Paneah, Boris A new approach to the stability of linear functional operators. (English) Zbl 1207.39046 Aequationes Math. 78, No. 1-2, 45-61 (2009). Summary: We discuss a new approach to the stability problem for an arbitrary linear functional operator \({\mathcal{P}} : C(I, B) \rightarrow C(D, B)\) of the form \({\mathcal{P}}F : = {\sum{{c}_j}}(x)F(a_j(x)), x \in D\), with \(D\) a compact or noncompact subset in \({\mathbb{R}}^n, I \subset {\mathbb{R}}\) an interval, and \(B\) a Banach space. We define strong stability of the operator \({\mathcal{P}}\) as an arbitrary nearness of a function \(F\) to the kernel of the operator \({\mathcal{P}}\) under condition of the smallness of \({\mathcal{P}}F(x)\) at points of some one-dimensional submanifold \(\Gamma \subset D\). Such a stability turns out to be equivalent to some nonstandard a priori estimate for the \({\mathcal{P}}\). This estimate is obtained in the work by functional analytic methods for an extensive class of operators \({\mathcal{P}}\) which has never been studied earlier. Cited in 16 Documents MSC: 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges Keywords:linear functional equation; Ulam stability; admissible curve; overdetermined equation; Banach space; strong stability; a priori estimate PDFBibTeX XMLCite \textit{B. Paneah}, Aequationes Math. 78, No. 1--2, 45--61 (2009; Zbl 1207.39046) Full Text: DOI