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...

A characterization of the total variation $TV\left(u,\mathrm{\Omega }\right)$ of the Jacobian determinant $detDu$ is obtained for some classes of functions $u:\mathrm{\Omega }\to {\mathbb{R}}^n$ outside the traditional regularity space $W^{1,n}\left(\mathrm{\Omega };{\mathbb{R}}^n\right)$. In particular, explicit formulas are deduced for functions that are locally Lipschitz continuous away from a given one point singularity $x_0\in \mathrm{\Omega }$. Relations between $TV\left(u,\mathrm{\Omega }\right)$ and the distributional determinant $\text{Det}Du$ are established, and an integral representation is obtained for the relaxed energy of certain polyconvex functionals at maps $u\in W^{1,p}\left(\mathrm{\Omega };{\mathbb{R}}^n\right)\cap W^{1,\mathrm{\infty }}\left(\mathrm{\Omega }\left\{x_0\right\};{\mathbb{R}}^n\right)$.