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On Ulam Stability of a Generalization of the Fréchet Functional Equation on a Restricted Domain

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Ulam Type Stability

Abstract

In this paper we prove the Ulam type stability of a generalization of the Fréchet functional equation on a restricted domain. In the proofs the main tool is a fixed point theorem for some function spaces.

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References

  1. Alsina, A., Sikorska, J., Tomás, M.S.: Norm Derivatives and Characterizations of Inner Product Spaces. World Scientific Publishing, Singapore (2010)

    MATH  Google Scholar 

  2. Badora, R., Brzdek, J.: Fixed points of a mapping and Hyers-Ulam stability. J. Math. Anal. Appl. 413, 450–457 (2014)

    Article  MathSciNet  Google Scholar 

  3. Bahyrycz, A., Olko, J.: Hyperstability of general linear functional equation. Aequationes Math. 90, 527–540 (2016)

    Article  MathSciNet  Google Scholar 

  4. Bahyrycz, A., Brzdek, J., Piszczek, M., Sikorska, J.: Hyperstability of the Fréchet equation and a characterization of inner product spaces. J. Funct. Spaces Appl. 2013, 6 pp. (2013). Article ID 496361

    Google Scholar 

  5. Bahyrycz, A., Brzdek, J., Leśniak, Z.: On approximate solutions of the generalized Volterra integral equation. Nonlinear Anal. RWA 20, 59–66 (2014)

    Article  MathSciNet  Google Scholar 

  6. Bahyrycz, A., Brzdek, J., Jabłońska, E., Malejki, R.: Ulam’s stability of a generalization of the Frechet functional equation. J. Math. Anal. Appl. 442, 537–553 (2016)

    Article  MathSciNet  Google Scholar 

  7. Bahyrycz, A., Ciepliński, K., Olko, J.: On Hyers-Ulam stability of two functional equations in non-Archimedean spaces. J. Fixed Point Theory Appl. 18, 433–444 (2016)

    Article  MathSciNet  Google Scholar 

  8. Brillouët-Belluot, N., Brzdek, J., Ciepliński, K.: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012, 41 pp. (2012). Article ID 716936

    Google Scholar 

  9. Brzdek, J.: Remarks on hyperstability of the Cauchy functional equation. Aequationes Math. 86, 255–267 (2013)

    Article  MathSciNet  Google Scholar 

  10. Brzdek, J.: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hungar. 141, 58–67 (2013)

    Article  MathSciNet  Google Scholar 

  11. Brzdek, J., Ciepliński, K.: A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal. 74(18), 6861–6867 (2011)

    Article  MathSciNet  Google Scholar 

  12. Brzdek, J., Chudziak, J., Páles, Z.: A fixed point approach to stability of functional equations. Nonlinear Anal. 74, 6728–6732 (2011)

    Article  MathSciNet  Google Scholar 

  13. Brzdek, J., Jabłońska, E., Moslehian, M.S., Pacho, P.: On stability of a functional equation of quadratic type. Acta Math. Hungar. 149, 160–169 (2016)

    Article  MathSciNet  Google Scholar 

  14. Brzdek, J., Leśniak, Z., Malejki, R.: On the generalized Fréchet functional equation with constant coefficients and its stability. Aequationes Math. 92, 355–373 (2018)

    Article  MathSciNet  Google Scholar 

  15. Brzdȩk, J., Popa, D., Raşa, I., Xu, B.: Ulam Stability of Operators. Elsevier, Oxford (2018)

    Google Scholar 

  16. Cădariu, L., Găvruţa, L., Găvruţa, P.: Fixed points and generalized Hyers-Ulam stability. Abstr. Appl. Anal. 2012, 10 pp. (2012). Article ID 712743

    Google Scholar 

  17. Dragomir, S.S.: Some characterizations of inner product spaces and applications. Studia Univ. Babes-Bolyai Math. 34, 50–55 (1989)

    MathSciNet  MATH  Google Scholar 

  18. Fechner, W.: On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings. J. Math. Anal. Appl. 322, 774–786 (2006)

    Article  MathSciNet  Google Scholar 

  19. Fréchet, M.: Sur la définition axiomatique d’une classe d’espaces vectoriels distanciés applicables vectoriellement sur l’espace de Hilbert. Ann. Math. (2) 36, 705–718 (1935)

    Google Scholar 

  20. Gselmann, E.: Hyperstability of a functional equation. Acta Math. Hungar. 124, 179–188 (2009)

    Article  MathSciNet  Google Scholar 

  21. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Boston (1998)

    Book  Google Scholar 

  22. Jordan, P., von Neumann, J.: On inner products in linear, metric spaces. Ann. Math. (2) 36, 719–723 (1935)

    Google Scholar 

  23. Jung, S.-M.: On the Hyers-Ulam stability of the functional equation that have the quadratic property. J. Math. Anal. Appl. 222, 126–137 (1998)

    Article  MathSciNet  Google Scholar 

  24. Jung, S.-M., Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer Optimization and Its Applications, vol. 48. Springer, New York (2011)

    Google Scholar 

  25. Kannappan, P.: Functional Equations and Inequalities with Applications. Springer Monographs in Mathematics. Springer, New York (2009)

    Google Scholar 

  26. Lee, Y.-H.: On the Hyers-Ulam-Rassias stability of the generalized polynomial function of degree 2. J. Chuncheong Math. Soc. 22, 201–209 (2009)

    Google Scholar 

  27. Maksa, G., Páles, Z.: Hyperstability of a class of linear functional equations. Acta Math. Acad. Paedag. Nyìregyháziensis 17, 107–112 (2001)

    MathSciNet  MATH  Google Scholar 

  28. Malejki, R.: Stability of a generalization of the Fréchet functional equation. Ann. Univ. Paedagog. Crac. Stud. Math. 14, 69–79 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Moslehian, M.S., Rassias, J.M.: A characterization of inner product spaces concerning an Euler-Lagrange identity. Commun. Math. Anal. 8, 16–21 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Nikodem, K., Pales, Z.: Characterizations of inner product spaces by strongly convex functions. Banach J. Math. Anal. 5, 83–87 (2011)

    Article  MathSciNet  Google Scholar 

  31. Piszczek, M.: Remark on hyperstability of the general linear equation. Aequationes Math. 88, 163–168 (2014)

    Article  MathSciNet  Google Scholar 

  32. Popa, D., Raşa, I.: The Fréchet functional equation with application to the stability of certain operators. J. Approx. Theory 164, 138–144 (2012)

    Article  MathSciNet  Google Scholar 

  33. Rassias, Th.M.: New characterizations of inner product spaces. Bull Sci. Math. (2) 108, 95–99 (1984)

    Google Scholar 

  34. 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 

  35. Zhang, D.: On hyperstability of generalised linear functional equations in several variables. Bull. Aust. Math. Soc. 92, 259–267 (2015)

    Article  MathSciNet  Google Scholar 

  36. Zhang, D.: On Hyers-Ulam stability of generalized linear functional equation and its induced Hyers-Ulam programming problem. Aequationes Math. 90, 559–568 (2016)

    Article  MathSciNet  Google Scholar 

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Correspondence to Renata Malejki .

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Malejki, R. (2019). On Ulam Stability of a Generalization of the Fréchet Functional Equation on a Restricted Domain. In: Brzdęk, J., Popa, D., Rassias, T. (eds) Ulam Type Stability . Springer, Cham. https://doi.org/10.1007/978-3-030-28972-0_11

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