Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings

Habtu Zegeye, Naseer Shahzad

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

Let K be a nonempty closed convex subset of a real reflexive Banach space E that has weakly continuous duality mapping Jφ for some gauge φ. Let Ti : K → K, i = 1, 2, ..., be a family of quasi-nonexpansive mappings with F {colon equals} ∩i ≥ 1 F (Ti) ≠ 0{combining long solidus overlay} which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For given x0 ∈ K, let {xn} be generated by the algorithm xn + 1 {colon equals} αn f (xn) + (1 - αn) Tn (xn), n ≥ 0, where f : K → K is a contraction mapping and {αn} ⊆ (0, 1) a sequence satisfying certain conditions. Suppose that {xn} satisfies condition (A). Then it is proved that {xn} converges strongly to a common fixed point over(x, ̄) = Q f (over(x, ̄)) of a family Ti, i = 1, 2, .... Moreover, over(x, ̄) is the unique solution in F to a certain variational inequality.

Original languageEnglish
Pages (from-to)2005-2012
Number of pages8
JournalNonlinear Analysis, Theory, Methods and Applications
Volume68
Issue number7
DOIs
Publication statusPublished - Apr 1 2008

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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