On existence and continuation of mild solutions of functional-differential equations of neutral type in Banach spaces

Abstract

The main object of research of this work is infinite-dimensional functional-differential equations of neutral type in Banach spaces. The conditions for the existence of a mild solution to the initial problem and its extension to the boundary of the domain are established. The presence of a delay in the derivative leads to the appearance of singular terms. The research methods are related to the technique of strong semigroups and fractional powers of the operator. The existence proof is based on the representation of the initial problem in abstract operator form with the further use of Krasnoselsky's theorem on a fixed point. For this purpose, the original operator is represented as the sum of the compression operator and the compact operator. The obtained abstract result is applied to the functional-differential equation in partial derivatives of the reaction-diffusion type.