Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 018909, 10 pages

Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces

Naseer Shahzad1 and Aniefiok Udomene2

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.


Suppose K is a nonempty closed convex subset of a real Banach space E. Let S,T:KK be two asymptotically quasi-nonexpansive maps with sequences {un},{vn}[0,) such that n=1un< and n=1vn<, and F=F(S)F(T):={xK:Sx=Tx=x}. Suppose {xn} is generated iteratively by x1K,xn+1=(1αn)xn+αnSn[(1βn)xn+βnTnxn],n1 where {αn} and {βn} are real sequences in [0,1]. It is proved that (a) {xn} converges strongly to some xF if and only if liminfnd(xn,F)=0; (b) if X is uniformly convex and if either T or S is compact, then {xn} converges strongly to some xF. Furthermore, if X is uniformly convex, either T or S is compact and {xn} is generated by x1K,xn+1=αnxn+βnSn[αnxn+βnTnxn+γnzn]+γnzn,n1, where {zn}, {zn} are bounded, {αn},{βn},{γn},{αn},{βn},{γn} are real sequences in [0,1] such that αn+βn+γn=1=αn+βn+γn and {γn}, {γn} are summable; it is established that the sequence {xn} (with error member terms) converges strongly to some xF.