Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 018909, 10 pages
doi:10.1155/FPTA/2006/18909
Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces
1Department of Mathematics, King Abdul Aziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics/Statistics/Computer Science, University of Port Harcourt, PMB, Port Harcourt 5323, Nigeria
Received 21 April 2005; Revised 13 July 2005; Accepted 18 July 2005
Copyright © 2006 Naseer Shahzad and Aniefiok Udomene. 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 subset of a real Banach
space E. Let S,T:K→K be two asymptotically
quasi-nonexpansive maps with sequences {un},{vn}⊂[0,∞) such that ∑n=1∞un<∞ and ∑n=1∞vn<∞, and F=F(S)∩F(T):={x∈K:Sx=Tx=x}≠∅. Suppose {xn} is generated iteratively by x1∈K,xn+1=(1−αn)xn+αnSn[(1−βn)xn+βnTnxn],n≥1 where {αn} and {βn} are real sequences in [0,1]. It is proved that (a) {xn} converges strongly to some x∗∈F if and only if liminfn→∞d(xn,F)=0; (b) if X is uniformly convex and if either T or S is compact, then {xn} converges strongly to some x∗∈F. Furthermore, if X is uniformly convex, either T or S is compact and {xn} is generated by x1∈K,xn+1=αnxn+βnSn[α′nxn+β′nTnxn+γ′nz′n]+γnzn,n≥1, where {zn}, {z′n} are bounded, {αn},{βn},{γn},{α′n},{β′n},{γ′n} are real sequences in [0,1] such that αn+βn+γn=1=α′n+β′n+γ′n and {γn}, {γ′n} are summable; it is established that the
sequence {xn} (with error member terms) converges strongly
to some x∗∈F.