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

[摘要] LetKbe a nonempty, closed, and convex subset of a real uniformly convex Banach spaceE. Let{Tλ}λ∈Λand{Sλ}λ∈Λbe two infinite families of asymptotically nonexpansive mappings fromKto itself withF:={x∈K:Tλx=x=Sλx,λ∈Λ}≠∅. For an arbitrary initial pointx0∈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,…, whereTn*=TλinandSn*=Sλinwithinandmnsatisfying 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∈Λand{Sλ}λ∈Λ,respectively;{αn},{βn},{γn},{αn′},{βn′},and{γn′}are sequences in[δ,1-δ]for someδ∈(0,1), satisfyingαn+βn+γn=1andαn′+βn′+γn′=1. Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings{Tλ}λ∈Λand{Sλ}λ∈Λis obtained. The results extend those of the authors whose related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.

[学科分类] 应用数学