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Abstract
AbstractIn this paper we deal with stationary points (also known as endpoints) of nonexpansive set-valued mappings and show that the existence of such points under certain conditions follows as a consequence of the existence of approximate stationary sequences. In particular we provide abstract extensions of well-known fixed point theorems.
Highlights
Introduction and preliminaries LetX be a Banach space and C be a nonempty subset of X
The first relevant work for existence of fixed points for nonexpansive set-valued mappings was provided by Markin [ ] in
The problem of the existence of stationary points has remained almost unexplored for nonexpansive mappings, it being the case that most results about them require contractive like conditions on the mapping as is the case in [ – ]
Summary
Introduction and preliminaries LetX be a Banach space and C be a nonempty subset of X. Theorem Let X be a Banach space and C a nonempty, weakly compact, and convex subset of X with normal structure. Theorem Let C be a nonempty, weakly compact, and convex subset of Banach space X and T : C → C \ {∅} be a stationary point free nonexpansive set-valued mapping.
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