• Title, Summary, Keyword: minimax theorem

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A NON-COMPACT GENERALIZATION OF HORVATH'S INTERSECTION THEOREM$^*$

  • Kim, Won-Kyu
    • Bulletin of the Korean Mathematical Society
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    • v.32 no.2
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    • pp.153-162
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    • 1995
  • Ky Fan's minimax inequality is an important tool in nonlinear functional analysis and its applications, e.g. game theory and economic theory. Since Fan gave his minimax inequality in [2], various extensions of this interesting result have been obtained (see [4,11] and the references therein). Using Fan's minimax inequality, Ha [6] obtained a non-compact version of Sion's minimax theorem in topological vector spaces, and next Geraghty-Lin [3], Granas-Liu [4], Shih-Tan [11], Simons [12], Lin-Quan [10], Park-Bae-Kang [17], Bae-Kim-Tan [1] further generalize Fan's minimax theorem in more general settings. In [9], using the concept of submaximum, Komiya proved a topological minimax theorem which also generalized Sion's minimax theorem and another minimax theorem of Ha in [5] without using linear structures. And next Lin-Quan [10] further generalizes his result to two function versions and non-compact topological settings.

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GENERALIZED MINIMAX THEOREMS IN GENERALIZED CONVEX SPACES

  • Kim, Hoon-Joo
    • Honam Mathematical Journal
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    • v.31 no.4
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    • pp.559-578
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    • 2009
  • In this work, we obtain intersection theorem, analytic alternative and von Neumann type minimax theorem in G-convex spaces. We also generalize Ky Fan minimax inequality to acyclic versions in G-convex spaces. The result is applied to formulate acyclic versions of other minimax results, a theorem of systems of inequalities and analytic alternative.

A TWO-FUNCTION MINIMAX THEOREM

  • Kim, Won Kyu;Kum, Sangho
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.3
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    • pp.321-326
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    • 2008
  • In this note, using the separation theorem for convex sets, we will give a two functions version generalization of Fan's minimax theorem by relaxing the convexlike assumption to the weak convexlike condition.

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MULTIPLICITY RESULTS FOR THE ELLIPTIC SYSTEM USING THE MINIMAX THEOREM

  • Nam, Hyewon
    • Korean Journal of Mathematics
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    • v.16 no.4
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    • pp.511-526
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    • 2008
  • In this paper, we consider an elliptic system of three equations using the minimax theorem. We prove the existence of two solutions for suitable forcing terms, under a condition on the linear part which prevents resonance with eigenvalues of the operator.

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GENERALIZATIONS OF THE NASH EQUILIBRIUM THEOREM ON GENERALIZED CONVEX SPACES

  • Park, Se-Hie
    • Journal of the Korean Mathematical Society
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    • v.38 no.4
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    • pp.697-709
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    • 2001
  • Generalized forms of the von neumann-Sion type minimax theorem, the Fan-Ma intersection theorem, the Fan-a type analytic alternative, and the Nash-Ma equilibrium theorem hold for generalized convex spaces without having any linear structure.

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COINCIDENCE THEOREMS FOR NONCOMPACT ℜℭ-MAPS IN ABSTRACT CONVEX SPACES WITH APPLICATIONS

  • Yang, Ming-Ge;Huang, Nan-Jing
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.6
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    • pp.1147-1161
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    • 2012
  • In this paper, a coincidence theorem for a compact ${\Re}\mathfrak{C}$-map is proved in an abstract convex space. Several more general coincidence theorems for noncompact ${\Re}\mathfrak{C}$-maps are derived in abstract convex spaces. Some examples are given to illustrate our coincidence theorems. As applications, an alternative theorem concerning the existence of maximal elements, an alternative theorem concerning equilibrium problems and a minimax inequality for three functions are proved in abstract convex spaces.

FIXED POINTS AND ALTERNATIVE PRINCIPLES

  • Park, Se-Hie;Kim, Hoon-Joo
    • Honam Mathematical Journal
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    • v.34 no.3
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    • pp.439-449
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    • 2012
  • In a recent paper, M. Balaj [B] established an alternative principle. The principle was applied to a matching theorem of Ky Fan type, an analytic alternative, a minimax inequality, and existence of solutions of a vector equilibrium theorem. Based on the first author's fixed point theorems, in the present paper, we obtain generalizations of the main result of Balaj [B] and their applications.

FIXED POINT THEOREMS, SECTION PROPERTIES AND MINIMAX INEQUALITIES ON K-G-CONVEX SPACES

  • Balaj, Mircea
    • Journal of the Korean Mathematical Society
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    • v.39 no.3
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    • pp.387-395
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    • 2002
  • In [11] Kim obtained fixed point theorems for maps defined on some “locally G-convex”subsets of a generalized convex space. Theorem 2 in Kim's article determines us to introduce, in this paper, the notion of K-G-convex space. In this framework we obtain fixed point theorems, section properties and minimax inequalities.