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A Strong Convergence Theorem for Solving Pseudo-monotone Variational Inequalities Using Projection Methods

Author

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  • Lateef Olakunle Jolaoso

    (University of KwaZulu-Natal)

  • Adeolu Taiwo

    (University of KwaZulu-Natal)

  • Timilehin Opeyemi Alakoya

    (University of KwaZulu-Natal)

  • Oluwatosin Temitope Mewomo

    (University of KwaZulu-Natal)

Abstract

Several iterative methods have been proposed in the literature for solving the variational inequalities in Hilbert or Banach spaces, where the underlying operator A is monotone and Lipschitz continuous. However, there are very few methods known for solving the variational inequalities, when the Lipschitz continuity of A is dispensed with. In this article, we introduce a projection-type algorithm for finding a common solution of the variational inequalities and fixed point problem in a reflexive Banach space, where A is pseudo-monotone and not necessarily Lipschitz continuous. Also, we present an application of our result to approximating solution of pseudo-monotone equilibrium problem in a reflexive Banach space. Finally, we present some numerical examples to illustrate the performance of our method as well as comparing it with related method in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:3:d:10.1007_s10957-020-01672-3
    DOI: 10.1007/s10957-020-01672-3
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    References listed on IDEAS

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    1. Gang Cai & Aviv Gibali & Olaniyi S. Iyiola & Yekini Shehu, 2018. "A New Double-Projection Method for Solving Variational Inequalities in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 178(1), pages 219-239, July.
    2. L. C. Ceng & M. Teboulle & J. C. Yao, 2010. "Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 19-31, July.
    3. Jen-Chih Yao, 1994. "Variational Inequalities with Generalized Monotone Operators," Mathematics of Operations Research, INFORMS, vol. 19(3), pages 691-705, August.
    4. Rapeepan Kraikaew & Satit Saejung, 2014. "Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 399-412, November.
    5. 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.
    6. Alfredo Iusem & Mostafa Nasri, 2011. "Korpelevich’s method for variational inequality problems in Banach spaces," Journal of Global Optimization, Springer, vol. 50(1), pages 59-76, May.
    7. Y. Censor & A. Gibali & S. Reich, 2011. "The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 318-335, February.
    8. H. Iiduka, 2009. "Hybrid Conjugate Gradient Method for a Convex Optimization Problem over the Fixed-Point Set of a Nonexpansive Mapping," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 463-475, March.
    9. Dan Butnariu & Elena Resmerita, 2006. "Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces," Abstract and Applied Analysis, Hindawi, vol. 2006, pages 1-39, February.
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    Cited by:

    1. Annel Thembinkosi Bokodisa & Lateef Olakunle Jolaoso & Maggie Aphane, 2021. "Halpern-Subgradient Extragradient Method for Solving Equilibrium and Common Fixed Point Problems in Reflexive Banach Spaces," Mathematics, MDPI, vol. 9(7), pages 1-24, March.
    2. 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.
    3. Timilehin O. Alakoya & Oluwatosin T. Mewomo & Yekini Shehu, 2022. "Strong convergence results for quasimonotone variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(2), pages 249-279, April.
    4. Lateef Olakunle Jolaoso & Maggie Aphane, 2020. "A Generalized Viscosity Inertial Projection and Contraction Method for Pseudomonotone Variational Inequality and Fixed Point Problems," Mathematics, MDPI, vol. 8(11), pages 1-29, November.
    5. 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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