IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v271y2015icp251-258.html

Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces

Author

Listed:
  • Nakajo, Kazuhide

Abstract

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E, A be a single valued monotone and Lipschitz continuous mapping of C into E* and T be a single valued relatively nonexpansive mapping of C into itself. In this paper, we consider the composition and the convex combination of T and the gradient projection method for A which Goldstein (1964) proposed and proved the strong convergence to a common element of solutions of the variational inequality problem for A and fixed points of T by the hybrid method in mathematical programming (Haugazeau, 1968). And we get several results which improve the well-known results in a 2-uniformly convex and uniformly smooth Banach space and a Hilbert space.

Suggested Citation

  • Nakajo, Kazuhide, 2015. "Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 251-258.
    • Handle: RePEc:eee:apmaco:v:271:y:2015:i:c:p:251-258
    DOI: 10.1016/j.amc.2015.08.096
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300315011522
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2015.08.096?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Heinz H. Bauschke & Patrick L. Combettes, 2001. "A Weak-to-Strong Convergence Principle for Fejér-Monotone Methods in Hilbert Spaces," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 248-264, May.
    2. W. Takahashi & M. Toyoda, 2003. "Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 118(2), pages 417-428, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ying Liu & Hang Kong, 2019. "Strong convergence theorems for relatively nonexpansive mappings and Lipschitz-continuous monotone mappings in Banach spaces," Indian Journal of Pure and Applied Mathematics, Springer, vol. 50(4), pages 1049-1065, December.
    2. Hammed Anuoluwapo Abass & Olawale Kazeem Oyewole & Seithuti Philemon Moshokoa & Abubakar Adamu, 2024. "An Adapted Proximal Point Algorithm Utilizing the Golden Ratio Technique for Solving Equilibrium Problems in Banach Spaces," Mathematics, MDPI, vol. 12(23), pages 1-17, November.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. H. Zegeye & N. Shahzad, 2012. "Convergence Theorems for a Common Point of Solutions of Equilibrium and Fixed Point of Relatively Nonexpansive Multivalued Mapping Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    2. Lu-Chuan Ceng & Ching-Feng Wen, 2012. "Hybrid Extragradient Iterative Algorithms for Variational Inequalities, Variational Inclusions, and Fixed‐Point Problems," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    3. Yanlai Song & Omar Bazighifan, 2022. "Regularization Method for the Variational Inequality Problem over the Set of Solutions to the Generalized Equilibrium Problem," Mathematics, MDPI, vol. 10(14), pages 1-20, July.
    4. S. Plubtieng & T. Thammathiwat, 2010. "A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities," Journal of Global Optimization, Springer, vol. 46(3), pages 447-464, March.
    5. Yanlai Song & Omar Bazighifan, 2022. "Two Regularization Methods for the Variational Inequality Problem over the Set of Solutions of the Generalized Mixed Equilibrium Problem," Mathematics, MDPI, vol. 10(16), pages 1-20, August.
    6. A. Tada & W. Takahashi, 2007. "Weak and Strong Convergence Theorems for a Nonexpansive Mapping and an Equilibrium Problem," Journal of Optimization Theory and Applications, Springer, vol. 133(3), pages 359-370, June.
    7. 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).
    8. Lili Chen & Ni Yang & Jing Zhou, 2020. "Common Attractive Points of Generalized Hybrid Multi-Valued Mappings and Applications," Mathematics, MDPI, vol. 8(8), pages 1-15, August.
    9. Yuanheng Wang & Mingyue Yuan & Bingnan Jiang, 2021. "Multi-Step Inertial Hybrid and Shrinking Tseng’s Algorithm with Meir–Keeler Contractions for Variational Inclusion Problems," Mathematics, MDPI, vol. 9(13), pages 1-13, July.
    10. A. Moudafi, 2011. "Split Monotone Variational Inclusions," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 275-283, August.
    11. L. Zeng & J. Yao, 2009. "A hybrid extragradient method for general variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(1), pages 141-158, March.
    12. Jing Zhao & Hang Zhang, 2014. "Solving Split Common Fixed‐Point Problem of Firmly Quasi‐Nonexpansive Mappings without Prior Knowledge of Operators Norms," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    13. Yasunori Kimura & Kazuhide Nakajo, 2013. "The Problem of Image Recovery by the Metric Projections in Banach Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    14. Yekini Shehu, 2012. "Iterative method for fixed point problem, variational inequality and generalized mixed equilibrium problems with applications," Journal of Global Optimization, Springer, vol. 52(1), pages 57-77, January.
    15. J. W. Peng, 2010. "Iterative Algorithms for Mixed Equilibrium Problems, Strict Pseudocontractions and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 144(1), pages 107-119, January.
    16. Ming Tian & Meng-Ying Tong, 2019. "Extension and Application of the Yamada Iteration Algorithm in Hilbert Spaces," Mathematics, MDPI, vol. 7(3), pages 1-13, February.
    17. Shin-ya Matsushita & Li Xu, 2014. "On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 701-715, June.
    18. Yanlai Song & Luchuan Ceng, 2013. "A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces," Journal of Global Optimization, Springer, vol. 57(4), pages 1327-1348, December.
    19. Yazheng Dang & Yan Gao, 2013. "Inertial Iteration for Split Common Fixed‐Point Problem for Quasi‐Nonexpansive Operators," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    20. Z. Y. Huang & M. A. Noor & E. Al-Said, 2010. "On an Open Question of Takahashi for Nonexpansive Mappings and Inverse Strongly Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 194-204, October.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:271:y:2015:i:c:p:251-258. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.