Fixed points of continuous nonlinear mappings in Banach spaces

Abstract

The purpose in this article is to discuss under what conditions a nonexpansive or a condensing mapping has a fixed point when its domain is a closed, convex subset of a Hilbert space and its range is not necessarily contained within its domain. When the mapping is completely continuous and its domain is a closed, convex subset of a uniformly convex Banach space a similar result is also investigated. We also deal with the existence of fixed points of condensing mappings whose domain is the closure of an open subset of an arbitrary Banach space