Solution and approximation of radical quintic functional equation related to quintic mapping in quasi-\(\beta \)-Banach spaces

  • Iz-iddine EL-Fassi
Original Paper


Let \(\mathbb {R}\) denote the set of real numbers. The purpose of the present paper is first to introduce and solve the radical quintic functional equation
$$\begin{aligned} f\left( \root 5 \of {x^5+y^5}\right) = f(x)+f(y),\;\;\;x,y\in \mathbb {R}, \end{aligned}$$
for f a mapping from \(\mathbb {R}\) into a vector space. We also establish stability results in quasi-\(\beta \)-Banach spaces, and then the stability by using subadditive and subquadratic functions in (\(\beta , p\))-Banach spaces for this functional equation. In addition, we also present a counterexample that does not satisfy the stability based on Ulam’s question.


Radical functional equations Subadditive and subquadratic functions Quasi-\(\beta \)-normed spaces Stability Counterexample 

Mathematics Subject Classification

65Q20 39B82 39B62 46H25 


  1. 1.
    Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn 2, 64–66 (1950)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, vol. 1, Colloq. Publ., vol. 48, Amer. Math. Soc., Providence (2000)Google Scholar
  3. 3.
    Bourgin, D.G.: Approximately isometric and multiplicative transformations on continuous function rings. Duke Math. J. 16, 385–397 (1949)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bourgin, D.G.: Classes of transformations and bordering transformations. Bull. Am. Math. Soc. 57, 223–237 (1951)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brzdęk, J.: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hungar. 141(1–2), 58–67 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    EL-Fassi, Iz., Kabbaj, S.: Non-Archimedean Random Stability of \(\sigma \)-Quadratic Functional Equation. Thai J. Math 14, 151–165 (2016)Google Scholar
  7. 7.
    EL-Fassi, Iz., Kabbaj, S.: On the generalized orthogonal stability of the Pexiderized quadratic functional equation in modular space. Math. Slovaca 67(1), 165–178 (2017)Google Scholar
  8. 8.
    Eshaghi Gordji, M., Khodaei, H., Ebadian, A., Kim, G. H.: Nearly radical quadratic functional equations in \(p\)-2-normed spaces, Abstr. Appl. Anal. 2012, Article ID 896032 (2012)Google Scholar
  9. 9.
    Forti, G.L.: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equation. J. Math. Anal. Appl. 295, 127–133 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gǎvruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gajda, Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431–434 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gajda, Z., Ger, R.: Subadditive Multifunctions and Hyers-Ulam Stability. In: Walter, W. (ed.) General Inequalities 5. International Series of Numerical Mathematics, vol. 80, pp. 281–291. Birkhäuser, Basel (1987)Google Scholar
  13. 13.
    Gordji, M.E., Khodaei, H., Rassias, Th.M.: A functional equation having monomials and its stability, Springer Optimization and Its Applications, 96 (2014)Google Scholar
  14. 14.
    Gruber, P.M.: Stability of isometries. Trans. Am. Math. Soc. 245, 263–277 (1978)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jun, K.-W., Kim, H.-M.: On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces. J. Math. Anal. Appl. 332, 1335–1350 (2007)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jung, S.M., Rassias, M.Th, Mortici, C.: On a functional equation of trigonometric type. Appl. Math. Comput. 252, 293–303 (2015)MathSciNetMATHGoogle Scholar
  19. 19.
    Lee, Y.H.: On the stability of the monomial functional equation. Bull. Korean Math. Soc. 45, 397–403 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Rassias, J.M., Kim, H.-M.: Generalized Hyers-Ulam stability for general additive functional equations in quasi-\(\beta \)-normed spaces. J. Math. Anal. Appl. 356, 302–309 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Rassias, ThM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Rassias, ThM: On a modified Hyers-Ulam sequence. J. Math. Anal. Appl. 158, 106–113 (1991)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rassias, ThM, Šemrl, P.: On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proc. Am. Math. Soc. 114(2), 989–993 (1992)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rassias, ThM: Solution of a functional equation problem of Steven Butler. Octogon Math. Mag. 12, 152–153 (2004)Google Scholar
  25. 25.
    Rolewicz, S.: Metric linear spaces. Reidel/PWN-Polish Sci. Publ, Dordrecht (1984)MATHGoogle Scholar
  26. 26.
    Ulam, S.M.: Problems in Modern Mathematics. Chapter IV, Science Editions, Wiley, New York (1960)Google Scholar
  27. 27.
    Wang, L.G., Liu, B.: The Hyers-Ulam stability of a functional equation deriving from quadratic and cubic functions in Quasi-\(\beta \)-normed spaces. Acta. Math. Sin.-English Ser 26(12), 2335–2348 (2010)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Xu, T.Z., Rassias, J.M., Rassias, M.J., Xu, W.X.: A fixed point approach to the stability of quintic and sextic functional equations in quasi-\(\beta \)-normed spaces, J. Inequal. Appl., Article ID 423231, 23 (2010)Google Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesIbn Tofail UniversityKenitraMorocco

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