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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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