonlocal Functional Differential Equations of Neutral Type

Abstract

This paper is concerned with the existence of mild and strong solutions on the interval $[0,T]$ for some neutral partial differential equations with nonlocal conditions. The linear part of the equations is assumed to generate a compact analytic semigroup of bounded linear operators, whereas the nonlinear part satisfies the Carathëodory condition and is bounded by some suitable functions. We first employ the Schauder fixed-point theorem to prove the existence of solution on the interval $[\delta ,T]$ for $\delta >0$ that is small enough, and, then, by letting $\delta \to 0$ and using a diagonal argument, we have the existence results on the interval $[0,T]$. This approach allows one to drop the compactness assumption on a nonlocal condition, which generalizes recent conclusions on this topic. The obtained results will be applied to a class of functional partial differential equations with nonlocal conditions.