Let \(E\) be a Hausdorff topological vector space, \(X\subset E\) a nonempty convex subset. A multi-valued mapping \(G: X\to 2^ E\) is called KKM, if \(co\{x_ 1,x_ 2,\dots,x_ n\}\subset\bigcup^ n_{i=1}G(X_ i)\) for each finite subset \(\{x_ 1,\dots,x_ n\}\subset X\). A classic Ky Fan theorem says that if \(G\) is a KKM mapping with nonempty closed values and there exists an \(x_ 0\in X\) such that \(G(x_ 0)\) is compact, then \(\bigcap_{x\in X}G(x)\neq\emptyset\). This paper proposes the concept of a generalized KKM mapping. \(G: X\to 2^ E\) is called a generalized KKM mapping, if for any finite set \(\{x_ 1,\dots,x_ n\}\subset X\), there exists a finite subset \(\{y_ 1,\dots,y_ n\}\subset E\) such that for any subset \(\{y_ 1,\dots,y_{i_ k}\}\subset\{y_ 1,\dots,y_ n\}\), \(1\leq k\leq n\), we have \(co\{y_{i_ 1},\dots,y_{i_ k}\}\subset\bigcup^ k_{j=1}G(x_{i_ j})\). Under this definition, the Ky Fan theorem may be improved as that
\(\bigcap_{x\in X}G(x)\neq\emptyset\Leftrightarrow G\) is a generalized KKM mapping.
By using this result, Ky Fan’s minimax inequality, the Browder-Hartman- Stampacchia variational inequality theorem and others may be also generalized.