Viscosity approximation methods for nonexpansive semigroups in CAT ( 0 ) spaces
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RESEARCH
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Viscosity approximation methods for nonexpansive semigroups in CAT() spaces Rabian Wangkeeree1,2* and Pakkapon Preechasilp1 *
Correspondence: [email protected] 1 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand 2 Centre of Excellence in Mathematics (CHE), Si Ayutthaya Road, Bangkok, 10400, Thailand
Abstract In this paper, we study the strong convergence of Moudafi’s viscosity approximation methods for approximating a common fixed point of a one-parameter continuous semigroup of nonexpansive mappings in CAT(0) spaces. We prove that the proposed iterative scheme converges strongly to a common fixed point of a one-parameter continuous semigroup of nonexpansive mappings which is also a unique solution of the variational inequality. The results presented in this paper extend and enrich the existing literature. Keywords: viscosity approximation method; nonexpansive semigroup; variational inequality; CAT(0) space; common fixed point
1 Introduction Let (X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [, l] ⊂ R to X such that c() = x, c(l) = y, and d(c(t), c(t )) = |t – t | for all t, t ∈ [, l]. In particular, c is an isometry and d(x, y) = l. The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic segment is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y ⊆ X is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle (x , x , x ) in a geodesic metric space (X, d) consists of three points x , x , x in X (the vertices of ) and a geodesic segment between each pair of vertices (the edges of ). A comparison triangle for the geodesic triangle (x , x , x ) in (X, d) is a triangle ¯ , x , x ) := (¯x , x¯ , x¯ ) in the Euclidean plane E such that dE (¯xi , x¯ j ) = d(xi , xj ) for all (x i, j ∈ {, , }. A geodesic space is said to be a CAT() space if all geodesic triangles of appropriate size satisfy the following comparison axiom. ¯ be a comparison triangle for . CAT(): Let be a geodesic triangle in X and let Then is said to satisfy the CAT() inequality if for all x, y ∈ and all comparison points ¯ x¯ , y¯ ∈ , d(x, y) ≤ dE (¯x, y¯ ). © 2013 Wangkeeree and Preechasilp; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution 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.
Wangkeeree and Preechasilp Fixed Point Theory and Applications 2013, 2013:160 http://www.fixedpointtheoryandapplications.com/content/2013/1/160
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If x, y , y are points in a
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