Strong and weak convergence theorems for nonself-asymptotically perturbed nonexpansive mappings

Suppose that K1K1 and K2K2 are nonempty closed convex subsets of a real uniformly convex Banach space EE which are also nonexpansive retracts of EE with retractions PP and QQ, respectively. Let T1:K1→ET1:K1→E and T2:K2→ET2:K2→E be two nonself asymptotically perturbed PP-nonexpansive and QQ-nonexpansive mappings satisfying the ball condition with sequences {kn}{kn}, {ln}⊂[1−ϵ,∞){ln}⊂[1−ϵ,∞), limn→∞kn=1−ϵlimn→∞kn=1−ϵ, limn→∞ln=1−ϵlimn→∞ln=1−ϵ, F(T1)∩F(T2)={x∈K1∩K2:T1x=T2x=x}≠0̸F(T1)∩F(T2)={x∈K1∩K2:T1x=T2x=x}≠0̸, respectively, such that K2⊇(1−λ)K1+λT1(K1)K2⊇(1−λ)K1+λT1(K1) for each λ∈[ϵ,1−ϵ)λ∈[ϵ,1−ϵ) for some ϵ>0ϵ>0. Suppose that {xn}{xn} is generated iteratively by {x1∈K1∩K2xn+1=P((1−αn)xn+αnT2(QT2)n−1yn)∈K1,yn=(1−βn)xn+βnT1(PT1)n−1xn∈K2 for each n≥1n≥1, where {αn}{αn} and {βn}{βn} are two real sequences in [ϵ,1−ϵ)[ϵ,1−ϵ) for some ϵ>0ϵ>0. (1) If one of T1T1 and T2T2 is completely continuous or demicompact and ∑n=1∞kn′<∞, ∑n=1∞ln′<∞, where kn′=(1+kn)(supi≥1ki)−1 and ln′=(1+ln)(supi≥1li)−1, then strong convergence theorems of both {xn}{xn} and {yn}{yn} to some q∈F(T1)∩F(T2)q∈F(T1)∩F(T2) are obtained. (2) If EE is real uniformly convex Banach space satisfying Opial’s condition, then weak convergence of both {xn}{xn} and {yn}{yn} to some q∈F(T1)∩F(T2)q∈F(T1)∩F(T2) are obtained.