An Implicit Iteration Process for Common Fixed Points of Two Infinite Families of Asymptotically Nonexpansive Mappings in Banach Spaces

Let K be a nonempty, closed, and convex subset of a real uniformly convex Banach space E. Let {Tλ} λ∈Λ and {Sλ} λ∈Λ be two infinite families of asymptotically nonexpansive mappings from K to itself with F : = {x ∈ K : Tλx = x = Sλx, λ ∈ Λ} ≠ ∅. For an arbitrary initial point x0 ∈ K, {xn} is defined as follows: xn=αnxn-1+βn(Tn-1*) mn-1xn-1+γn(Tn*) mnyn, yn=αn′xn+βn′(Sn-1*) mn-1xn-1+γn′(Sn*) mnxn, n = 1,2, 3, …, where Tn*=Tλin and Sn*=Sλin with in and mn satisfying the positive integer equation: n = i + (m − 1)m/2, m ≥ i; {Tλi} i=1∞ and {Sλi} i=1∞ are two countable subsets of {Tλ} λ∈Λ and {Sλ} λ∈Λ, respectively; {αn}, {βn}, {γn}, {αn′}, {βn′}, and {γn′} are sequences in [δ, 1 − δ] for some δ ∈ (0,1), satisfying αn + βn + γn = 1 and αn′+βn′+γn′=1. Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings {Tλ} λ∈Λ and {Sλ} λ∈Λ is obtained. The results extend those of the authors whose related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.