Critical types of Krasnoselskii fixed point theorems in weak topologies

Abstract

In this note, by means of the technique of measures of weak noncompactness, we establish a generalized form of fixed point theorem for the sum of T + S in weak topology setups of a metrizable locally convex space, where S is not weakly compact, I − T allows to be noninvertible, and T is not necessarily continuous. The obtained results unify and significantly extend a lot of previously known extensions of Krasnoselskii fixed-point theorems. The analysis presented here reveals the essential characteristics of the Krasnoselskii type fixed-point theorem in weak topology settings.

Keywords: Fixed point, noncompact mapping, multi-valued mapping