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The Viscosity Approximation Forward‐Backward Splitting Method for Zeros of the Sum of Monotone Operators

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  • Oganeditse Aaron Boikanyo

Abstract

We investigate the convergence analysis of the following general inexact algorithm for approximating a zero of the sum of a cocoercive operator A and maximal monotone operators B with D(B) ⊂ H: xn+1 = αnf(xn) + γnxn + δn(I + rnB) −1(I − rnA)xn + en, for n = 1,2, …, for given x1 in a real Hilbert space H, where (αn), (γn), and (δn) are sequences in (0,1) with αn + γn + δn = 1 for all n ≥ 1, (en) denotes the error sequence, and f : H → H is a contraction. The algorithm is known to converge under the following assumptions on δn and en: (i) (δn) is bounded below away from 0 and above away from 1 and (ii) (en) is summable in norm. In this paper, we show that these conditions can further be relaxed to, respectively, the following: (i) (δn) is bounded below away from 0 and above away from 3/2 and (ii) (en) is square summable in norm; and we still obtain strong convergence results.

Suggested Citation

  • Oganeditse Aaron Boikanyo, 2016. "The Viscosity Approximation Forward‐Backward Splitting Method for Zeros of the Sum of Monotone Operators," Abstract and Applied Analysis, John Wiley & Sons, vol. 2016(1).
    • Handle: RePEc:wly:jnlaaa:v:2016:y:2016:i:1:n:2371857
    DOI: 10.1155/2016/2371857
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    References listed on IDEAS

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    1. Genaro López & Victoria Martín-Márquez & Fenghui Wang & Hong-Kun Xu, 2012. "Forward-Backward Splitting Methods for Accretive Operators in Banach Spaces," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-25, July.
    2. Genaro López & Victoria Martín-Márquez & Fenghui Wang & Hong-Kun Xu, 2012. "Forward‐Backward Splitting Methods for Accretive Operators in Banach Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    3. Jonathan Eckstein & Michael C. Ferris, 1998. "Operator-Splitting Methods for Monotone Affine Variational Inequalities, with a Parallel Application to Optimal Control," INFORMS Journal on Computing, INFORMS, vol. 10(2), pages 218-235, May.
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