KKM Theory Research Articles

Existence Results for Generalized Vector Equilibrium Problems with Multivalued Mappings via KKM Theory

We first define upper sign continuity for a set-valued mapping and then we consider two types of generalized vector equilibrium problems in topological vector spaces and provide sufficient conditions under which the solution sets are nonempty and compact. Finally, we give an application of our main results. The paper generalizes and improves results obtained by Fang and Huang in (2005).

Equilibrium existence theorems in KKM spaces

An abstract convex space satisfying the KKM principle is called a KKM space. This class of spaces contains G -convex spaces properly. In this work, we show that a large number of results in KKM theory on G -convex spaces also hold on KKM spaces. Examples of such results are theorems of Sperner and Alexandroff–Pasynkoff, Fan–Browder type fixed point theorems, Horvath type fixed point theorems, Ky Fan type minimax inequalities, variational inequalities, von Neumann type minimax theorems, Nash type equilibrium theorems, and Himmelberg type fixed point theorems.

Elements of the KKM theory for generalized convex spaces

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 andG-convex space versions of the von Neumann type minimax theorem and the Nash equilibrium theorem as typical examples of applications of our theory.

Foundations of the KKM Theory on Generalized Convex Spaces

Recently, we introduced a new concept of a generalized convex space. In this paper, from a coincidence theorem, we deduce far-reaching generalizations of the KKM theorem, the matching theorem, a whole intersection property, an analytic alternative, the Ky Fan minimax inequality, geometric or section properties, and others on generalized convex spaces.

Almost fixed point and best approximations theorems in H-spaces

Using the methods of KKM theory, almost fixed point and best approximations theorems in H-spaces are proved.