Fixed Point Theory and Applications
Volume 2006 (2006), Issue 1, Pages Article 18909, 10 p.
doi:10.1155/FPTA/2006/18909

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

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.