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→1kin→1 as n→∞n→∞, i=1,2,…,mi=1,2,…,m, and ∑n=1∞(kin−1)<∞. Let {αin}n=1∞ be a sequence in [ϵ,1−ϵ],ϵ∈(0,1)[ϵ,1−ϵ],ϵ∈(0,1), 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=P[(1−α1n)xn+α1nT1(PT1)n−1yn+m−2],yn+m−2=P[(1−α2n)xn+α2nT2(PT2)n−1yn+m−3],⋮yn=P[(1−αmn)xn+αmnTm(PTm)n−1xn],n⩾1. Let ⋂i=1mF(Ti)≠∅. Strong and weak convergence of the sequence {xn}{xn} to a common fixed point of the family {Ti}i=1m are proved. Furthermore, if T1,T2,…,TmT1,T2,…,Tm are nonexpansive mappings and the dual E∗E∗ of E satisfies the Kadec–Klee property, weak convergence theorem is also proved.