Journal of Applied Mathematics

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

Wei-Qi Deng and Peng Bai

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Abstract

Let K be a nonempty, closed, and convex subset of a real uniformly convex Banach space E. Let {Tλ}λΛ and {Sλ}λΛ be two infinite families of asymptotically nonexpansive mappings from K to itself with F:={xK:Tλx=x=Sλx,λΛ}. For an arbitrary initial point x0K, {xn} is defined as follows: xn=αnxn-1+βn(Tn-1*)mn-1xn-1+γn(Tn*)mnyn, yn=αnxn+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, mi; {Tλi}i=1 and {Sλi}i=1 are two countable subsets of {Tλ}λΛ and {Sλ}λΛ, 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}λΛ and {Sλ}λΛ is obtained. The results extend those of the authors whose related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 602582, 6 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807843

Digital Object Identifier
doi:10.1155/2013/602582

Mathematical Reviews number (MathSciNet)
MR3029966

Zentralblatt MATH identifier
1266.47090

Citation

Deng, Wei-Qi; Bai, Peng. An Implicit Iteration Process for Common Fixed Points of Two Infinite Families of Asymptotically Nonexpansive Mappings in Banach Spaces. J. Appl. Math. 2013 (2013), Article ID 602582, 6 pages. doi:10.1155/2013/602582. https://projecteuclid.org/euclid.jam/1394807843


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