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A new approach to the stability of linear functional operators. (English) Zbl 1207.39046
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.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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