• Bulletin of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.153-162
• /
• 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.

• Honam Mathematical Journal
• /
• v.31 no.4
• /
• pp.559-578
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.47 no.5
• /
• pp.1037-1040
• /
• 2010

For convex subsets X of a topological vector space E, we show that a KKM principle implies a Fan-Browder type fixed point theorem and that this theorem implies generalized forms of the Sion minimax theorem.

• Journal of the Chungcheong Mathematical Society
• /
• v.21 no.3
• /
• pp.321-326
• /
• 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.

• Korean Journal of Mathematics
• /
• v.16 no.4
• /
• pp.511-526
• /
• 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.

• Journal of the Korean Mathematical Society
• /
• v.38 no.4
• /
• pp.697-709
• /
• 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.

• Journal of the Korean Mathematical Society
• /
• v.34 no.3
• /
• pp.719-730
• /
• 1997

From a minimax inequality related to admissible multimaps, we deduce generalized versions of lopsided saddle point theorems, fixed point theorems, existence of maximizable linear functionals, the Walras excess demand theorem, and the Gale-Nikaido-Debreu theorem.

• Journal of applied mathematics & informatics
• /
• v.7 no.1
• /
• pp.1-28
• /
• 2000

In the present paper, we introduce fundamental results in the KKM theory for G-convex spaces which are equivalent to the Brouwer theorem, the Sperner lemma, and the KKM theorem. Those results are all abstract versions of known corresponding ones for convex subsets of topological vector spaces. Some earlier applications of those results are indicated. Finally, We give a new proof of the Himmelberg fixed point theorem and G-convex space versions of the von Neumann type minimax theorem and the Nash equilibrium theorem as typical examples of applications of our theory.

• Honam Mathematical Journal
• /
• v.34 no.3
• /
• pp.439-449
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.49 no.6
• /
• pp.1147-1161
• /
• 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.