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A comparison among methods for proving stability

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Abstract

The aim of this work is to present and compare the most used methods for proving Hyers–Ulam stability of various classes of functional equations.

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References

  1. Aczél, J.: Lectures on Functional Equations and their Applications. Academic Press, New York (1966)

    MATH  Google Scholar 

  2. Aczél, J., Dhombres, J.: Functional Equations in Several Variables. Encyclopedia Math. Appl. 31, Cambridge Universtiy Press, Cambridge (1989)

  3. Badora, R., Brzdȩk, J.: A note on a fixed point theorem and the Hyers–Ulam stability. J. Differ. Equ. Appl. 18, 1115–1119 (2012)

    Article  MathSciNet  Google Scholar 

  4. Baker, J.A.: The stability of certain functional equations. Proc. Amer. Math. Soc. 112, 729–732 (1991)

    Article  MathSciNet  Google Scholar 

  5. Borelli, C.: On Hyers–Ulam stability for a class of functional equations. Aequ. Math. 54, 74–86 (1997)

    Article  MathSciNet  Google Scholar 

  6. Burger, M., Ozawa, N., Thom, A.: On Ulam stability. Israel J. Math. 193(1), 109–129 (2013)

    Article  MathSciNet  Google Scholar 

  7. Cadăriu, L., Radu, V.: On the stability of the Cauchy functional equation: a fixed point approach. Iteration theory (ECIT ’02), pp. 43–52, Grazer Math. Ber. Vol. 346, Karl-Franzens-Univ. Graz, Graz (2004)

  8. De Chiffre, M., Ozawa, N., Thom, A.: Operator algebraic approach to inverse and stability theorems for amenable groups. Mathematika 65(1), 98–118 (2019)

    Article  MathSciNet  Google Scholar 

  9. Forti, G.L.: Hyers-Ulam stability of functional equations in several variables. Aequat. Math. 50, 143–190 (1995)

    Article  MathSciNet  Google Scholar 

  10. Forti, G.L.: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations. J. Math. Anal. Appl. 295, 127–133 (2004)

    Article  MathSciNet  Google Scholar 

  11. Forti, G.L.: Elementary remarks on Ulam–Hyers stability of linear functional equations. J. Math. Anal. Appl. 328, 109–118 (2007)

    Article  MathSciNet  Google Scholar 

  12. Forti, G.L., Sikorska, J.: Variations on the Drygas equation and its stability. Nonlinear Anal. 74, 343–350 (2011)

    Article  MathSciNet  Google Scholar 

  13. Ger, R.: A survey of recent results on stability of functional equations. In: Proceedings of the 4th Internatioal Conference on Functional Equations and Inequalities, Pedagogical University in Cracow, pp. 5–36 (1994)

  14. Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941)

    Article  MathSciNet  Google Scholar 

  15. Hyers, D.H., Isac, G., Rassias, T.M.: Stability of functional equations in several variables. Progress in Nonlinear Differential Equations and their Applications, 34. Birkhauser Boston, Inc., Boston, MA, (1998)

  16. Levi-Civita, T.: Sulle funzioni che ammettono una formula d’addizione del tipo \(f(x+y)=\sum _{i=1}^n X_i(x)Y_i(y)\). Atti Accad. Naz. Lincei Rend. 22, 181–183 (1913)

    MATH  Google Scholar 

  17. Rademacher, H., Toeplitz, O.: The spanning circle of a finite set of points. In: The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton University Press, Princeton, NJ, pp. 103–110 (1957)

  18. Rassias, Th.M.: (Ed.), Handbook of functional equations. Stability theory. Springer Optimization and Its Applications, 96. Springer, New York, (2014)

    MATH  Google Scholar 

  19. Ryll-Nardzewski, C.: On fixed points of semigroups of endomorphisms of linear spaces. In: Proceeding of the Fifth Berkeley Symposium Mathematicla Statistics and Probability, pp. 55–61 (1967)

  20. Shtern, A.I.: Quasirepresentations of amenable groups: results, errors, and hopes. Russ. J. Math. Phys. 20(2), 239–253 (2013)

    Article  MathSciNet  Google Scholar 

  21. Shulman, E.: Functional equations of homological type. Ph.D. Thesis, Moscow Institute of Electronics and Mathematics, Moscow (Russian) (1994)

  22. Shulman, E.: Group representations and stability of functional equations. J. Lond. Math. Soc. 54(2), 111–120 (1996)

    Article  MathSciNet  Google Scholar 

  23. Shulman, E.: Decomposable functions and representations of topological semigroups. Aequ. Math. 79(1–2), 13–21 (2010)

    Article  MathSciNet  Google Scholar 

  24. Shulman, E.: Subadditive set-functions on semigroups, applications to group representations and functional equations. J. Funct. Anal. 263, 1468–1484 (2012)

    Article  MathSciNet  Google Scholar 

  25. Shulman, E.: On nearly subadditive maps. Funct. Anal. Appl. 50(1), 80–82 (2016)

    Article  MathSciNet  Google Scholar 

  26. Shulman, E.: On widths of invariant sets. Modern methods in operator theory and harmonic analysis. Proc. Math. Stat. 291, 249–261 (2019)

    Google Scholar 

  27. Sikorska, J.: On a direct method for proving the Hyers–Ulam stability of functional equations. J. Math. Anal. Appl. 372, 99–109 (2010)

    Article  MathSciNet  Google Scholar 

  28. Stäckel, P.: Sulla equazione funzionale \(f(x+y)=\sum _{i=1}^n X_i(x)Y_i(y)\). Atti Accad. Naz. Lincei Rend. 22(5), 392–393 (1913)

    MATH  Google Scholar 

  29. Stéphanos, C.: Sur une catégorie d’équations fonctionnelles. Rend. Circ. Mat. Palermo 18(1), 360–362 (1904)

    Article  Google Scholar 

  30. Székelyhidi, L.: Note on a stability theorem. Canad. Math. Bull. 25(4), 500–501 (1982)

    Article  MathSciNet  Google Scholar 

  31. Tabor, Ja: Locally expanding mappings and hyperbolicity. Topol. Methods Nonlinear Anal. 30, 335–343 (2007)

    MathSciNet  MATH  Google Scholar 

  32. Tabor, Ja, Tabor, Jó: General stability of functional equations of linear type. J. Math. Anal. Appl. 328, 192–200 (2007)

    Article  MathSciNet  Google Scholar 

  33. Trif, T.: On the stability of a general gamma-type functional equation. Publ. Math. Debrecen 60, 47–61 (2002)

    MathSciNet  MATH  Google Scholar 

  34. Yang, D.: The stability of the quadratic equation on amenable groups. J. Math. Anal. Appl. 291, 666–672 (2004)

    Article  MathSciNet  Google Scholar 

  35. Yang, D.: The quadratic functional equation on groups. Publ. Math. Debrecen 66, 327–348 (2005)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133, Milano, Italy

    Gian Luigi Forti

  2. University of Silesia, 14 Bankowa Str., 40-007, Katowice, Poland

    Ekaterina Shulman

  3. Vologda State University, 6 S. Orlova Str., Vologda, Russia, 160035

    Ekaterina Shulman

Authors
  1. Gian Luigi Forti
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  2. Ekaterina Shulman
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Correspondence to Gian Luigi Forti.

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Dedicated to Professor János Aczél on the occasion of his 95th birthday, with esteem and gratitude.

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Forti, G.L., Shulman, E. A comparison among methods for proving stability. Aequat. Math. 94, 547–574 (2020). https://doi.org/10.1007/s00010-019-00679-0

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  • DOI: https://doi.org/10.1007/s00010-019-00679-0

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