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Minimax Inequalities and Variational Equations

  • Maria Isabel BerenguerEmail author
  • Domingo Gámez
  • A. I. Garralda–Guillem
  • M. Ruiz Galán
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

In this paper we study some weak conditions guaranteeing the validity of several minimax inequalities and illustrate the possibilities of such a tool for characterizing the existence of solutions of certain variational equations.

Keywords

Minimax inequalities Variational equations 

References

  1. 1.
    Baye, M., Tian, G., Zhou, J.: Characterizations of the existence of equilibria in games with discontinuous and nonquasiconcave payoffs. Rev. Econ. Stud. 60, 935–948 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berenguer, M. I., Gámez, D., Garralda-Guillem, A. I., Ruiz Galán, M.: A discrete characterization of the solvability of equilibrium problems. Submitted for publicationGoogle Scholar
  3. 3.
    Boffi, D., et al.: Mixed Finite Elements, Compatibility Conditions and Applications. Lecture Notes in Mathematics, vol. 1939. Springer-Verlag, Heidelberg (2008)Google Scholar
  4. 4.
    Borwein, J.M., Giladi, O.: Some remarks on convex analysis in topological groups. J. Convex. Anal. 23, 313–332 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Deng, X.T., Li, Z.F., Wang, S.Y.: A minimax portfolio selection strategy with equilibrium. Eur. J. Oper. Res. 166, 278–292 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fan, K.: Minimax theorems. Proc. Nat. Acad. Sci. USA 39, 42–47 (1953)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Garralda Guillem, A.I., Ruiz Galán, R.: Mixed variational formulations in locally convex spaces. J. Math. Anal. Appl. 414, 825–849 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Garralda Guillem, A.I., Ruiz Galán, R.: A minimax approach for the study of systems of variational equations and related Galerkin schemes. J. Comput. Appl. Math. 354, 103–111 (2019)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gatica, G.N.: A Simple Introduction to the Mixed Finite Element Method Theory and Applications. Springer Briefs in Mathematics. Springer, Cham (2014)CrossRefGoogle Scholar
  10. 10.
    Grossmann, C., Roos, H.G., Stynes, M.: Numerical Treatment of Partial Differential Equations. Springer-Verlag, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Kassay, G., Kolumbán, J.: On a generalized sup-inf problem. J. Optim. Theory Appl. 91, 651–670 (1996)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Khanh, P.Q., Quan, N.H.: General existence theorems, alternative theorems and applications to minimax problems. Nonlinear Anal. 72, 2706–2715 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kenmochi, N.: Monotonicity and compactness methods for nonlinear variational inequalities, Handbook of differential equations: stationary partial differential equations, IV, pp. 203–298. Elsevier/North-Holland, Amsterdam (2007)CrossRefGoogle Scholar
  14. 14.
    Kunze, H., La Torre, D., Levere, K., Ruiz Galán, M.: Inverse problems via the “generalized collage theorem” for vector-valued Lax-Milgram-based variational problems. Math. Probl. Eng. 8 (2015). Article ID 764643Google Scholar
  15. 15.
    Polyanskiy, Y.: Saddle point in the minimax converse for channel coding. IEEE Trans. Inf. Theory 59, 2576–2595 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ricceri, B.: On a minimax theorem: an improvement, a new proof and an overview of its applications. Minimax Theory Appl. 2, 99–152 (2017)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ruiz Galán, M.: A concave-convex Ky Fan minimax inequality. Minimax Theory Appl. 1, 11–124 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ruiz Galán, M.: The Gordan theorem and its implications for minimax theory. J. Nonlinear Convex Anal. 17, 2385–2405 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ruiz Galán, M.: An intrinsic notion of convexity for minimax. J. Convex Anal. 21, 1105–1139 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Ruiz Galán, M.: A version of the Lax-Milgram theorem for locally convex spaces. J. Convex Anal. 16, 993–1002 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Saint Raymond, J.: A new minimax theorem for linear operators. Minimax Theory Appl. 3, 131–160 (2018)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Simons, S.: Minimax and Monotonicity. Lecture Notes in Mathematics, vol. 1693. Springer-Verlag, Heidelberg (1998)Google Scholar
  23. 23.
    Simons, S.: Minimax theorems and their proofs. In: Minimax and applications. Nonconvex Optimization and its Applications, pp. 1–23. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of Granada, E.T.S. Ingeniería de EdificacióónGranadaSpain

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