A class of ``onto'' multifunctions
The authors generalize the following classical theorem: ``If \(C\) is a compact convex set in \(\mathbb{R}^ n\) and \(f: C\to \mathbb{R}^ n\) is a continuous function that coincides with the identity on the boundary of \(C\), then \(C\subseteq f(C):= \{f(x): x\in C\}\).'' Their generalization allows noncompact sets \(C\), multivalued functions \(f\), and rather general boundary behaviour, but retains the key conclusion, namely, \[ C\subseteq f(C):=\bigcup \{f(x): x\in C\}.\tag{1} \] For any convex subset \(C\) of \(\mathbb{R}^ n\), one may define a multifunction \(\sigma: C\to C\) by taking \(\sigma(x)\) as the smallest face of \(C\) containing \(x\), and call a vector \(c\) in \(\mathbb{R}^ n\) a coercive direction for \(C\) if the set of \(x\) in \(C\) minimizing \(c^ T x\) is nonempty and bounded. Main Theorem: Let \(C\) be a nonempty closed and pointed convex subset of \(\mathbb{R}^ n\). Let \(f: C\to \mathbb{R}^ n\) be a set-valued mapping with nonempty convex values. Assume that \(f\) is upper continuous, in the sense that \(\limsup_{y\to x} f(y)\subseteq f(x)\) for every \(x\) in \(C\). Then a sufficient condition for (1) is that there exist a selection \(s(x)\in f(x)\) for all \(x\in C\) and a coercive direction \(c\) for \(C\) such that (a) \(s(x)\) lies in the affine hull of \(\sigma(x)\) for each \(x\) in \(C\); (b) if \(\| s(x_ k)\|\to\infty\) for some sequence \(\{x_ k\}\) in \(C\), then \(\| x_ k\|\to\infty\); (c) if \(\| x_ k\|\to\infty\) for some sequence \(\{x_ k\}\) in \(C\), then \(c^ T s(x_ k)\to+\infty\). The authors draw a number of interesting corollaries from their main result, including the classical theorem quoted above. They take care to show by example that none of their hypotheses can be discarded, and present a detailed analysis of their theorem's application to the ``congestion attainability problem''. In this problem, a ``Kelly network'' of queues with \(p\) single-server stations and clients divided into \(n\) classes is allowed to converge to steady state. The main theorem implies that once the service rate at each station is prescribed, the number of customers of each class in the system at steady state can be assigned arbitrarily, simply by varying the \(n\)-vector of arrival rates for the various customer classes.
scientific article; zbMATH DE number 513285