Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings

Suppose that K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E  . Let T1,T2:K→E be two nonself asymptotically nonexpansive mappings with sequences {kn},{ln}⊂[1,∞){kn},{ln}⊂[1,∞), limn→∞kn=1limn→∞kn=1, limn→∞ln=1limn→∞ln=1, F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠∅, respectively. Suppose {xn}{xn} is generated iteratively by{x1∈K,xn+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. (1) Strong convergence theorems of {xn}{xn} to some q∈F(T1)∩F(T2)q∈F(T1)∩F(T2) are obtained under conditions that one of T1T1 and T2T2 is completely continuous or demicompact and ∑n=1∞(kn−1)<∞, ∑n=1∞(ln−1)<∞. (2) If E   is real uniformly convex Banach space satisfying Opial's condition, then weak convergence of {xn}{xn} to some q∈F(T1)∩F(T2)q∈F(T1)∩F(T2) is obtained.