Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space whose dual space E∗ satisfies the Kadec–Klee property, K be a closed convex nonempty subset of E. Let T1,T2, . . . , Tm :K → K 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) <∞. For arbitrary ∈ (0, 1), let {αin}∞n=1 be a sequence in [ , 1− ], for each i ∈ {1, 2, . . . , m} (respectively). Let {xn} be a sequence generated for m 2 by ⎧⎪ ⎪⎪⎪⎪⎨⎪ ⎪⎪⎪⎪⎩ x1 ∈ K, xn+1 = (1−α1n)xn +α1nT n 1 yn+m−2, yn+m−2 = (1−α2n)xn + α2nT n 2 yn+m−3, ... yn = (1−αmn)xn + αmnT nm xn, n 1. Let m i=1 F(Ti ) = ∅. Then, {xn} converges weakly to a common fixed point of the family {Ti }m i=1. Under some appropriate condition on the family {Ti }m i=1, a strong convergence theorem is also proved. © 2006 Elsevier Inc. All rights reserved.