Sehgal, V. M.; Singh, S. P. A generalization to multifunctions of Fan’s best approximation theorem. (English) Zbl 0672.47043 Proc. Am. Math. Soc. 102, No. 3, 534-537 (1988). Let E be a locally convex Hausdorff topological vector space, S a nonempty subset of E and p a continuous seminorm on E. The following Reich’s theorem is well-known: If S is approximatively compact and f: \(S\to E\) is continuous with f(S) relatively compact then there exists \(x\in S\) satisfying \(p(f(x)-x)=\min \{p(f(x)-y)|\) \(y\in S\}.\) The authors generalize this result for multifunctions. Reviewer: Ioan A.Rus Cited in 1 ReviewCited in 68 Documents MSC: 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47H05 Monotone operators and generalizations Keywords:condensing mappings; locally convex Hausdorff topological vector space; Reich’s theorem; approximatively compact; multifunctions PDFBibTeX XMLCite \textit{V. M. Sehgal} and \textit{S. P. Singh}, Proc. Am. Math. Soc. 102, No. 3, 534--537 (1988; Zbl 0672.47043) Full Text: DOI