Convergence theorems of common fixed points for some semigroups of nonexpansive mappings in complete CAT(0) spaces

Lin Lai-Jiu, Chuang Chih-Sheng, Yu, Yu Zenn-Tsun

Description

In this paper, we consider some iteration processes for one-parameter continuous semigroups of nonexpansive mappings in a nonempty compact convex subset C of a complete CAT(0) space X and prove that the proposed sequence converges to a common fixed point for these semigroups of nonexpansive mappings. Note that our results generalize Cho et al. result (Nonlinear Anal. 74:6050-6059, 2011) and related results.

Sujets

• CAT(k) space

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 4
• Langue: English

Extrait

Lin et al.Fixed Point Theory and Applications2012,2012:155 http://www.fixedpointtheoryandapplications.com/content/2012/1/155

R E S E A R C HOpen Access Convergence theorems of common fixed points for some semigroups of nonexpansive mappings in complete CAT(0) spaces 1* 12 Lai-Jiu Lin, Chih-Sheng Chuangand Zenn-Tsun Yu

* Correspondence: [e-mail] 1 Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan Full list of author information is available at the end of the article

Abstract In this paper, we consider some iteration processes for one-parameter continuous semigroups of nonexpansive mappings in a nonempty compact convex subsetCof a complete CAT(0) spaceXand prove that the proposed sequence converges to a common fixed point for these semigroups of nonexpansive mappings. Note that our results generalize Choet al.result (Nonlinear Anal. 74:6050-6059, 2011) and related results. Keywords:common fixed point; CAT(0) space; nonexpansive semigroup; implicit iteration process

1 Introduction Fixed point theory in CAT() spaces was first studied by Kirk [, ]. He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT() space always has a fixed point. Since then, the fixed point theory for single-valued and multivalued mappings in CAT() spaces has been rapidly developed, and many papers have appeared; for example, one can see [–] and related references. Let (X,d) be a metric space. A geodesic path joiningx∈Xtoy∈X(or, more briefly, a geodesic fromxtoy) is a mapcfrom a closed interval [,]⊆RtoXsuch thatc() =x,   c() =y, andd(c(t),c(t)) =|t–t|for allt,t∈[,]. In particular,cis an isometry and d(x,y) =. The imageαofcis called a geodesic (or metric) segment joiningxandy. When it is unique, this geodesic is denoted by [x,y]. The space (X,d) is said to be a geodesic space if every two points ofXare joined by a geodesic, andXis said to be uniquely geodesic if there is exactly one geodesic joiningxandyfor eachx,y∈X. A subsetY⊆Xis said to be convex ifYincludes every geodesic segment joining any two of its points. A geodesic triangle(x  ,x  ,x  ) in a geodesic space (X,d) consists of three pointsx  , x  , andx  inX(the vertices ofand a geodesic segment between each pair of vertices (the edge of)). A comparison triangle for geodesic triangle(x  ,x  ,x  ) in (X,d) is a , ¯ x) =d( triangle(x  ,x  ,x  ) :=(x ¯  ,x ¯  , ¯ x  ) in the Euclidean planeE  such thatd E  ( ¯ x i j x i ,x j ) fori,j∈ {, , }. A geodesic space is said to be a CAT() space if all geodesic triangles of appropriate size satisfy the following comparison axiom. CAT(): Letbe a geodesic triangle inX, and letbe a comparison triangle for. Thenis said to satisfy the CAT() inequality if for allx,y∈and all comparison points

©2012 Lin et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.