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

Abstract EN

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.