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

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Asymptotically Nonexpansive Mapping
Banach spaces
Strong Theorems
Set theory
Strong Convergence
Convergence Theorem
Semigroup
Converge
Strongly Continuous Semigroups
Uniformly Convex
Differentiable
Fixed point
Banach space
Norm
Closed
Subset

All Science Journal Classification (ASJC) codes

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

Cite this

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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)un+αnu0, 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}.",
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Strong convergence theorems for continuous semigroups of asymptotically nonexpansive mappings. / Zegeye, Habtu; Shahzad, Naseer.

In: Numerical Functional Analysis and Optimization, Vol. 30, No. 7-8, 01.07.2009, p. 833-848.

Research output: Contribution to journalArticle

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AB - 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)un+αnu0, 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}.

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