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

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 [ ,1− ] for some > 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.