Existence of Some Semilinear Nonlocal Functional Differential Equations of Neutral Type

Abstract EN

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 [δ,T] for δ>0 that is small enough, and, then, by letting δ→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.