Abstract:We introduce some condition on mappings. The condition is weaker than nonexpansiveness and stronger than quasinonexpansiveness. We present fixed point theorems and convergence theorems for mappings satisfying the condition.
“…As remarked in [6], the following theorem proved in [15] in the setting of a complete CAT(0) space (see also [17]) holds in more general contexts. We formulate this result in the framework of geodesic Ptolemy spaces with a uniformly continuous midpoint map since the proof only requires the uniqueness of asymptotic centers and the convexity of the metric, conditions which are satisfied in such a setting.…”
Section: Properties Of Geodesic Ptolemy Spacesmentioning
We prove that geodesic Ptolemy spaces with a continuous midpoint map are strictly convex. Moreover, we show that geodesic Ptolemy spaces with a uniformly continuous midpoint map are reflexive and that in such a setting bounded sequences have unique asymptotic centers. These properties will then be applied to yield a series of fixed point results specific to CAT(0) spaces.
“…As remarked in [6], the following theorem proved in [15] in the setting of a complete CAT(0) space (see also [17]) holds in more general contexts. We formulate this result in the framework of geodesic Ptolemy spaces with a uniformly continuous midpoint map since the proof only requires the uniqueness of asymptotic centers and the convexity of the metric, conditions which are satisfied in such a setting.…”
Section: Properties Of Geodesic Ptolemy Spacesmentioning
We prove that geodesic Ptolemy spaces with a continuous midpoint map are strictly convex. Moreover, we show that geodesic Ptolemy spaces with a uniformly continuous midpoint map are reflexive and that in such a setting bounded sequences have unique asymptotic centers. These properties will then be applied to yield a series of fixed point results specific to CAT(0) spaces.
“…In 2008, Suzuki [17] introduced a condition on mappings, called (C) which is weaker than nonexpansiveness and stronger than quasi-nonexpansiveness. A multivalued mapping T : E !…”
Section: (E) It Is Obvious That P (E) Cb(e)mentioning
ISA YILDIRIMAbstract. The aim of this paper is to introduce an iterative process with errors for a …nite family of multivalued mappings satisfying the condition (C) which is weaker than nonexpansiveness. We also prove some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces.
“…The nonexpansiveness condition (C), also known as Suzuki condition, was introduced in [65] to study mild nonexpansiveness conditions which still imply existence of fixed points.…”
Section: Theorem 519 Let M Be a Complete R-tree And K A Closed Boundmentioning
In this survey we present an exposition of the development during the last decade of metric fixed point theory on hyperconvex metric spaces. Therefore we mainly cover results where the conditions on the mappings are metric. We will recall results about proximinal nonexpansive retractions and their impact into the theory of best approximation and best proximity pairs. A central role in this survey will be also played by some recent developments on R-trees. Finally, some considerations and new results on the extension of compact mappings will be shown.
Mathematics Subject Classification
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