Journal of Applied Mathematics
Volume 2012 (2012), Article ID 421050, 11 pages
http://dx.doi.org/10.1155/2012/421050
Research Article

Strong Convergence of Viscosity Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

Luo Yi Shi and Ru Dong Chen

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Received 30 March 2012; Accepted 27 April 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Luo Yi Shi and Ru Dong Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self-mapping 𝑇 of a closed convex subset 𝐶 of a CAT(0) space 𝑋 . Suppose that the set Fix ( 𝑇 ) of fixed points of 𝑇 is nonempty. For a contraction 𝑓 on 𝐶 and 𝑡 ∈ ( 0 , 1 ) , let 𝑥 𝑡 ∈ 𝐶 be the unique fixed point of the contraction 𝑥 ↦ 𝑡 𝑓 ( 𝑥 ) ⊕ ( 1 − 𝑡 ) 𝑇 𝑥 . We will show that if 𝑋 is a CAT(0) space satisfying some property, then { 𝑥 𝑡 } converge strongly to a fixed point of 𝑇 which solves some variational inequality. Consider also the iteration process { 𝑥 𝑛 } , where 𝑥 0 ∈ 𝐶 is arbitrary and 𝑥 𝑛 + 1 = 𝛼 𝑛 𝑓 ( 𝑥 𝑛 ) ⊕ ( 1 − 𝛼 𝑛 ) 𝑇 𝑥 𝑛 for 𝑛 ≥ 1 , where { 𝛼 𝑛 } ⊂ ( 0 , 1 ) . It is shown that under certain appropriate conditions on 𝛼 𝑛 , { 𝑥 𝑛 } converge strongly to a fixed point of 𝑇 which solves some variational inequality.