Strong convergence theorems for continuous semigroups of asymptotically nonexpansive mappings

Habtu Zegeye, Naseer Shahzad

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Let K be a nonempty closed convex and bounded subset of a real Banach space E. Let T ℑ: ={T(t): t ∈ ℝ+} be a strongly continuous semigroup of asymptotically nonexpansive self-mappings on K with a sequence {Lt} ⊂ [1,∞ ). Then, for a given μ0 ∈ K and sn ∈(0, 1), tn > 0 there exists a sequence {un} ⊂ K such that un=(1-αn)T(t n)unnu0, for each n,∈ Ndbl; atisfying||un-T (t)un||→0 as→n ∞, for any t+, where [image omitted]. If, in addition, E is uniformly convex with uniformly G[image omitted]teaux differentiable norm, then it is proved that F(ℑ) ≠ ∅ and the sequence {un} converges strongly to a point of F(ℑ) under certain mild conditions on {Lt}, {t n} and {sn}. Moreover, it is proved that an explicit sequence {xn} generated from x1 ∈ K by xn+1: α nu0+(1-αnn)T(tn)x n, n ≥1,converges to a fixed point of T under appropriate assumption imposed upon the sequence {xn}.

Original languageEnglish
Pages (from-to)833-848
Number of pages16
JournalNumerical Functional Analysis and Optimization
Volume30
Issue number7-8
DOIs
Publication statusPublished - Jul 1 2009

All Science Journal Classification (ASJC) codes

  • Analysis
  • Signal Processing
  • Computer Science Applications
  • Control and Optimization

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