Note on Kkm Maps and Applications

In 1929, the KKMmap was introduced by Knaster et al. [13] and it provides the foundation for many well-known existence results, such as Ky Fan’s minimax inequality theorem, Ky Fan-Browder’s fixed point theorem, Nash’s equilibrium theorem, HartmanStampacchia’s variational inequality theorem and many others (see [1, 2, 5–12, 14–17]). The central idea of applying KKM theory to prove that a family of sets has nonempty intersection is to find a suitable space and a mapping defined on that space such that this mapping is a KKM mapping and the original family of sets has finite intersection property provided the resulted family of sets by this mapping has finite intersection property. Based this idea, we first introduce a large class of mappings that can be interpreted as KKMmappings, then we apply the KKM technique to study fixed point theory, minimax inequality and coincidence theorem. A new concept on lower (upper) semi-continuous function is given and some new results on Fan-Browder’s fixed point theorem, Fan’s minimax theorem and coincidence theorem are obtained.