Explicit iteration method for common fixed points of a finite family of nonself asymptotically nonexpansive mappings

Suppose that KK is a nonempty closed convex subset of a real uniformly convex Banach space EE, which is also a nonexpansive retract of EE with nonexpansive retraction PP. Let {Ti:i∈I}{Ti:i∈I} be NN nonself asymptotically nonexpansive mappings from KK to EE such that F={x∈K:Tix=x,i∈I}≠ϕF={x∈K:Tix=x,i∈I}≠ϕ, where I={1,2,…,N}I={1,2,…,N}. From arbitrary x0∈Kx0∈K, {xn}{xn} is defined by xn=P((1−αn)xn−1+αnTn(PTn)m−1xn−1),n≥1 where n=(m−1)N+in=(m−1)N+i, Tn=Tn(modN)=TiTn=Tn(modN)=Ti, i∈Ii∈I, the modNmodN function takes values in II, {αn}{αn} is a real sequence in [δ,1−δ][δ,1−δ] for some δ∈(0,1)δ∈(0,1). Some strong and weak convergence theorems of {xn}{xn} to some q∈Fq∈F are obtained under some suitable conditions in real uniformly convex Banach spaces.