Fixed Point and Related Theorems for Set-Valued Mappings

In this chapter, we focus in the discussion of fixed point theory for set, valued mappings by using Knaster-Kuratowski-Mazurkiewicz (KKM) principle in both topological vector spaces and hyperconvex metric spaces. In particular, the fixed point theory of set-valued mappings of Browder-Fan and Fan-Glicksberg type has been extensively studied in the setting of locally convex spaces, H-spaces, G-convex spaces and metric hyperconvex spaces. By using its own feature of hyperconvex metric spaces being a special class of H-spaces, we also establish its general KKM theory and then its various applications. In section 2, we first discuss some recent developments of KKM theory itself and the general Ky Fan minimax principle is given in section 3. In sections 4 and 5, two types of Ky Fan minimax inequalities and their equivalent fixed point forms for set-valued mappings are given. In section 6, the general Fan-Glicksberg type fixed point theorem is discussed in G-Convex spaces. These spaces include locally convex H-spaces, locally convex topological vector spaces and metric hyperconvex metric spaces as special cases. Finally, the general KKM theory and its various applications in metric hyperconvex spaces and the generic stability of fixed points are discussed in section 7.