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In the present work we discuss a new approach to the stability problem for an arbitrary linear functional operator P : C(I, B) → C(D, B) of the form PF : = Σc j (x)F a j (x) , x ∈ D, with D a compact or noncompact subset in R n , I ⊂ R an interval, and B a Banach space. We define strong stability of the operator P as an arbitrary nearness of a function F to the kernel of the operator P under condition of the smallness of PF (x) at points of some one-dimensional submanifold Γ ⊂ D. Such a stability turns out to be equivalent to some nonstandard a priori estimate for the P. This estimate is obtained in the work by functional analytic methods for an extensive class of operators P which has never been studied earlier.

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