A new iterative process for common fixed points of finite families of non-self-asymptotically non-expansive mappings

Let EE be a real uniformly convex Banach space, KK be a closed convex non-empty subset of EE which is also a non-expansive retract with retraction PP. Let T1,T2,…,Tr:K→ET1,T2,…,Tr:K→E be asymptotically non-expansive mappings of KK into EE with sequences (respectively), {kjn}n=1∞ satisfying kjn→1kjn→1 as n→∞n→∞, j=1,2,…,rj=1,2,…,r such that ∑n=1∞(kjn−1)<∞. Let {αjn}n=1∞ be a sequence in [ε,1−ε][ε,1−ε] for some ε∈(0,1)ε∈(0,1), for each j∈{1,2,…,r}j∈{1,2,…,r}. Let {xn}{xn} be a sequence generalized for r≥2r≥2 by {x1∈K,xn+1=P((1−α1n)yn+r−2+α1nT1(PT1)n−1yn+r−2),yn+r−2=P((1−α2n)yn+r−3+α2nT2(PT2)n−1yn+r−3),⋮yn+1=P((1−α(r−1)n)yn+α(r−1)nTr−1(PTr−1)n−1yn),yn=P((1−αrn)xn+αrnTr(PTr)n−1xn),n≥1. Let ∩j=1rF(Tj)≠∅. Strong and weak convergence of the sequence {xn}{xn} to a common fixed point of family {Tj}j=1r are proved.