Advertisement

Acta Mathematica Scientia

, Volume 39, Issue 4, pp 1017–1032 | Cite as

Hyperstability of the Generalized Cauchy-Jensen Functional Equation in Ultrametric Spaces

  • Prondanai KaskasemEmail author
  • Chakkrid Kiln-EamEmail author
Article
  • 16 Downloads

Abstract

In this article, we prove hyperstability results of the generalized Cauchy-Jensen functional equation
$$\alpha f\left( {{{x + y} \over \alpha } + z} \right) = f(x) + f(y) + \alpha f(z)$$
for any fixed positive integer α ≥ 2 in ultrametric Banach spaces by using fixed point method.

Key words

Hyperstability generalized Cauchy-Jensen functional equation ultrametric space 

2010 MR Subject Classification

39B52 39B82 47H10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors would like to thank Science Achievement Scholarship of Thailand, which provides funding for research.

References

  1. [1]
    Almahalebi M. Generalized hyperstability of the Jensen functional equation in ultrametric spaces. J Anal, 2017. Doi: https://doi.org/10.1007/s41478-017-0060-7
  2. [2]
    Almahalebi M. Non-Archimedean hyperstability of a Cauchy-Jensen type functional equation. J Class Anal, 2017, 11(2): 159–170MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Almahalebi M. On the stability of a generalization of Jensen functional equation. Acta Math Hungar, 2018, 154(1): 187–198MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Almahalebi M. Stability of a generalization of Cauchys and the quadratic functional equations. J Fixed Point Theory Appl, 2018, 20(12). Doi. https://doi.org/10.1007/s11784-018-0503-z
  5. [5]
    Almahalbi M, Chahbi A. Hyperstability of the Jensen functional equation in ultrametric spaces. Aequat Math, 2017, 91: 647–661MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Almahalebi M, Charifi A, Kabbaj S. Hyperstability of a Cauchy functional equation. J Nonlinear Anal Optim, 2015, 6(2): 127–137MathSciNetzbMATHGoogle Scholar
  7. [7]
    Aribou Y, Almahalebi M, Kabbaj S. Hyperstability of cubic functional equation in ultrametric spaces. Proyecciones J of Math, 2017, 36(3): 461–484MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Bahyrycz A, Brzdęk J, Piszczek M. On Approximately p-Wright Affine Functions in Ultrametric Spaces. J Funct Spaces, 2013. Art ID 723545Google Scholar
  9. [9]
    Bahyrycz A, Piszczek M. Hyperstability of the Jensen functional equation. Acta Math Hungar, 2014, 142(2): 353–365MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Bourgin D G. Approximately isometric and multiplicative transformations on continuous function rings. Duke Math J, 1949, 16: 385–397MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Brzdęk J, Chudziak J, Páles Z. A fixed point approach to stability of functional equations. Nonlinear Anal, 2011, 74: 6728–6732MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Brzdęk J, Ciepliñski K. A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal, 2011, 74: 6861–6867MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Brzdęk J, Ciepliñski K. Hyperstability and superstability. Abstr Appl Anal, 2013. Article ID 401756Google Scholar
  14. [14]
    Brzdęk J. A hyperstability result for the Cauchy equation. Bull Aust Math Soc, 2014, 89: 33–40MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Brzdęk J. Hyperstability of the Cauchy equation on restricted domains. Acta Math Hungar, 2013, 141: 58–67MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Brzdęk J. Remarks on hyperstability of the Cauchy functional equation. Aequat Math, 2013, 86: 255–267MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Brzdęk J. Stability of additivity and fixed point methods. Fixed Point Theory Appl, 2013: 285Google Scholar
  18. [18]
    Diagana T, Ramarosan F. Non-Archimedean Operator Theory. SpringerBrieft in Mathematics. DOI  https://doi.org/10.1007/978-3-319-27323-52_2, 2016
  19. [19]
    EL-Fassi I, Brzdęk J, Chahbi A, Kabbaj S. On the Hyperstability of the biadditive functional equation. Acta Mathematica Scientia, 2017, 37B(6): 1727–1739MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    EL-Fassi I, Kabbaj S, Charifi A. Hyperstability of Cauchy-Jensen functional equations. Indag Math, 2016, 27: 855–867MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    EL-Fassi I. On a New Type of Hyperstability for Radical Cubic Functional Equation in Non-Archimedean Metric Spaces. Results Math, 2017, 72: 991–1005MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Gao Z X, Cao H X, Zheng W T, Xu L. Generalized Hyers-Ulam-Rassias stability of functional inequalities and functional equations. J Math Inequal, 2009, 3(1): 63–77MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Gselmann E. Hyperstability of a functional equation. Acta Math Hungar, 2009, 124: 179–188MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Găvruţa P. A generalization of the Hyers-Ulam-Rassias stability od approximately additive mapping. J Math Anal Appl, 1994, 184: 431–436MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Hensel K. Uber eine neue begründung der theorie der algebraischen zahlen. Jahresbericht der Deutschen Mathematiker-Vereinigung, 1899, 6: 83–88zbMATHGoogle Scholar
  26. [26]
    Hyers D H. On the stability of the linear functional equation. Proc Natl Acad Sci USA, 1941, 27(4): 222–224MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Khrennikov A. Non-Archimedean Analysis, Quantum Paradoxes, Dynamical Systems and Biological Models. Dordrecht: Kluwer Academic Publishers, 1997CrossRefzbMATHGoogle Scholar
  28. [28]
    Maksa G, Páles Z. Hyperstability of a class of linear functional equations. Acta Math, 2001, 17(2): 107–112MathSciNetzbMATHGoogle Scholar
  29. [29]
    Moszner Z. Stability has many names. Aequat Math, 2016, 90: 983–999MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Piszczek M. Remark on hyperstability of the general linear equation. Aequat Math, 2014, 88(1): 163–168MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Rassias Th M. On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc, 1978, 72(2): 297–300MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Sirouni M, Kabbaj S. A fixed point approach to the hyperstability of Drygas functional equation in metric spaces. J Math Comput Sci, 2014, 4(4): 705–715Google Scholar
  33. [33]
    Ulam SM. A Collection of Mathematical Problems//Interscience Tracts in Pure and Applied Mathematics. No 8. New York, NY, USA: Interscience Publishers, 1960Google Scholar
  34. [34]
    Zhang D. On Hyperstability of generalized linear functional equations in several variables. Bull Aust Math Soc, 2015, 92(2): 259–267MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceNaresuan UniversityPhitsanulokThailand
  2. 2.Research Center for Academic Excellence in MathematicsNaresuan UniversityPhitsanulokThailand

Personalised recommendations