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Generalized Equilibrium Problems

Author

Listed:
  • Mircea Balaj

    (Department of Mathematics, University of Oradea, 410087 Oradea, Romania)

  • Dan Florin Serac

    (Department of Mathematics, University of Oradea, 410087 Oradea, Romania)

Abstract

If X is a convex subset of a topological vector space and f is a real bifunction defined on X × X , the problem of finding a point x 0 ∈ X such that f ( x 0 , y ) ≥ 0 for all y ∈ X , is called an equilibrium problem. When the bifunction f is defined on the cartesian product of two distinct sets X and Y we will call it a generalized equilibrium problem. In this paper, we study the existence of the solutions, first for generalized equilibrium problems and then for equilibrium problems. In the obtained results, apart from the bifunction f , another bifunction is introduced, the two being linked by a certain compatibility condition. The particularity of the equilibrium theorems established in the last section consists of the fact that the classical equilibrium condition ( f ( x , x ) = 0 , for all x ∈ X ) is missing. The given applications refer to the Minty variational inequality problem and quasi-equilibrium problems.

Suggested Citation

  • Mircea Balaj & Dan Florin Serac, 2023. "Generalized Equilibrium Problems," Mathematics, MDPI, vol. 11(9), pages 1-11, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2146-:d:1138725
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    References listed on IDEAS

    as
    1. Rabia Nessah & Guoqiang Tian, 2013. "Existence of Solution of Minimax Inequalities, Equilibria in Games and Fixed Points Without Convexity and Compactness Assumptions," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 75-95, April.
    2. Gábor Kassay, 2010. "On Equilibrium Problems," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Rentsen Enkhbat & Ider Tseveendorj (ed.), Optimization and Optimal Control, pages 55-83, Springer.
    3. M. Bianchi & R. Pini, 2005. "Coercivity Conditions for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 79-92, January.
    4. Ravi P. Agarwal & Mircea Balaj & Donal O’Regan, 2017. "Common Fixed Point Theorems in Topological Vector Spaces via Intersection Theorems," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 443-458, May.
    5. Mircea Balaj & Marco Castellani & Massimiliano Giuli, 2023. "New criteria for existence of solutions for equilibrium problems," Computational Management Science, Springer, vol. 20(1), pages 1-16, December.
    6. Yannelis, Nicholas C. & Prabhakar, N. D., 1983. "Existence of maximal elements and equilibria in linear topological spaces," Journal of Mathematical Economics, Elsevier, vol. 12(3), pages 233-245, December.
    7. Tian, Guoqiang, 1994. "Generalized KKM theorem, minimax inequalities and their applications," MPRA Paper 41217, University Library of Munich, Germany.
    8. Ravi P. Agarwal & Mircea Balaj & Donal O’Regan, 2018. "Intersection Theorems with Applications in Optimization," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 761-777, December.
    9. 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.
    10. Mircea Balaj, 2022. "Scalar and vector equilibrium problems with pairs of bifunctions," Journal of Global Optimization, Springer, vol. 84(3), pages 739-753, November.
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