A New Iterative Algorithm for Approximating Common Fixed Points for Asymptotically Nonexpansive Mappings

Abstract

Suppose that K is a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and asymptotically nonexpansive mappings with respect to P with sequences {Kn},{ln}⊂[1,∞), limn→∞kn=1, limn→∞ln=1, F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠∅, respectively. Suppose that {xn} is a sequence in K generated iteratively by x1∈K, xn+1=αnxn+βn(PT1)nxn+γn(PT2)nxn, for all n≥1, where {αn}, {βn}, and {γn} are three real sequences in [ε,1−ε] for some ε>0 which satisfy condition αn+βn+γn=1. Then, we have the following. (1) If one of T1 and T2 is completely continuous or demicompact and ∑n=1∞(kn−1)<∞,∑n=1∞(ln−1)<∞, then the strong convergence of {xn} to some q∈F(T1)∩F(T2) is established. (2) If E is a real uniformly convex Banach space satisfying Opial's condition or whose norm is Fréchet differentiable, then the weak convergence of {xn} to some q∈F(T1)∩F(T2) is proved.