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

摘要

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 be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . For arbitrary (0,1), let be a sequence in [,1?], for each i{1,2,…,m} (respectively). Let {xn} be a sequence generated for m2 by {x1∈K, xn+1=(1-α1n)xn+a1nTn1Yn+m-2, yn+m-2=(1-α2n)xn=α2nTn2yn+m-3, n≥1. yn=(1-αmn)xn+αmnTnmxn, Let . Then, {xn} converges weakly to a common fixed point of the family . Under some appropriate condition on the family , a strong convergence theorem is also proved.