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∗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→1kin→1 as n→∞n→∞, i=1,2,…,mi=1,2,…,m, and ∑n=1∞(kin−1)<∞. For arbitrary ϵ∈(0,1)ϵ∈(0,1), let {αin}n=1∞ be a sequence in [ϵ,1−ϵ][ϵ,1−ϵ], for each i∈{1,2,…,m}i∈{1,2,…,m} (respectively). Let {xn}{xn} be a sequence generated for m⩾2m⩾2 by{x1∈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 ⋂i=1mF(Ti)≠∅. Then, {xn}{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 proved.