The KKM principle in abstract convex spaces: Equivalent formulations and applications

Generalized equilibrium problems in abstract convex spaces

“…Let A WD conv.A/ for each A 2 E, where conv.A/ is intersection of all closed convex subsets of the R tree E, that contains A. .E; / is an abstract convex space which satisfies the partial KKM principle, see [15]. is nonempty and closed.…”

“…In specially, .RI co/ is an abstract convex space. Several examples of abstract convex spaces are given in [15] and the references therein.…”

“…Theorem 1 ( [15], Generalized Partial KKM Principle). Let .E; DI / be a compact abstract convex space satisfying the partial KKM principle and G W D ( E a mapping such that:…”

Applications of some basic theorems in the KKM theory

“…Recently, the KKM theory is extended to abstract convex spaces by the author and we obtained new results in such frame; see [3] and the references therein. In fact, there are very large numbers of equivalent formulations, generalizations, and applications of the KKM theorem.…”

“…In Section 2, we introduce basic concepts on our abstract convex spaces and a basic KKM theorem by following [3]. We give also the new concepts of intersectionally closed and unionly open sets due to Luc et al [4].…”