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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
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    6. 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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