The essential components of coincident points for weakly inward and outward set-valued mappings
The paper concerns a semicontinuous dependence on \(y=(f,g)\) of the coincident set \(F(y)=\{x\in X;f(x)\cap g(x)\neq\emptyset\}\) where \(f:X\to X\) and \(g:X\to X\) are set-valued mappings. Precisely, \(X\) is a nonempty compact convex set of a normed space, \(f\) and \(g\) are upper semicontinuous, with nonempty closed convex values, \(f-g\) is weakly inward, and on this domain \(Y\) of \(F\) there is considered an appropriate Hausdorff metric \(\rho\). The main result asserts, for each \(y\in Y\), there exists a connected component of \(F(y)\) which is an essential one, that is, for each open set \(O\supseteq F(y)\), there exists \(\delta>0\) such that \(F(y')\cap O\neq\emptyset\) for all \(y'\in Y\) with \(\rho(y',y)<\delta\).
scientific article; zbMATH DE number 1369360