Skip to main content
SpringerLink
Log in

The Hyers–Ulam stability of an additive-quadratic s-functional inequality in Banach spaces

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

For any fixed \(s \in \left\{ z \in \mathbb {C} : z \ne 0 \text { and } |z| <1 \right\} ,\) we consider the following functional inequality:

$$\begin{aligned}&\nonumber \Vert f(a+a', c+c') + f(a+a', c-c') + f(a-a', c+c') + f(a-a', c-c')\nonumber \\&\quad -4f(a,c)-4f(a,c')\Vert \le \Bigg \Vert s \Bigg (2f\left( a+a', c-c'\right) + 2f\left( a-a', c+c'\right) \nonumber \\&\quad - 4f(a,c )-4f(a,c')+ 4 f(a',c')\Bigg )\Bigg \Vert . \end{aligned}$$
(1)

In this paper, we obtain the Hyers–Ulam stability of the proposed functional inequality using the direct and fixed point methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

  2. Amyari, M., Baak, C., Moslehian, M.: Nearly ternary derivations. Taiwan. J. Math. 11, 1417–1424 (2007)

    Article  MathSciNet  Google Scholar 

  3. Cădariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Art. ID 4

  4. Cădariu, L., Radu, V.: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346, 43–52 (2004)

    MathSciNet  Google Scholar 

  5. Cădariu, L., Radu, V.: Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. 2008, Art. ID 749392 (2008)

  6. Cho, Y.J., Park, C., Saadati, R.: Functional inequalities in non-Archimedean Banach spaces. Appl. Math. Lett. 23, 1238–1242 (2010)

    Article  MathSciNet  Google Scholar 

  7. Diaz, J., 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)

    Article  MathSciNet  Google Scholar 

  8. El-Fassi, I.: Generalized hyperstability of a Drygas functional equation on a restricted domain using Brzdek’s fixed point theorem. J. Fixed Point Theory Appl. 19, 2529–2540 (2017)

    Article  MathSciNet  Google Scholar 

  9. Eshaghi Gordji, M., Fazeli, A., Park, C.: 3-Lie multipliers on Banach 3-Lie algebras, Int. J. Geom. Meth. Mod. Phys. 9 (2012), no. 7, Art. ID 1250052. 15 pp

  10. Eshaghi Gordji, M., Ghaemi, M.B., Alizadeh, B.: A fixed point method for perturbation of higher ring derivationsin non-Archimedean Banach algebras. Int. J. Geom. Meth. Mod. Phys. 8(7), 1611–1625 (2011)

    Article  Google Scholar 

  11. Gǎvruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

    Article  MathSciNet  Google Scholar 

  12. Hwang, I., Park, C.: Ulam stability of an additive-quadratic functional equation in Banach spaces. J. Math. Inequal. 14, 421–436 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  14. Hyers, D.H., Ulam, S.M.: Approximately convex functions. Proc. Am. Math. Soc. 3, 821–828 (1952)

    Article  MathSciNet  Google Scholar 

  15. Hyers, D.H.: The stability of homomorphisms and related topics. Glob. Anal. Anal. Manifolds. 75, 140–153 (1983)

    MathSciNet  Google Scholar 

  16. Hyers, D.H., Rassias, Th.M.: Approximate homomorphisms. Aequationes Math. 44, 125–153 (1992)

    Article  MathSciNet  Google Scholar 

  17. Isac, G., Rassias, Th.M.: Stability of \(\psi \)-additive mappings: Applications to nonlinear analysis. Int. J. Math. Math. Sci. 19, 219–228 (1996)

    Article  MathSciNet  Google Scholar 

  18. Jung, S., Th, M., Rassias and C. Mortici, : On a functional equation of trigonometric type. Appl. Math. Comput. 252, 294–303 (2015)

  19. Jung, S., Popa, D., Rassias, MTh.: On the stability of the linear functional equation in a single variable on complete metric spaces. J. Global Optim. 59, 165–171 (2014)

    Article  MathSciNet  Google Scholar 

  20. Lee, Y., Jung, S., Rassias, MTh.: Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation. J. Math. Inequal. 12, 43–61 (2018)

    Article  MathSciNet  Google Scholar 

  21. Miheţ, D., Radu, V.: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343, 567–572 (2008)

    Article  MathSciNet  Google Scholar 

  22. Nikoufar, I.: Jordan \((\theta, \phi )\)-derivations on Hilbert \(C^*\)-modules. Indag. Math. 26, 421–430 (2015)

    Article  MathSciNet  Google Scholar 

  23. Park, C.: Additive \(\rho \)-functional inequalities and equations. J. Math. Inequal. 9, 17–26 (2015)

    Article  MathSciNet  Google Scholar 

  24. Park, C.: Additive \(\rho \)-functional inequalities in non-Archimedean normed spaces. J. Math. Inequal. 9, 397–407 (2015)

    Article  MathSciNet  Google Scholar 

  25. Park, C.: Fixed point method for set-valued functional equations. J. Fixed Point Theory Appl. 19, 2297–2308 (2017)

    Article  MathSciNet  Google Scholar 

  26. Park, C.: Biderivations and bihomomorphisms in Banach algebras. Filomat 33, 2317–2328 (2019)

    Article  MathSciNet  Google Scholar 

  27. Park, C., Paokanta, S., Suparatulatorn, R.: Ulam stability of bihomomorphisms and biderivations in Banach algebras, J. Fixed Point Theory Appl. 22 (2020), Paper No. 27

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Ulam, S.M.: A Collection of the Mathematical Problems. Interscience Publ, New York (1960)

    Google Scholar 

  31. Wang, Z.: Stability of two types of cubic fuzzy set-valued functional equations. Res. Math. 70, 1–14 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was partially supported by Chiang Mai University.

Author information

Authors and Affiliations

  1. Advanced Research Center for Computational Simulation, Chiang Mai University, Chiang Mai, 50200, Thailand

    Tanadon Chaobankoh & Raweerote Suparatulatorn

  2. Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Tanadon Chaobankoh & Raweerote Suparatulatorn

  3. Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, Korea

    Choonkil Park

  4. Department of Mathematics Education, Gyeongsang National University, Jinju, 52828, Korea

    Yeol Je Cho

  5. Center for General Education, China Medical University, Taichung, 40402, Taiwan

    Yeol Je Cho

Authors
  1. Tanadon Chaobankoh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raweerote Suparatulatorn
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Choonkil Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yeol Je Cho
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raweerote Suparatulatorn.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chaobankoh, T., Suparatulatorn, R., Park, C. et al. The Hyers–Ulam stability of an additive-quadratic s-functional inequality in Banach spaces. Ricerche mat 73, 1029–1044 (2024). https://doi.org/10.1007/s11587-021-00648-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-021-00648-3

Keywords

Mathematics Subject Classification

Search

Navigation