`Journal of Applied Mathematics and Stochastic AnalysisVolume 2006 (2006), Article ID 21961, 13 pagesdoi:10.1155/JAMSA/2006/21961`

# Yongfu Su and Xiaolong Qin

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

Received 23 March 2006; Revised 9 June 2006; Accepted 10 July 2006

Copyright © 2006 Yongfu Su and Xiaolong Qin. 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

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T:KE be an asymptotically nonexpansive mapping with {kn}[1,) such that n=1(kn1)< and F(T) is nonempty, where F(T) denotes the fixed points set of T. Let {αn}, {αn'}, and {αn''} be real sequences in (0,1) and εαn,αn',αn''1ε for all n and some ε>0. Starting from arbitrary x1K, define the sequence {xn} by x1K, zn=P(αn''T(PT)n1xn+(1αn'')xn), yn=P(αn'T(PT)n1zn+(1αn')xn), xn+1=P(αnT(PT)n1yn+(1αn)xn). (i) If the dual E* of E has the Kadec-Klee property, then { xn} converges weakly to a fixed point pF(T); (ii) if T satisfies condition (A), then {xn} converges strongly to a fixed point pF(T).