Applications of Convex-valued KKM maps (Study on Nonlinear Analysis and Convex Analysis)

Abstract

This is to introduce the contents of two articles of Granas and Lassonde [2, 3] on applications of some elementary principles of convex analysis. In the first article, they presented a geometric approach in the theory of minimax inequalities, which has numerous applications in different areas of mathematics. In the second article, they complement and elucidate the preceding approach within the context of complete metric spaces. In this paper, we give abstract convex space versions of the basic results of [2, 3], and, as the supplements of overviews on recently developed KKM theory in [5, 8], we introduce applications appeared in [2, 3]. Consequently, many of known results in the traditional convex analysis can be deduced from the KKM theory.