Strong convergence theorems for a semigroup of asymptotically nonexpansive mappings

H. Zegeye, N. Shahzad, O. A. Daman

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:={T(t):t≥0} be a strongly continuous semigroup of asymptotically nonexpansive mappings from K into K with a sequence {Lt}∪[1,∞). Suppose F(T)≠Ø. Then, for a given uεK there exists a sequence {un}∪K such that un=(1-αn)1tn∫0tnT(s)unds+αnu, for nεN, where tnεR+, {αn}∪(0,1) and {Lt} satisfy certain conditions. Suppose, in addition, that E is reflexive strictly convex with a Gâteaux differentiable norm. Then, the sequence {un} converges strongly to a point of F(T). Furthermore, an explicit sequence {xn} which converges strongly to a fixed point of T is proved.

Original languageEnglish
Pages (from-to)2077-2086
Number of pages10
JournalMathematical and Computer Modelling
Volume54
Issue number9-10
DOIs
Publication statusPublished - Nov 1 2011

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

All Science Journal Classification (ASJC) codes

  • Modelling and Simulation
  • Computer Science Applications

Cite this

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Strong convergence theorems for a semigroup of asymptotically nonexpansive mappings. / Zegeye, H.; Shahzad, N.; Daman, O. A.

In: Mathematical and Computer Modelling, Vol. 54, No. 9-10, 01.11.2011, p. 2077-2086.

Research output: Contribution to journalArticle

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T1 - Strong convergence theorems for a semigroup of asymptotically nonexpansive mappings

AU - Zegeye, H.

AU - Shahzad, N.

AU - Daman, O. A.

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N2 - Let K be a nonempty closed convex subset of a real Banach space E. Let T:={T(t):t≥0} be a strongly continuous semigroup of asymptotically nonexpansive mappings from K into K with a sequence {Lt}∪[1,∞). Suppose F(T)≠Ø. Then, for a given uεK there exists a sequence {un}∪K such that un=(1-αn)1tn∫0tnT(s)unds+αnu, for nεN, where tnεR+, {αn}∪(0,1) and {Lt} satisfy certain conditions. Suppose, in addition, that E is reflexive strictly convex with a Gâteaux differentiable norm. Then, the sequence {un} converges strongly to a point of F(T). Furthermore, an explicit sequence {xn} which converges strongly to a fixed point of T is proved.

AB - Let K be a nonempty closed convex subset of a real Banach space E. Let T:={T(t):t≥0} be a strongly continuous semigroup of asymptotically nonexpansive mappings from K into K with a sequence {Lt}∪[1,∞). Suppose F(T)≠Ø. Then, for a given uεK there exists a sequence {un}∪K such that un=(1-αn)1tn∫0tnT(s)unds+αnu, for nεN, where tnεR+, {αn}∪(0,1) and {Lt} satisfy certain conditions. Suppose, in addition, that E is reflexive strictly convex with a Gâteaux differentiable norm. Then, the sequence {un} converges strongly to a point of F(T). Furthermore, an explicit sequence {xn} which converges strongly to a fixed point of T is proved.

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