A remark on intersection of convex sets.
In this short note, the author shows that if \(\{G_\gamma:\gamma\in\Gamma\}\) is a collection of convex subsets of a real normed linear space with nonempty intersection and if \((G)^\varepsilon\) denotes the \(\varepsilon\)-neighborhood of a set \(G\), then either the intersection \(\bigcap_{\gamma\in\Gamma} (G_\gamma)^\varepsilon\) is bounded for each \(\varepsilon>0\) or the intersection is unbounded for each \(\varepsilon>0\). The author uses this to prove a result about fixed points of continuous pseudo-contractive mappings from closed convex subsets of a Hilbert space into the Hilbert space. In particular, if such a map \(T:K\to H\) satisfies a certain weak inward condition and if there exists a bounded subset \(A\) of \(K\) such that \(K \cap (\bigcap_{a\in A} G(a,T(a))) \neq \emptyset\) (where \(G(a,T(a))\) denotes the closed half-space of points in \(H\) closer to \(T(a)\) than to \(a\)) and \(\bigcap_{a\in A} (G(a,T(a)))^\varepsilon\) is bounded for some \(\varepsilon > 0\), then \(T\) has a fixed point in \(K\). Reviewer's remark: There is a misstatement in the proof of the theorem: the last line on page 777 states that an unbounded set is contained in a bounded set. If the reader ignores this line, the next inclusion holds with \(\delta\) equal to the diameter of \(A-T(A)\).
scientific article; zbMATH DE number 1976026