Banach Journal of Mathematical Analysis

Some remarks on stability and solvability of linear functional equations

Boris Paneah

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In the present work we continue studying the solvability of the linear functional equations ${\sum_{j=1}^N }c_j F\circ a_j=H$ and also the strong and weak stability of the corresponding operator $\mathcal{P}$ (see the definition below). By analogy with the Cauchy and Jensen operators once more model operator $\widehat{\mathcal{P}}$ is considered, and the stability problems as well as some solvability problems for $\widehat{\mathcal{P}}$ are studied. Several unsolved problem of a general character are formulated.

Article information

Banach J. Math. Anal., Volume 1, Number 1 (2007), 56-65.

First available in Project Euclid: 21 April 2009

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Zentralblatt MATH identifier

Primary: 39B22: Equations for real functions [See also 26A51, 26B25]
Secondary: 39B52: Equations for functions with more general domains and/or ranges

strong weak and Ulam stability Cauchy operator Jensen operator a priori estimate


Paneah, Boris. Some remarks on stability and solvability of linear functional equations. Banach J. Math. Anal. 1 (2007), no. 1, 56--65. doi:10.15352/bjma/1240321555.

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  • B. Paneah, On the solvability of functional equations associated with dynamical systems with two generators, Funct. Anal. Its Appl., 37(1) (2003), 46–60.
  • B. Paneah, Dynamical approach to some problems in integral geometry, Trans. Amer. Math. Soc., 356 (2003), pp. 2757–2780.
  • B. Paneah, Dynamical systems and functional equations related to boundary problems for hyperbolic differential operators, Doklady Mathematics, 72, (2005), 949–953.
  • B. Paneah, On the general theory of the Cauchy type functional equations with applications in analysis, Aequationes Math., 74 (2007), 119–157.
  • B. Paneah, Another approach to the stability of linear functional operators, Preprint 2006/13, ISSN 14437 - 739X, Institut fur Matematik, Uni Potsdam, (2006).
  • B. Paneah, On the stability of the linear functional operators structurally associated with the Jensen operator, Iteration Theory (ECTT'06), (to appear).