Critical type of Krasnosel’skii fixed point theorem

Abstract

In this paper, by means of the technique of measures of noncompactness, we establish a generalized form of the fixed point theorem for the sum of $T+S$, where $S$ is noncompact, $I-T$ may not be injective, and $T$ is not necessarily continuous. The obtained results unify and significantly extend a number of previously known generalizations of the Krasnosel’skii fixed point theorem. The analysis presented here reveals the essential characteristics of the Krasnosel’skii type fixed point theorem in strong topology setups. Further, the results are used to prove the existence of periodic solutions of a nonlinear neutral differential equation with delay in the critical case.