### $(n+1)$-families of sets in general position.

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In this paper, we establish two constant selection theorems for a map whose dual is upper or lower semicontinuous. As applications, matching theorems, analytic alternatives, and minimax inequalities are obtained.

In this paper the main result in [1], concerning $(n+1)$-families of sets in general position in ${\mathbf{R}}^{n}$, is generalized. Finally we prove the following theorem: If $\{{A}_{1},{A}_{2},\cdots ,{A}_{n+1}\}$ is a family of compact convexly connected sets in general position in ${\mathbf{R}}^{n}$, then for each proper subset $I$ of $\{1,2,\cdots ,n+1\}$ the set of hyperplanes separating $\cup \{{A}_{i}:i\in I\}$ and $\cup \{{A}_{j}:j\in \overline{I}\}$ is homeomorphic to ${S}_{n}^{+}$.

We obtain generalizations of the Fan's matching theorem for an open (or closed) covering related to an admissible map. Each of these is restated as a KKM theorem. Finally, applications concerning coincidence theorems and section results are given.

In this paper fixed point theorems for maps with nonempty convex values and having the local intersection property are given. As applications several minimax inequalities are obtained.

Given a nonempty convex set $X$ in a locally convex Hausdorff topological vector space, a nonempty set $Y$ and two set-valued mappings $T:X\rightrightarrows X$, $S:Y\rightrightarrows X$ we prove that under suitable conditions one can find an $x\in X$ which is simultaneously a fixed point for $T$ and a common point for the family of values of $S$. Applying our intersection theorem we establish a common fixed point theorem, a saddle point theorem, as well as existence results for the solutions of some equilibrium and complementarity problems.

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