On an open problem of Kyung Soo Kim

Bancha Panyanak

doi:10.1186/s13663-015-0438-7

Abstract

AbstractWe prove a convergence theorem of the Mann iteration scheme for a uniformlyL-Lipschitzian asymptotically demicontractive mapping in a$\operatorname{CAT}(\kappa)$CAT(κ)space with$\kappa>0$κ>0. We also obtain a convergence theorem of the Ishikawa iteration scheme for a uniformlyL-Lipschitzian asymptotically hemicontractive mapping. Our results provide a complete solution to an open problem raised by Kim (Abstr. Appl. Anal. 2013:381715, 2013).

Highlights

Speaking, CAT(κ) spaces are geodesic spaces of bounded curvature and generalizations of Riemannian manifolds of sectional curvature bounded above
The letters C, A, and T stand for Cartan, Alexandrov, and Toponogov, who have made important contributions to the understanding of curvature via inequalities for the distance function, and κ is a real number that we impose it as the curvature bound of the space
Fixed point theory in CAT(κ) spaces was first studied by Kirk [, ]

Summary

Introduction

CAT(κ) spaces are geodesic spaces of bounded curvature and generalizations of Riemannian manifolds of sectional curvature bounded above. In , Schu [ ] proved the strong convergence of Mann iteration for asymptotically nonexpansive mappings in Hilbert spaces. Qihou [ ] extended Schu’s result to the general setting of asymptotically demicontractive mappings and obtained the strong convergence of Ishikawa iteration for asymptotically hemicontractive mappings. Theorem A Let (X, ρ) be a complete CAT( ) space, C be a nonempty bounded closed convex subset of X, and T : C → C be a completely continuous and uniformly L-Lipschitzian. Theorem B Let (X, ρ) be a complete CAT( ) space, let C be a nonempty bounded closed convex subset of X, and let T : C → C be a completely continuous and uniformly L-Lipschitzian asymptotically hemicontractive mapping with sequence {an} in [ , ∞) such that.

Given x

If tn

Let κ

The conclusion follows from

Then i αnl βnl