Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings

Suppose that K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E. Let T1,T2 :K →E be two nonself asymptotically nonexpansive mappings with sequences {kn}, {ln} ⊂ [1,∞), limn→∞kn = 1, limn→∞ln = 1, F(T1) ∩ F(T2) = {x ∈ K: T1x = T2x = x} = ∅, respectively. Suppose {xn} is generated iteratively by x1 ∈ K, xn+1 = P((1− αn)xn +αnT1(P T1)n−1yn), yn = P((1−βn)xn +βnT2(P T2)n−1xn), n 1, where {αn} and {βn} are two real sequences in [ , 1 − ] for some >0. (1) Strong convergence theorems of {xn} to some q ∈ F(T1) ∩ F(T2) are obtained under conditions that one of T1 and T2 is completely continuous or demicompact and ∞n=1(kn−1) <∞, ∞n=1(ln−1) <∞. (2) If E is real uniformly convex Banach space satisfying Opial’s condition, then weak convergence of {xn} to some q ∈ F(T1) ∩ F(T2) is obtained. © 2005 Elsevier Inc. All rights reserved.