Approximation and Léray--Schauder type results for multimaps in the S-KKM class (Q2455386)

Having as starting point the well-known result of \textit{Ky Fan} [Math.\ Z.\ 112, 234--240 (1969; Zbl 0185.39503)] that, if \(C\) is a nonempty, compact, convex subset of a normed space \(E\), then for any continuous mapping \(f\) from \(C\) to \(E\) there exists an \(x_0\in C\) such that \[ \left\| x_0-f(x_0)\right\|=\inf_{y\in C}\left\| f(x_0)-y\right\|, \] the author establishes Fan type approximation results for \(s\)-KKM multimaps in Hausdorff locally convex spaces and then, as an application, obtains a Leray--Schauder type result. Approximation results in normed spaces are also obtained as corollaries of the main results.