Convergence theorems for strongly continuous semi-groups of asymptotically nonexpansive mappings

Habtu Zegeye, Naseer Shahzad

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Let K be a nonempty closed convex subset of a real Banach space E. Let T {colon equals} {T (t) : t ∈ R+} be a strongly continuous semi-group of asymptotically nonexpansive mappings from K into K with a sequence {Lt} ⊂ [1, ∞). Suppose F (T) ≠ 0{combining long solidus overlay}. Then, for a given u0 ∈ K and tn > 0 there exists a sequence {un} ⊂ K such that un = (1 - αn) T (tn) un + αn u0, for n ∈ N such that {αn} ⊂ (0, 1) and Ltn - 1 < αn, where tn ∈ R+. Suppose, in addition, that E is reflexive strictly convex with a uniformly Gâteaux differentiable norm and that limn → ∞ tn = ∞, limn → ∞ αn = limn → ∞ frac(Ltn - 1, αn) = 0. Then the sequence {un} converges strongly to a point of F (T). Moreover, it is proved that an explicit sequence {xn} generated from x1 ∈ K by xn + 1 {colon equals} αn u + (1 - αn) T (tn) xn, n ≥ 1, converges to a fixed point of T.

Original languageEnglish
Pages (from-to)2308-2315
Number of pages8
JournalNonlinear Analysis, Theory, Methods and Applications
Volume71
Issue number5-6
DOIs
Publication statusPublished - Sep 1 2009

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Asymptotically Nonexpansive Mapping
Strongly Continuous Semigroups
Banach spaces
Set theory
Convergence Theorem
Converge
Strictly Convex
Overlay
Differentiable
Fixed point
Banach space
Norm
Closed
Subset

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "Let K be a nonempty closed convex subset of a real Banach space E. Let T {colon equals} {T (t) : t ∈ R+} be a strongly continuous semi-group of asymptotically nonexpansive mappings from K into K with a sequence {Lt} ⊂ [1, ∞). Suppose F (T) ≠ 0{combining long solidus overlay}. Then, for a given u0 ∈ K and tn > 0 there exists a sequence {un} ⊂ K such that un = (1 - αn) T (tn) un + αn u0, for n ∈ N such that {αn} ⊂ (0, 1) and Ltn - 1 < αn, where tn ∈ R+. Suppose, in addition, that E is reflexive strictly convex with a uniformly G{\^a}teaux differentiable norm and that limn → ∞ tn = ∞, limn → ∞ αn = limn → ∞ frac(Ltn - 1, αn) = 0. Then the sequence {un} converges strongly to a point of F (T). Moreover, it is proved that an explicit sequence {xn} generated from x1 ∈ K by xn + 1 {colon equals} αn u + (1 - αn) T (tn) xn, n ≥ 1, converges to a fixed point of T.",
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Convergence theorems for strongly continuous semi-groups of asymptotically nonexpansive mappings. / Zegeye, Habtu; Shahzad, Naseer.

In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 71, No. 5-6, 01.09.2009, p. 2308-2315.

Research output: Contribution to journalArticle

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