Strong convergence theorems for maximal monotone mappings in Banach spaces

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

Let E be a uniformly convex and 2-uniformly smooth real Banach space with dual E*. Let A : E* → E be a Lipschitz continuous monotone mapping with A-1 (0) ≠ ∅. For given u, x1 ∈ E, let {xn} be generated by the algorithm xn + 1 : = βn u + (1 - βn) (xn - αn A J xn), n ≥ 1, where J is the normalized duality mapping from E into E* and {λn} and {θn} are real sequences in (0, 1) satisfying certain conditions. Then it is proved that, under some mild conditions, {xn} converges strongly to x* ∈ E where J x* ∈ A-1 (0). Finally, we apply our convergence theorems to the convex minimization problems.

Original languageEnglish
Pages (from-to)663-671
Number of pages9
JournalJournal of Mathematical Analysis and Applications
Volume343
Issue number2
DOIs
Publication statusPublished - Jul 15 2008

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Maximal Monotone Mapping
Banach spaces
Strong Theorems
Strong Convergence
Convergence Theorem
Normalized Duality Mapping
Banach space
Monotone Mapping
Convex Minimization
Uniformly Convex
Minimization Problem
Lipschitz
Converge

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "Let E be a uniformly convex and 2-uniformly smooth real Banach space with dual E*. Let A : E* → E be a Lipschitz continuous monotone mapping with A-1 (0) ≠ ∅. For given u, x1 ∈ E, let {xn} be generated by the algorithm xn + 1 : = βn u + (1 - βn) (xn - αn A J xn), n ≥ 1, where J is the normalized duality mapping from E into E* and {λn} and {θn} are real sequences in (0, 1) satisfying certain conditions. Then it is proved that, under some mild conditions, {xn} converges strongly to x* ∈ E where J x* ∈ A-1 (0). Finally, we apply our convergence theorems to the convex minimization problems.",
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Strong convergence theorems for maximal monotone mappings in Banach spaces. / Zegeye, Habtu.

In: Journal of Mathematical Analysis and Applications, Vol. 343, No. 2, 15.07.2008, p. 663-671.

Research output: Contribution to journalArticle

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AB - Let E be a uniformly convex and 2-uniformly smooth real Banach space with dual E*. Let A : E* → E be a Lipschitz continuous monotone mapping with A-1 (0) ≠ ∅. For given u, x1 ∈ E, let {xn} be generated by the algorithm xn + 1 : = βn u + (1 - βn) (xn - αn A J xn), n ≥ 1, where J is the normalized duality mapping from E into E* and {λn} and {θn} are real sequences in (0, 1) satisfying certain conditions. Then it is proved that, under some mild conditions, {xn} converges strongly to x* ∈ E where J x* ∈ A-1 (0). Finally, we apply our convergence theorems to the convex minimization problems.

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