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AIMS Mathematics

2021, Volume 6, Issue 4: 3160-3176. doi: 10.3934/math.2021191
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Strongly essential set of vector Ky Fan's points problem and its applications

  • 1.
    School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China
  • 2.
    School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, China
  • 3.
    College of Mathematical and Information Science, Guiyang University, Guiyang, Guizhou 550005, China
  • 4.
    School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou 550018, China
  • Received: 12 November 2020 Accepted: 05 January 2021 Published: 15 January 2021

  • MSC : 46T20, 49J53, 91A10

  • In this paper, several existence results of strongly essential set of the solution set for Ky Fan's section problems and vector Ky Fan's point problems are obtained. Firstly, two kinds of strongly essential sets of Ky Fan's section problems are defined, and some further results on existence of the strongly essential component of solutions set of Ky Fan's section problems are proved, which generalize the conclusion in [22], and further generalize the conclusions in [21,28]. Secondly, based on the above results, two classes of stronger perturbations of vector-valued inequality functions are proposed respectively, and several existence results of the strongly essential component of set of vector Ky Fan's points are obtained. By comparing several metrics, we give some strong and weak relationships among the various metrics involved in the text. The main results of this paper actually generalize the relevant conclusions in the current literature. Finally, as an application, we obtain an existence result of the strongly essential component of weakly Pareto-Nash equilibrium for multiobjective games.

    Citation: Dejin Zhang, Shuwen Xiang, Xicai Deng, Yanlong Yang. Strongly essential set of vector Ky Fan's points problem and its applications[J]. AIMS Mathematics, 2021, 6(4): 3160-3176. doi: 10.3934/math.2021191

    Related Papers:

  • In this paper, several existence results of strongly essential set of the solution set for Ky Fan's section problems and vector Ky Fan's point problems are obtained. Firstly, two kinds of strongly essential sets of Ky Fan's section problems are defined, and some further results on existence of the strongly essential component of solutions set of Ky Fan's section problems are proved, which generalize the conclusion in [22], and further generalize the conclusions in [21,28]. Secondly, based on the above results, two classes of stronger perturbations of vector-valued inequality functions are proposed respectively, and several existence results of the strongly essential component of set of vector Ky Fan's points are obtained. By comparing several metrics, we give some strong and weak relationships among the various metrics involved in the text. The main results of this paper actually generalize the relevant conclusions in the current literature. Finally, as an application, we obtain an existence result of the strongly essential component of weakly Pareto-Nash equilibrium for multiobjective games.


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    [1] C. D. Aliprantis, K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3 Eds., Berlin: Springer-Verlag, 2006.
    [2] L. Q. Anh, N. V. Hung, The existence and stability of solutions for symmetric generalized quasivariational inclusion problems, Filomat, 29 (2015), 2147–2165.
    [3] L. Q. Anh, N. V. Hung, Stability of solution mappings for parametric bilevel vector equilibrium problems, Comput. Appl. Math., 37 (2018), 1537–1549.
    [4] Q. H. Ansari, J. C. Yao, Systems of vector quasi-equilibrium problems and their applications, in Variational Analysis and Generalized Differentiation in Optimization and Control, New York: Springer, 47 (2010), 1–42.
    [5] H. Br$\acute{e}$zis, L. Nirenberg, G. Stampacchia, A remark on Ky Fan's minimax principle, Boll. Un. Mat. Ital., 6 (1972), 293–300.
    [6] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123–145.
    [7] S. S. Chang, Salahuddin, G. Wang, On the existence theorems of solutions for generalized vector variational inequalities, J. Inequal. Appl., 2015 (2015), 365.
    [8] G. Y. Chen, Existence of solutions for a vector variational inequalities: An extension of Hartman-Stampacchia theorems, J. Optim. Theory Appl., 74 (1992), 445–456.
    [9] G. Y. Chen, X. X. Huang, X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Berlin: Springer-Verlag, 2005.
    [10] X. P. Ding, KKM type theorems and minimax inequalities and saddle point theorems in generalized convex spaces, Acta Math. Sin., 47 (2004), 711–722.
    [11] K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Annl., 142 (1961), 305–310.
    [12] K. Fan, A minimax inequality and applications, Inequalities, 3 (1972), 103–113.
    [13] F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Springer Science & Business Media, 2013.
    [14] N. J. Huang, On the generalized implicit quasivariational inequalities, J. Math. Anal. Appl., 216 (1997), 197–210.
    [15] N. V. Hung, D. O'Regan, Bilevel equilibrium problems with lower and upper bounds in locally convex Hausdorff topological vector spaces, Topol. Appl., 2020 (2020), 106939.
    [16] N. V. Hung, V. M. Tam, D. O'Regan, Y. J. Cho, A new class of generalized multiobjective games in bounded rationality with fuzzy mappings: Structural $(\lambda, \epsilon)$-stability and $(\lambda, \epsilon)$-robustness to $\epsilon$-equilibria, J. Comput. Appl. Math., 372 (2020), 112735.
    [17] W. S. Jia, S. W. Xiang, J. H. He, Y. L. Yang, The existence and stability for weakly Ky Fan's points of set-valued mappings, J. Inequal. Appl., 2012 (2012), 199.
    [18] E. Klein, A. C. Thompson, Theory of Correspondences, New York: A Wiley-Inter Science Publication, 1984.
    [19] U. Mosco, Implicit variational problems and quasivariational inequalities, in Nonlinear Operators and the Calculus of Variations, Berlin: Springer-Verlag, (1976), 83–156.
    [20] K. K. Tan, J. Yu, X. Z. Yuan, The stability of Ky Fan's points, Proc. Amer. Math. Soc., 123 (1995), 1511–1519.
    [21] S. W. Xiang, J. H. He, C. W. Liu, W. S. Jia, Y. L. Yang, Some further results on the stability of Ky Fan's points, J. Inequal. Appl., 2017 (2017), 1–12.
    [22] S. W. Xiang, D. J. Zhang, R. Li, Y. L. Yang, Strongly essential set of Ky Fan's points and the stability of Nash equilibrium, J. Math. Anal. Appl., 459 (2018), 839–851.
    [23] Z. Yang, On existence and essential stability of solutions of symmetric variational relation problems, J. Inequalities Appl., 2014 (2014), 1–13.
    [24] J. Yu, Q. Luo, On essential components of the solution set of generalized games, J. Math. Anal. Appl., 230 (1999), 303–310.
    [25] J. Yu, D. T. Peng, Solvability of vector Ky Fan inequalities with applications, J. Syst. Sci. Complexity, 26 (2016), 978–990.
    [26] J. Yu, S. W. Xiang, On essential components of the set of Nash equilibrium points, Nonlinear Anal.: Theory Methods Appl., 38 (1999), 259–264.
    [27] J. Yu, H. Yang, Essential components of the set of weakly Pareto-Nash equilibrium points, Appl. Math. Lett., 15 (2002), 553–560.
    [28] Y. H. Zhou, S. W. Xiang, H. Yang, Stability of solutions for Ky Fan's section theorem with some applications, Nonlinear Anal.: Theory Methods Appl., 62 (2005), 1127–1136.
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