On the Ulam–Hyers stability of the complex functional equation \(\varvec{F(z)+F(2z)+\cdots +F(nz)=0}\)


In the present paper we prove that the complex functional equation \(F(z)+F(2z)+\cdots +F(nz)=0\), \(n\ge 2\), \(z\in {\mathbb {C}}{\setminus }( -\infty ,0] \), is stable in the generalized Hyers–Ulam sense.

This is a preview of subscription content, access via your institution.

Fig. 1


  1. 1.

    Agarwal, R.P., Xu, B., Zhang, Z.: Stability of functional equations in single variable. J. Math. Anal. Appl. 288(2), 852–869 (2003)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bahyrycz, A., Olko, J.: On stability of the general linear equation. Aequationes Math. 89(6), 1461–1474 (2015)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Baker, J.A.: A general functional equation and its stability. Proc. Am. Math. Soc. 133(6), 1657–1664 (2005)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Batko, B., Brzdȩk, J.: A fixed point theorem and the Hyers–Ulam stability in Riesz spaces. Adv. Differ. Equ. 2013, 138 (2013)

    MathSciNet  MATH  Google Scholar 

  5. 5.

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

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Bodaghi, A.: Stability of a quartic functional equation. Sci. World J. Article ID 752146 (2014)

  7. 7.

    Brzdȩk, J., Ciepliński, K.: A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal. 74(18), 6861–6867 (2001)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Cădariu, L., Radu, V.: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. Article ID 749392 (2018)

  9. 9.

    Cădariu, L., Radu, V.: On the stability of the Cauchy functional equation: a fixed point approach. In: Iteration Theory (ECIT ’02), vol. 346 of Grazer Mathematische Berichte, pp. 43–52. Karl-Franzens-Universitaet Graz, Graz (2004)

  10. 10.

    Ciepliński, K.: Applications of fixed point theorems to the Hyers–Ulam stability of functional equations—a survey. Ann. Funct. Anal. 3(1), 151–164 (2012)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Diaz, J.B., Margolis, B.: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Ferrando, J.C., López-Pellicer, M.: Descriptive topology and functional analysis. In: Springer Proceedings in Mathematics and Statistics 80 (2014)

  13. 13.

    Ferrando, J.C.: Descriptive topology and functional analysis II. Springer Proceedings in Mathematics and Statistics 286 (2019)

  14. 14.

    Gao, Z.X., Cao, H.X., Zheng, W.T., Xu, L.: Generalized Hyers–Ulam–Rassias stability of functional inequalities and functional equation. J. Math. Inequal. 3(1), 63–77 (2009)

    MathSciNet  MATH  Google Scholar 

  15. 15.

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

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Jung, S.M.: Hyers–Ulam–Rassias stability of functional equations in nonlinear analysis. In: Springer Optimization and its Applications, 48. New York (2011)

  17. 17.

    Jung, S.M., Lee, Z.F.: A fixed point approach to the stability of quadratic functional equation with involution. Fixed Point Theory Appl. Article ID 732086 (2008)

  18. 18.

    Jung, S.M.: A fixed point approach to the stability of isometries. J. Math. Anal. Appl. 329(2), 879–890 (2007)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Jung, S.M., Kim, T.S.: A fixed point approach to the stability of the cubic functional equation. Boletín de la Sociedad Matemática Mexicana 12(1), 51–57 (2006)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Mora, G., Cherruault, Y., Ziadi, A.: Functional equations generating space-densifying curves. Comput. Math. Appl. 39(9–10), 45–55 (2000)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Mora, G.: An estimate of the lower bound of the real parts of the zeros of the partial sums of the Riemann zeta function. J. Math. Anal. Appl. 427(1), 428–439 (2015)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Mora, G.: A note on the functional equation \(F(z)+F(2z)+\cdots + F(nz)=0\). J. Math. Anal. Appl. 340(1), 466–475 (2008)

    MathSciNet  MATH  Google Scholar 

  23. 23.

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

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Moszner, Z.: On the normal stability of functional equations. Ann. Math. Sil. 30(1), 111–128 (2016)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Park, C., Kim, S.O.: Quadratic \(\alpha \)-functional equations. Int. J. Nonlinear Anal. Appl. 8(1), 1–9 (2017)

    MathSciNet  Google Scholar 

  26. 26.

    Radu, V.: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4(1), 91–96 (2003)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Rassias, T.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251(1), 264–284 (2000)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Son, E., Lee, J., Kim, H.M.: Stability of quadratic functional equations via the fixed point and direct method. J. Inequal. Appl. Article ID 635720 (2010)

  29. 29.

    Ulam, S.M.: A collection of mathematical problems. Interscience Publishers, New York (1940)

    MATH  Google Scholar 

  30. 30.

    Xu, T.Z., Rassias, T.M.: On the Hyers–Ulam stability of a general mixed additive and cubic functional equation in \(n\)-Banach spaces. Abstr. Appl. Anal. Article ID 926390 (2012)

  31. 31.

    Zhang, D.H., Cao, H.X.: Stability of functional equations in several variables. Acta Math. Sin. (Engl. Ser.) 23(2), 321–326 (2007)

    MathSciNet  MATH  Google Scholar 

Download references


To the anonymous referee, for his/her useful comments and suggestions to improve the quality of the paper.

Author information



Corresponding author

Correspondence to G. García.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

García, G., Mora, G. On the Ulam–Hyers stability of the complex functional equation \(\varvec{F(z)+F(2z)+\cdots +F(nz)=0}\). Aequat. Math. 94, 899–911 (2020). https://doi.org/10.1007/s00010-019-00693-2

Download citation


  • Stability
  • Functional equations
  • Complex variable functions
  • Metric fixed point

Mathematics Subject Classification

  • 37L15
  • 30D05 (Primary)
  • 39B32
  • 47H10 (Secondary)