Common fixed points of two multivalued nonexpansive maps in Kohlenbach hyperbolic spaces

Safeer Hussain Khan,Mujahid Abbas

doi:10.1186/1687-1812-2014-181

Safeer Hussain Khan, Mujahid Abbas

Open Access

https://doi.org/10.1186/1687-1812-2014-181

Copy DOI

Abstract

In this paper, we introduce an iteration scheme for two multivalued maps in Kohlenbach hyperbolic spaces. This extends the single-valued iteration process due to Agarwal et al. (J. Nonlinear Convex Anal. 8(1):61-79, 2007). Using this new algorithm, we approximate common fixed points of two multivalued mappings through � -convergence and strong convergence under some weaker conditions. A necessary and sufficient condition is given for strong convergence. MSC: Primary 47A06; 47H09; 47H10; secondary 49M05

Highlights

Introduction and preliminaries A subsetK of a metric space X is proximinal if for each x ∈ X, there exists an element k ∈ K such that d(x, K) = inf d(x, y) : y ∈ K = d(x, k).Let CB(K), C(K) and P(K) be the families of closed and bounded subsets, compact subsets and proximinal bounded subsets of K, respectively
The existence of fixed points for multivalued nonexpansive mappings in convex metric spaces has been shown by Shimizu and Takahashi [ ]
In order to get rid of the condition Tp = {p} for any p ∈ F(T), they used PT (x) := {y ∈ Tx : x – y = d(x, Tx)} for a multivalued map T : K → P(K) and proved some strong convergence results using Mann and Ishikawa type iterative algorithms

Summary

Introduction

The existence of fixed points for multivalued nonexpansive mappings in convex metric spaces has been shown by Shimizu and Takahashi [ ]. Since many authors have published papers on the existence and convergence of fixed points for multivalued nonexpansive maps in convex metric spaces.

Results

Conclusion