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Forward‐Backward Splitting Methods for Accretive Operators in Banach Spaces

Author

Listed:
  • Genaro López
  • Victoria Martín-Márquez
  • Fenghui Wang
  • Hong-Kun Xu

Abstract

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce two iterative forward‐backward splitting methods with relaxations and errors to find zeros of the sum of two accretive operators in the Banach spaces. We shall prove the weak and strong convergence of these methods under mild conditions. We also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem.

Suggested Citation

  • 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).
    • Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:109236
    DOI: 10.1155/2012/109236
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    Cited by:

    1. 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).
    2. Yanlai Song & Luchuan Ceng, 2014. "Weak and Strong Convergence Theorems for Zeroes of Accretive Operators in Banach Spaces," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    3. Puntita Sae-jia & Eakkpop Panyahan & Suthep Suantai, 2025. "A New Accelerated Forward–Backward Splitting Algorithm for Monotone Inclusions with Application to Data Classification," Mathematics, MDPI, vol. 13(17), pages 1-24, August.
    4. Hongwei Jiao & Fenghui Wang, 2014. "On an Iterative Method for Finding a Zero to the Sum of Two Maximal Monotone Operators," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).

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