Weak convergence theorems for asymptotically nonexpansive nonself-mappings

Suppose that KK is a nonempty closed convex subset of a real uniformly convex Banach space EE with PP as a nonexpansive retraction. Let T1,T2:K→ET1,T2:K→E be two asymptotically nonexpansive nonself-mappings with sequences {kn},{ln}⊂[1,∞){kn},{ln}⊂[1,∞) such that ∑n=1∞(kn−1)<∞ and ∑n=1∞(ln−1)<∞, respectively and F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠0̸F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠0̸. Suppose that {xn}{xn} is generated iteratively by {x1∈Kxn+1=P((1−αn)xn+αnT1(PT1)n−1yn)yn=P((1−βn)xn+βnT2(PT2)n−1xn),∀n≥1, where {αn}{αn} and {βn}{βn} are two real sequences in [ϵ,1−ϵ][ϵ,1−ϵ] for some ϵ>0ϵ>0. If EE also has a Fréchet differentiable norm or its dual E∗E∗ has the Kadec–Klee property, then weak convergence of {xn}{xn} to some q∈F(T1)∩F(T2)q∈F(T1)∩F(T2) are obtained.