An application of a random fixed point theorem to random best approximation
Let \(X\) be a Banach space and let \((\Omega, \mathcal{A})\) be a measurable space. Functions \(f,g\) from \(\Omega \times X \mapsto X\) are here called random operators. The authors show that if \(f\) anf \(g\) satisfy a long list of complicated conditions (impossible to summarize or explain here) then there is a mapping \(\xi\) from \(\Omega\) to a set \(P \subseteq X\) (\(P\) is the set of best approximations in a set \(S\) to a given point) with the property that \(\xi(\omega) = f(\omega, \xi(\omega)) = g(\omega, \xi(\omega))\) for all \(\omega \in \Omega\). Such a function \(\xi\) is here called a common random fixed point. The reviewer has the impression that, in any given instance, it may be easier to prove the result directly than verify the conditions of the theorem.
scientific article; zbMATH DE number 1462960