IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v189y2021i2d10.1007_s10957-021-01844-9.html
   My bibliography  Save this article

The Galerkin Method and Regularization for Variational Inequalities in Reflexive Banach Spaces

Author

Listed:
  • Bui Trong Kien

    (Quang Trung University)

  • Xiaolong Qin

    (Yibin University)

  • Ching-Feng Wen

    (Kaohsiung Medical University
    Kaohsiung Medical University Hospital)

  • Jen-Chih Yao

    (Center for General Education, China Medical University
    National Sun Yat-sen University)

Abstract

This paper studies the convergence of the Galerkin method and regularization for variational inequalities with pseudomonotone operators in the sense of Brézis. Namely, we prove that under certain conditions, the solutions of the Galerkin equations and regularized variational inequalities converge strongly to a solution of the original variational inequality in reflexive Banach spaces. An application for obstacle problems is given.

Suggested Citation

  • Bui Trong Kien & Xiaolong Qin & Ching-Feng Wen & Jen-Chih Yao, 2021. "The Galerkin Method and Regularization for Variational Inequalities in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 578-596, May.
    • Handle: RePEc:spr:joptap:v:189:y:2021:i:2:d:10.1007_s10957-021-01844-9
    DOI: 10.1007/s10957-021-01844-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-021-01844-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-021-01844-9?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 search for a different version of it.

    References listed on IDEAS

    as
    1. Jen-Chih Yao, 1994. "Variational Inequalities with Generalized Monotone Operators," Mathematics of Operations Research, INFORMS, vol. 19(3), pages 691-705, August.
    2. B. T. Kien & M. M. Wong & N. C. Wong & J. C. Yao, 2009. "Solution Existence of Variational Inequalities with Pseudomonotone Operators in the Sense of Brézis," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 249-263, February.
    Full references (including those not matched with items on IDEAS)

    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. Vu Thi Huong & Jen-Chih Yao & Nguyen Dong Yen, 2017. "On the Stability and Solution Sensitivity of a Consumer Problem," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 567-589, November.
    2. Lai-Jiu Lin & Zenn-Tsun Yu & Chih-Sheng Chuang, 2013. "Weak and strong convergence theorems for asymptotically pseudo-contraction mappings in the intermediate sense in Hilbert spaces," Journal of Global Optimization, Springer, vol. 56(1), pages 165-183, May.
    3. N. El Farouq, 2001. "Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method," Journal of Optimization Theory and Applications, Springer, vol. 111(2), pages 305-322, November.
    4. Lateef Olakunle Jolaoso & Adeolu Taiwo & Timilehin Opeyemi Alakoya & Oluwatosin Temitope Mewomo, 2020. "A Strong Convergence Theorem for Solving Pseudo-monotone Variational Inequalities Using Projection Methods," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 744-766, June.
    5. Y. B. Zhao & G. Isac, 2000. "Quasi-P*-Maps, P(τ, α, β)-Maps, Exceptional Family of Elements, and Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 105(1), pages 213-231, April.
    6. B. T. Kien & M. M. Wong & N. C. Wong & J. C. Yao, 2009. "Solution Existence of Variational Inequalities with Pseudomonotone Operators in the Sense of Brézis," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 249-263, February.
    7. Y. Chiang & J. C. Yao, 2004. "Vector Variational Inequalities and the (S)+ Condition," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 271-290, November.
    8. N. El Farouq, 2001. "Pseudomonotone Variational Inequalities: Convergence of Proximal Methods," Journal of Optimization Theory and Applications, Springer, vol. 109(2), pages 311-326, May.
    9. Yonghong Yao & Mihai Postolache, 2012. "Iterative Methods for Pseudomonotone Variational Inequalities and Fixed-Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 273-287, October.
    10. Mostafa Ghadampour & Ebrahim Soori & Ravi P. Agarwal & Donal O’Regan, 2022. "A Strong Convergence Theorem for Solving an Equilibrium Problem and a Fixed Point Problem Using the Bregman Distance," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 854-877, December.
    11. L. C. Zeng & J. C. Yao, 2007. "Existence Theorems for Variational Inequalities in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 132(2), pages 321-337, February.
    12. Yonghong Yao & Yeong-Cheng Liou & Shin Kang, 2013. "Two-step projection methods for a system of variational inequality problems in Banach spaces," Journal of Global Optimization, Springer, vol. 55(4), pages 801-811, April.
    13. L. C. Ceng & A. Petruşel, 2010. "Krasnoselski-Mann Iterations for Hierarchical Fixed Point Problems for a Finite Family of Nonself Mappings in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 617-639, September.
    14. Monica Bianchi & Siegfried Schaible, 2004. "Equilibrium Problems under Generalized Convexity and Generalized Monotonicity," Journal of Global Optimization, Springer, vol. 30(2), pages 121-134, November.
    15. L. C. Zeng & N. C. Wong & J. C. Yao, 2007. "Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 51-69, January.
    16. Fabián Flores-Bazán & Ruben López, 2005. "The Linear Complementarity Problem Under Asymptotic Analysis," Mathematics of Operations Research, INFORMS, vol. 30(1), pages 73-90, February.
    17. L. C. Ceng & S. Schaible & J. C. Yao, 2008. "Existence of Solutions for Generalized Vector Variational-like Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 137(1), pages 121-133, April.
    18. L. C. Zeng & S. Schaible & J. C. Yao, 2009. "Hybrid Steepest Descent Methods for Zeros of Nonlinear Operators with Applications to Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 141(1), pages 75-91, April.
    19. Daniel Steck, 2019. "Brezis Pseudomonotonicity is Strictly Weaker than Ky–Fan Hemicontinuity," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 318-323, April.
    20. Ahmed, Md. Tanveer & Kwon, Changhyun, 2012. "Pricing game of online display advertisement publishers," European Journal of Operational Research, Elsevier, vol. 219(2), pages 477-487.

    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:spr:joptap:v:189:y:2021:i:2:d:10.1007_s10957-021-01844-9. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.