Strong convergence theorems for maximal monotone mappings in Banach spaces

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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.

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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All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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