Hybrid viscosity approximation method for zeros of m-accretive operators in banach spaces

Lu Chuan Ceng, Qamrul Ansari, Siegfried Schaible, Jen Chih Yao

Research output: Contribution to journalJournal articlepeer-review

19 Scopus citations

Abstract

Let X be a uniformly smooth Banach space and A be an m-accretive operator on X with A -1 (0) ≠ θ. Assume that F: X → X is δstrongly accretive and λ-strictly pseudocontractive with δ+λ > 1. This article proposes hybrid viscosity approximation methods which combine viscosity approximation methods with hybrid steepest-descent methods. For each t∈ (0, 1) and each integer n≥0, let {x t,n} be defined byx t,n = tf(x t,n + (1-t)[Jr nx t,n-θ F(Jr nx t,n] where f: X → X is a contractive map, {r n}⊂[ε,∞) for some ε > 0 and {θ:t∈⊂(0, 1)}[0, 1) with lim t→t/t=0. We deduce that as t→0, {x t,n} converges strongly to a zero p of A, which is a unique solution of some variational inequality. On the other hand, given a point x 0∈ X and given sequences {λ n}, {μ n} in [0, 1], {α n}, {β n} in (0, 1], let the sequence {x n be generated by {y n= α nx n+(1 -α n)Jr n,x n+1 = β nf (x n)+(1-β n) [Jr n -y n-λμ nμ nF(Jr n y n)], ∀n ≥ 0. It is proven that under appropriate conditions {x n} converges strongly to the same zero p of A. The results presented here extend, improve and develop some very recent theorems in the literature to a great extent.

Original languageEnglish
Pages (from-to)142-165
Number of pages24
JournalNumerical Functional Analysis and Optimization
Volume33
Issue number2
DOIs
StatePublished - 1 Feb 2012
Externally publishedYes

Keywords

  • Contractive maps
  • Hybrid viscosity approximation methods
  • m-accretive operators
  • Uniformly smooth Banach space
  • Variational inequality

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