Bulletin of the
Korean Mathematical Society BKMS

In this paper we give two matching theorems of Ky Fan type concerning open or closed coverings of nonempty convex sets in a topological vector space. One of them will permit us to put in evidence, when $X$ and $Y$ are convex sets in topological vector spaces, a new subclass of $\mathbf{KKM}(X,Y)$ different by any admissible class $\frak{A}_c(X,Y)$. For this class of set-valued mappings we establish a KKM-type theorem which will be then used for obtaining existence theorems for the solutions of two types of simultaneous relation problems.

Keywords: matching theorem, KKM theorem, $\mathbf{KKM}(X,Y)$, $\frak{A}_c(X,Y)$, variational relation

MSC numbers: 47H10, 49J53