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Aequationes mathematicae

, 78:45 | Cite as

A new approach to the stability of linear functional operators

  • Boris Paneah
Article

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.

Mathematics Subject Classification (2000).

39B22 39B52 

Keywords.

Strong weak and Ulam stability a priori estimate admissible curve overdetermined equation 

Copyright information

© Birkhäuser Verlag, Basel 2009

Authors and Affiliations

  • Boris Paneah
    • 1
  1. 1.Department of MathematicsTechnionHaifaIsrael

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