Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space,K be a closed convex nonempty subset of E which is also a nonexpansive retract with retraction P. Let T1,T2, . . . , Tm :K →E be asymptotically nonexpansive mappings of K into E with sequences (respectively) {kin}∞n=1 satisfying kin →1 as n→∞, i = 1, 2, . . . , m, and ∞n=1(kin −1) <∞. Let {αin}∞n=1 be a sequence in [ , 1 − ], ∈ (0, 1), for each i ∈ {1, 2, . . . , m} (respectively). Let {xn} be a sequence generated for m 2 by ⎧⎪⎪ ⎪⎪⎪ ⎨⎪⎪⎪⎩ x1 ∈ K, xn+1 = P[(1−α1n)xn + α1nT1(P T1)n−1yn+m−2], yn+m−2 = P[(1− α2n)xn +α2nT2(P T2)n−1yn+m−3], ... yn = P[(1− αmn)xn + αmnTm(P Tm)n−1xn], n 1. Let m i=1 F(Ti ) = ∅. Strong and weak convergence of the sequence {xn} to a common fixed point of the family {Ti }m i=1 are proved. Furthermore, if T1,T2, . . . , Tm are nonexpansive mappings and the dual E∗ of E satisfies the Kadec–Klee property, weak convergence theorem is also proved. © 2006 Elsevier Inc. All rights reserved.