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 Σn=1∞(kin - 1) < ∞. For arbitrary ε element of (0, 1), let {αin}n=1∞ be a sequence in [ε, 1 - ε ], for each i element of { 1, 2 , . . . ,m} (respectively). Let {xn} be a sequence generated for m ≥ 2 by, x1 element of K, xn+1 = (1 - α1n)xn + α1nT1nyn+m-2, yn+m-2 = (1 - α2n)xn + α2nT2nyn+m-3, ..., yn = (1 - αmn)xn + αmnTmnxn , n ≥ 1. Let Intersectioni=1m F (Ti) ≠ 0 . Then, {xn} converges weakly to a common fixed point of the family {Ti}i=1m. Under some appropriate condition on the family {Ti}i=1m, a strong convergence theorem is also roved. (author)