### Maximal elements and equilibria of generalized games for $\mathcal{U}$-majorized and condensing correspondences.

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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...

The aim of this paper is to establish a random coincidence degree theory. This degree theory possesses all the usual properties of the deterministic degree theory such as existence of solutions, excision and Borsuk’s odd mapping theorem. Our degree theory provides a method for proving the existence of random solutions of the equation $Lx\in N(\omega ,x)$ where $L:\mathit{\text{dom}}\phantom{\rule{1.3pt}{0ex}}L\subset X\to Z$ is a linear Fredholm mapping of index zero and $N:\Omega \times \overline{G}\to {2}^{Z}$ is a noncompact Carathéodory mapping. Applications to random differential inclusions are also considered.

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