A new approach to the stability of linear functional operators

Summary.

In the present work 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.