Generalizations of the Nash Equilibrium Theorem in the KKM Theory

Abstract

The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. These results are compared with previously known cases for G-convex spaces. Consequently, our results unify and generalize most of previously known particular cases of the same nature. Finally, we add some detailed historical remarks on related topics.

Optimality and Duality Theorems in Nonsmooth Multiobjective Optimization

“…Several nonlinear analysis problems arise from areas of optimization theory, game theory, differential equations, mathematical physics, convex analysis and nonlinear functional analysis. Park [1][2][3] has devoted to the study of nonlinear analysis and his results had a strong influence on the research topics of equilibrium complementarity and optimization problems. Nonsmooth phenomena in mathematics and optimization occurs naturally and frequently.…”

Riemannian manifolds are KKM spaces

“…Recall that some corrections on [19] were made in [22]. For any geodesic convex subset X of a Riemannian manifold (M, g) with a nonempty set D ⊂ X, the KKM space (X, D; Γ) satisfies all results in [18], [19].…”

“…Since 2006, we have established the Knaster-Kuratowski-Mazurkiewicz (simply, KKM) theory on abstract convex spaces; for example, see [18,19]. In the present article, we are going to establish the basis of the KKM theory on Riemannian manifolds.…”

Fixed point theorems and $$L^{*}$$ L ∗ -operators

“…In this way, proving various types of fixed point theorems (of Tychonoff or Schauder type), along with a version of Nash's equilibrium theorem, and generalization of the Maynard-Smith theorem has become achievable within L * -spaces (see [7][8][9][10]). Since Park's partial KKM spaces are closely related to L * -spaces, many results obtained by S. Park in his development of the KKM theory carry out to L * -spaces (see [18][19][20][22][23][24][25][26][27][28][29]). …”