In this paper, we first establish the dual form of Knaster- Kuratowski-Mazurkiewicz principle which is a hyperconvex version of corresponding result due to Shih. Then Ky Fan type matching theorems for finitely closed and open covers are given. As applications, we establish some intersection theorems which are hyperconvex versions of corresponding results due to Alexandroff and Pasynkoff, Fan, Klee, Horvath and Lassonde. Then Ky Fan type best approximation theorem and Schauder-Tychonoff fixed point theorem for set-valued mappings (i.e., Fan-Glicksberg fixed point theorem) in hyperconvex spaces are also developed, and finally one unified form of Browder-Fan fixed point theorem for set-valued mappings in hyperconvex spaces is given. These results include corresponding results in the literature due to Khamsi, Kirk and Shin, Kirk et al. as special cases.