This paper is concerned with the existence of mild and strong solutions on the interval 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 for that is small enough, and, then, by letting and using a diagonal argument, we have the existence results on the interval . 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.
"Existence of Some Semilinear Nonlocal Functional Differential Equations of Neutral Type." Abstr. Appl. Anal. 2013 (SI45) 1 - 12, 2013. https://doi.org/10.1155/2013/503656