Abstract
AbstractWe present new results regarding fixed point sets of various set-valued mappings using the concept of fixed point iteration schemes and the newly defined concept of fixed point resolutions. In particular, we prove that the fixed point sets of certain nonexpansive set-valued mappings are contractible.
Highlights
In, Bruck [ ] gave an intriguing result on the structure of fixed point sets by proving that the fixed point set of a certain nonexpansive mapping is always a nonexpansive retract of its domain
In [ ], Chaoha introduced the notion of virtually nonexpansive mappings on metric spaces and proved that the fixed point set of a virtually nonexpansive mapping is always a retract of a certain subset called the convergence set
Chaoha and Chanthorn [ ] presented the concept of fixed point iteration schemes that unifies well-known iteration processes and showed that in regular spaces the fixed point set of a certain virtually stable scheme is a retract of its convergence set
Summary
In , Bruck [ ] gave an intriguing result on the structure of fixed point sets by proving that the fixed point set of a certain nonexpansive mapping is always a nonexpansive retract of its domain. Every selection of a quasi-nonexpansive set-valued mapping on a metric space satisfying the end-point condition is quasi-nonexpansive. To generalize such a result, we will consider a continuous set-valued mapping on a compact metric space as follows.
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