On a theorem of Cellina for set valued functions
\textit{A. Cellina} proved [Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat., VIII. Ser. 48, 412-416 (1970; Zbl 0198.280)] the following fundamental result about approximating a multifunction by continuous single valued functions. Cellina's Theorem. Let X and Y be normed linear spaces, and let C be a convex subset of X. Suppose \(\rho\) denotes the box metric in \(X\times Y\). Let \(\Gamma\) be a convex valued multifunction from C to Y. If \(\Gamma\) is Hausdorff upper semicontinuous and has totally bounded values, then for each \(\epsilon >0\) there exists a continuous function \(f: C\to Y\) for which \(h_{\rho}(f,\Gamma)<\epsilon\). The author proves the following extension of Cellina's result. Theorem. Let (X,d) be a metric space and let Y be a normed linear space. Suppose F is a continuous multifunction from X into Y with totally bounded starshaped values. If \(\Gamma\) maps isolated points to singletons, then for each \(\epsilon >0\), there exists a continuous function \(f: X\subset Y\) for which \(h_{\rho}(f,\Gamma)\leq \epsilon\). The author also proves the following sharp converse to Cellina's result. Theorem. Let X be a locally compact metric space and let Y be a complete metric space. Suppose \(\Gamma\) is a multifunction from X to Y with closed graph, and for each \(\epsilon >0\), there exists a continuous function \(f: X\subset Y\) for which \(h_{\rho}(f,\Gamma)<\epsilon\). Then \(\Gamma\) is Hausdorff upper semicontinuous and has totally bounded values (equivalently, \(\Gamma\) is upper semicontinuous and has compact values).
scientific article; zbMATH DE number 4058398