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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, Hindawi, vol. 2012, pages 1-25, July.
    • Handle: RePEc:hin:jnlaaa:109236
    DOI: 10.1155/2012/109236
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    Cited by:

    1. Cholamjiak, Watcharaporn & Dutta, Hemen, 2022. "Viscosity modification with parallel inertial two steps forward-backward splitting methods for inclusion problems applied to signal recovery," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    2. Lu-Chuan Ceng & Meijuan Shang, 2019. "Generalized Mann Viscosity Implicit Rules for Solving Systems of Variational Inequalities with Constraints of Variational Inclusions and Fixed Point Problems," Mathematics, MDPI, vol. 7(10), pages 1-18, October.
    3. Yekini Shehu & Aviv Gibali, 2020. "Inertial Krasnoselskii–Mann Method in Banach Spaces," Mathematics, MDPI, vol. 8(4), pages 1-13, April.
    4. Nattakarn Kaewyong & Kanokwan Sitthithakerngkiet, 2021. "Modified Tseng’s Method with Inertial Viscosity Type for Solving Inclusion Problems and Its Application to Image Restoration Problems," Mathematics, MDPI, vol. 9(10), pages 1-15, May.
    5. Yanlai Song & Mihai Postolache, 2021. "Modified Inertial Forward–Backward Algorithm in Banach Spaces and Its Application," Mathematics, MDPI, vol. 9(12), pages 1-17, June.
    6. Jenwit Puangpee & Suthep Suantai, 2020. "A New Accelerated Viscosity Iterative Method for an Infinite Family of Nonexpansive Mappings with Applications to Image Restoration Problems," Mathematics, MDPI, vol. 8(4), pages 1-20, April.
    7. Li Wei & Yingzi Shang & Ravi P. Agarwal, 2019. "New Inertial Forward-Backward Mid-Point Methods for Sum of Infinitely Many Accretive Mappings, Variational Inequalities, and Applications," Mathematics, MDPI, vol. 7(5), pages 1-19, May.
    8. Adamu, A. & Kitkuan, D. & Padcharoen, A. & Chidume, C.E. & Kumam, P., 2022. "Inertial viscosity-type iterative method for solving inclusion problems with applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 445-459.
    9. Chanjuan Pan & Yuanheng Wang, 2019. "Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces," Mathematics, MDPI, vol. 7(2), pages 1-12, February.
    10. Dang Hieu & Pham Ky Anh & Nguyen Hai Ha, 2021. "Regularization Proximal Method for Monotone Variational Inclusions," Networks and Spatial Economics, Springer, vol. 21(4), pages 905-932, December.
    11. Peeyada, Pronpat & Suparatulatorn, Raweerote & Cholamjiak, Watcharaporn, 2022. "An inertial Mann forward-backward splitting algorithm of variational inclusion problems and its applications," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    12. Prasit Cholamjiak & Suparat Kesornprom & Nattawut Pholasa, 2019. "Weak and Strong Convergence Theorems for the Inclusion Problem and the Fixed-Point Problem of Nonexpansive Mappings," Mathematics, MDPI, vol. 7(2), pages 1-19, February.
    13. Shamshad Husain & Mohammed Ahmed Osman Tom & Mubashshir U. Khairoowala & Mohd Furkan & Faizan Ahmad Khan, 2022. "Inertial Tseng Method for Solving the Variational Inequality Problem and Monotone Inclusion Problem in Real Hilbert Space," Mathematics, MDPI, vol. 10(17), pages 1-16, September.

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