Geodesic Ptolemy spaces and fixed points

Abstract

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.

Asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces

“…However, Busemann convexity implies the CAT(0) condition in the setting of Ptolemy spaces [20]. Other properties of geodesic Ptolemy spaces in connection to metric fixed point theory can be found in [18,19].…”

Fixed Points of Asymptotically Nonexpansive Mappings in the Intermediate Sense in Cat(0) Spaces

“…In recent years, CAT(0) spaces have attracted the attention of many authors as they have played a very important role in different aspects of geometry ( [12]). Kirk [18,19] showed that a nonexpansive mapping defined on a bounded closed convex subset of a complete CAT(0) space has a fixed point.…”

Convergence theorems for total asymptotically nonexpansive mappings in $\operatorname{CAT} (\kappa )$ spaces

“…Now we introduce the concept of model spaces M n κ . For more details on these spaces, the reader is referred to [4,14]. Let n ∈ N .…”

“…Taking {ν n = 0}, ζ = t, t ≥ 0, then (1.3) can be written as In recent years, CAT(0) spaces (the precise definition of a CAT(0) space is given below) have attracted the attention of many authors because they have played a very important role in different aspects of geometry [4]. Kirk [5,6] showed that a nonexpansive mapping defined on a bounded closed convex subset of a complete CAT(0) space has a fixed point.…”