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On a New Type of Hyperstability for Radical Cubic Functional Equation in Non-Archimedean Metric Spaces

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An Erratum to this article was published on 15 September 2017

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Abstract

Let \(\mathbb {R}\) be the set of real numbers, \((G,+)\) be a commutative group and d be a complete ultrametric on G that is invariant (i.e., \(d(x + z, y + z)= d(x, y\)) for \(x, y, z \in G\)). Under some weak natural assumptions on the function \(\gamma :{\mathbb {R}}^2\rightarrow [0,\infty )\), we study the generalised hyperstability results when \(f:\mathbb {R}\rightarrow G\) satisfy the following radical cubic inequality

$$\begin{aligned} d\big (f(\root 3 \of {x^3+y^3}),f(x)+f(y)\big ) \le \gamma (x,y), \quad x,y\in \mathbb {R}{\setminus }\{0\}, \end{aligned}$$

with \(x\ne -y\). The method is based on a quite recent fixed point theorem (cf. Brzdęk and Cieplińnski in Nonlinear Anal 74:6861–6867, 2011, Theorem 1) in some functions spaces.

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  • 15 September 2017

    An erratum to this article has been published.

References

  1. Alizadeh, Z., Ghazanfari, A.G.: On the stability of a radical cubic functional equation in quasi-\(\beta \)-spaces. J. Fixed Point Theory Appl. 18, 843–853 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bahyrycz, A., Piszczek, M.: Hyperstability of the Jensen functional equation. Acta Math. Hung. 142(2), 353–365 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bahyrycz, A., Olko, J.: Hyperstability of general linear functional equation. Aequationes Math. 89, 1461–1476 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourgin, D.G.: Approximately isometric and multiplicative transformations on continuous function rings. Duke Math. J. 16, 385–397 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourgin, D.G.: Classes of transformations and bordering transformations. Bull. Am. Math. Soc. 57, 223–237 (1951)

    Article  MathSciNet  MATH  Google Scholar 

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

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

  9. Brzdęk, J., Ciepliński, K.: Hyperstability and superstability, Abstr. Appl. Anal. 2013, p 13 (2013), Article ID 401756

  10. Brzdęk, J.: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hung. 141(1–2), 58–67 (2013)

  11. Brzdęk, J.: Stability of additivity and fixed point methods. Fixed Point Theory Appl. 2013, 9 (2013)

  12. Brzdęk, J.: A hyperstability result for the Cauchy equation. Bull. Aust. Math. Soc. 89, 33–40 (2014)

  13. Brzdęk, J.: Remarks on stability of some inhomogeneous functional equations. Aequationes Math. 89, 83–96 (2015)

  14. Ciepliński, K.: Stability of multi-additive mappings in non-Archimedean normed spaces. J. Math. Anal. Appl. 373, 376–383 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. EL-Fassi, Iz., Kabbaj, S.: On the hyperstability of a Cauchy-Jensen type functional equation in Banach spaces. Proyecciones J. Math 34, 359–375 (2015)

  16. EL-Fassi, Iz., Kabbaj, S., Charifi, A.: Hyperstability of Cauchy–Jensen functional equations. Indag. Math. 27, 855–867 (2016)

  17. EL-Fassi, Iz., Kabbaj, S.: Non-Archimedean random stability of \(\sigma \)-quadratic functional equation. Thai J. Math. 14, 151–165 (2016)

  18. Eshaghi Gordji, M., Savadkouhi, M.B.: Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Appl. Math. Lett. 23, 1198–1202 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gajda, Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431–434 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Hensel, K.: Uber eine neue begründung der theorie der algebraischen zahlen. Jahresbericht der Deutschen Mathematiker-Vereinigung 6, 83–88 (1899)

    MATH  Google Scholar 

  22. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. U.S.A. 27, 222–224 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  23. Khodaei, H., Eshaghi Gordji, M., Kim, S.S., Cho, Y.J.: Approximation of radical functional equations related to quadratic and quartic mappings. J. Math. Anal. Appl. 395, 284–297 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Khrennikov, A.: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publishers, Dordrecht (1997)

    Book  MATH  Google Scholar 

  25. Lee, Y.-H.: On the stability of the monomial functional equation. Bull. Korean Math. Soc. 45, 397–403 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  27. Mirmostafaee, A.K.: Hyers-Ulam stability of cubic mappings in non-Archimedean normed spaces. Kyngpook Math. J. 50, 315–327 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Moslehian, M.S., Rassias, T.M.: Stability of functional equations in non-Archimedean spaces. Appl. Anal. Discret. Math. 1, 325–334 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Moslehian, M.S., Sadeghi, G.: Stability of two types of cubic functional equations in non-Archimedean spaces. Real Anal. Exchange 33(2), 375–384 (2007–2008)

  30. Moszner, Z.: Stability has many names. Aequationes Math. 90, 983–999 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rassias, T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rassias, T.M.: On a modified Hyers-Ulam sequence. J. Math. Anal. Appl. 158, 106–113 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rassias, T.M., Semrl, P.: On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proc. Am. Math. Soc. 114, 989–993 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ulam, S.M.: Problems in Modern Mathematics, Chapter IV, Science Editions. Wiley, New York (1960)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Ibn Tofail University, BP 133 , Kenitra, Morocco

    Iz-iddine EL-Fassi

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  1. Iz-iddine EL-Fassi
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Correspondence to Iz-iddine EL-Fassi.

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An erratum to this article is available at https://doi.org/10.1007/s00025-017-0743-z.

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EL-Fassi, Ii. On a New Type of Hyperstability for Radical Cubic Functional Equation in Non-Archimedean Metric Spaces . Results Math 72, 991–1005 (2017). https://doi.org/10.1007/s00025-017-0716-2

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