Fixed Point Theory and ApplicationsVolume 2006 (2006), Article ID 018909, 10 pagesdoi:10.1155/FPTA/2006/18909

# 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

#### Abstract

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.