Strong and Δ-convergence for mixed type total asymptotically nonexpansive mappings in CAT(0) spaces

Lin Wang,Shih-sen Chang,Heung Wing Joseph Lee,Chi-kin Chan

doi:10.1186/1687-1812-2013-122

Abstract

It is our purpose in this paper first to introduce the class of total asymptotically nonexpansive nonself mappings and to prove the demiclosed principle for such mappings in CAT(0) spaces. Then, a new mixed Agarwal-O’Regan-Sahu type iterative scheme for approximating a common fixed point of two total asymptotically nonexpansive mappings and two total asymptotically nonexpansive nonself mappings is constructed. Under suitable conditions, some strong convergence theorems and Δ-convergence theorems are proved in a CAT(0) space. Our results improve and extend the corresponding results of Agarwal, O’Regan and Sahu (J. Nonlinear Convex Anal. 8(1):61-79, 2007), Guo et al. (Fixed Point Theory Appl. 2012:224, 2012. doi:10.1186/1687-1812-2012-224), Sahin et al. (Fixed Point Theory Appl. 2013:12, 2013. doi:10.1186/1687-1812-2013-12), Chang et al. (Appl. Math. Comput. 219:2611-2617, 2012), Khan and Abbas (Comput. Math. Appl. 61:109-116, 2011), Khan et al. (Nonlinear Anal. 74:783-791, 2011), Xu (Nonlinear Anal., Theory Methods Appl. 16(12):1139-1146, 1991), Chidume et al. (J. Math. Anal. Appl. 280:364-374, 2003) and others.MSC:47J05, 47H09, 49J25.

Highlights

Introduction and preliminariesLet (X, d) be a metric space and x, y ∈ X with d(x, y) = l
A metric space X is a geodesic space if every two points of X are joined by only one geodesic segment
A geodesic triangle (x, x, x ) in a geodesic space X consists of three points x, x, x of X and three geodesic segments joining each pair of vertices

Summary

If there exists a subsequence

[ ] Let X be a complete CAT( ) space, {xn} be a bounded sequence in X with A({xn}) = {p}, and {un} be a subsequence of {xn} with A({un}) = {u} and the sequence {d(xn, u)} converges, p = u. Let C be a bounded closed and convex subset of a complete CAT( ) X. ∅ and the following conditions are satisfied:. Since p ∈ F , p = Pp. In addition, since Si and Ti, i = , , are total asymptotically nonexpansive mappings, by the condition (iii), we have d(yn, p) = d P ( – βn)S nxn ⊕ βnT (PT )n– xn , Pp ≤ d ( – βn)S nxn ⊕ βnT (PT )n– xn, p ≤ ( – βn)d S nxn, p + βnd T (PT )n– xn, p. By Lemma . the limits limn→∞ d(xn, F ) and limn→∞ d(xn, p) exist for each p ∈ F . (II) we prove that lim n→∞

Si xn

Ti xni