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Existence Results for Mixed Equilibrium Problems Involving Set-Valued Operators with Applications to Quasi-Hemivariational Inequalities

Author

Listed:
  • Bijaya Kumar Sahu

    (Indian Institute of Technology Bhubaneswar)

  • Ouayl Chadli

    (Ibn Zohr University)

  • Ram N. Mohapatra

    (University of Central Florida)

  • Sabyasachi Pani

    (Indian Institute of Technology Bhubaneswar)

Abstract

In this paper, we study the existence of solutions for mixed equilibrium problems associated with a set-valued operator in the general setting of vector spaces in duality, and in particular in Banach spaces. We use a Galerkin-type method and the notion of pseudomonotonicity in the sense of Brézis for bifunctions. As application, we study the existence of solutions for quasi-hemivariational inequalities governed by a set-valued mapping and perturbed with a nonlinear term. Our main results can be applied to differential inclusions, evolution equations and evolution hemivariational inequalities.

Suggested Citation

  • Bijaya Kumar Sahu & Ouayl Chadli & Ram N. Mohapatra & Sabyasachi Pani, 2020. "Existence Results for Mixed Equilibrium Problems Involving Set-Valued Operators with Applications to Quasi-Hemivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 810-823, March.
    • Handle: RePEc:spr:joptap:v:184:y:2020:i:3:d:10.1007_s10957-019-01629-1
    DOI: 10.1007/s10957-019-01629-1
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    References listed on IDEAS

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    1. Nicuşor Costea & Vicenţiu Rădulescu, 2012. "Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term," Journal of Global Optimization, Springer, vol. 52(4), pages 743-756, April.
    2. Rabian Wangkeeree & Pakkapon Preechasilp, 2013. "Existence theorems of the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in Banach spaces," Journal of Global Optimization, Springer, vol. 57(4), pages 1447-1464, December.
    3. Y. L. Zhang & Y. R. He, 2013. "The Hemivariational Inequalities for an Upper Semicontinuous Set-valued Mapping," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 716-725, March.
    4. Nicuşor Costea & Daniel Alexandru Ion & Cezar Lupu, 2012. "Variational-Like Inequality Problems Involving Set-Valued Maps and Generalized Monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 79-99, October.
    5. O. Chadli & S. Schaible & J. C. Yao, 2004. "Regularized Equilibrium Problems with Application to Noncoercive Hemivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 571-596, June.
    6. Ouayl Chadli & Qamrul Hasan Ansari & Jen-Chih Yao, 2016. "Mixed Equilibrium Problems and Anti-periodic Solutions for Nonlinear Evolution Equations," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 410-440, February.
    7. 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.
    8. Bigi, Giancarlo & Castellani, Marco & Pappalardo, Massimo & Passacantando, Mauro, 2013. "Existence and solution methods for equilibria," European Journal of Operational Research, Elsevier, vol. 227(1), pages 1-11.
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