Skip to main content
SpringerLink
Log in

Stability and hyperstability of orthogonally \(*\)-m-homomorphisms in orthogonally Lie \(C^*\)-algebras: a fixed point approach

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

A Correction to this article was published on 28 June 2018

This article has been updated

Abstract

Recently Eshaghi et al. introduced orthogonal sets and proved the real generalization of the Banach fixed point theorem on these sets. In this paper, we prove the real generalization of Diaz–Margolis fixed point theorem on orthogonal sets. By using this fixed point theorem, we study the stability of orthogonally \(*\)-m-homomorphisms on Lie \(C^*\)-algebras associated with the following functional equation:

$$\begin{aligned} \begin{aligned}&f(2x+y)+f(2x-y)+(m-1)(m-2)(m-3)f(y)\\&\quad =2^{m-2}[f(x+y)+f(x-y)+6f(x)]. \end{aligned} \end{aligned}$$

for each \(m=1,2,3,4.\). Moreover, we establish the hyperstability of these functional equations by suitable control functions.

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

Change history

  • 28 June 2018

    The acknowledgment in the original article is incomplete.

References

  1. Fuchs, J.: Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  2. Facchi, P., Garnero, G., Ligab, M.: Quantum fluctuation relations. Int. J. Geom. Methods Mod. Phys. 14, 1740002 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. De Graaf, W.: Lie Algebras: Theory and Algorithms, vol. 56. Elsevier, Amsterdam (2000)

    MATH  Google Scholar 

  4. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. II. Physical applications. Ann. Phys. 111, 111–151 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fiorenza, D., Sati, H., Schreiber, U.: Super-Lie n-algebra extensions, higher WZW models and super-p-branes with tensor multiplet fields. Int. J. Geom. Methods Mod. Phys. 12, 1550018 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Iglesias, D., Marrero, J.C.: Generalized Lie bialgebroids and Jacobi structures. J. Geom. Phys. 40, 176–200 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Doebner, H.D.: In: Dobrev, V.K., Hilgert, J. (eds.) Lie Theory and Its Applications in Physics. World Scientific, Singapore (1996)

  8. Ulam, S.M.: Problems in Modern Mathematics, Science Edition. Wiley, New York (1964)

    MATH  Google Scholar 

  9. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126–130 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Jung, S.M.: Hyers–Ulam–Rassias stability of Jensen’s equation and its application. Proc. Am. Math. Soc. 126, 3137–3143 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Khodaei, H., Rassias, ThM: Approximately generalized additive functions in several variables. Int. J. Nonlinear Anal. Appl. 1, 22–41 (2010)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Margolis, B., Diaz, J.B.: A fixed point theorem of the alternative for contractions on the generalized complete metric space. Bull. Am. Math. Soc. 126, 305–309 (1968)

    MathSciNet  MATH  Google Scholar 

  18. Cǎdariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4, Article 4 (2003). http://jipam.vu.edu.au

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

    MathSciNet  MATH  Google Scholar 

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

  21. Gǎvruta, P., Gǎvruta, L.: A new method for the generalized Hyers–Ulam–Rassias stability. Int. J. Nonlinear Anal. Appl. 1, 11–18 (2010)

    MATH  Google Scholar 

  22. Khodaei, H., Gordji, M.E., Khodabakhsh, R.: Fixed points, Lie \(*\)-homomorphisms and Lie \(*\)-derivations on Lie \(C^*\)-algebras. Fixed Point Theory 14(2), 387–400 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Gordji, M.E., Askari, G., Ansari, N., Anastassiou, G.A., Park, C.: Stability and hyperstability of generalized orthogonally quadratic ternary homomorphisms in non-Archimedean ternary Banach algebras: a fixed point approach. J. Comput. Anal. Appl. 21(3), 515–520 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Park, C.: Lie \(*\)-homomorphism between Lie \(C^*\)-algebra and Lie \(*\)-derrivation on Lie \(C^*\)-algebra. J. Math. Anal. Appl. 15, 419–434 (2004)

    Article  MATH  Google Scholar 

  25. Gordji, M.E., Ghobadipour, N.: Stability of \((\alpha, \beta, \gamma )\)-derivations on Lie \(C^*\)-algebras. Int. J. Geom. Methods Mod. Phys. 7, 1093–1102 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Park, C.: Homomorphism between Lie \(JC^*\)-algebra and Cauchy–Rassias stability of \(JC^*\)-algebra derivation. J. Lie Theory. 15, 393–414 (2005)

    MathSciNet  Google Scholar 

  27. Bae, J.H., Park, W.G.: A functional equation having monomials as solutions. Appl. Math. Comput. 216, 87–94 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Lee, S.H., Im, S.M., Hawng, I.S.: Quartic functional equation. J. Math. Anal. Appl. 307, 387–394 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Kannappan, Pl: Functional Equations and Inequalities with Applications. Springer, New York (2009)

    Book  MATH  Google Scholar 

  31. Lee, Y.H., Jung, S.M., Rassias, MTh: On an n-dimensional mixed type additive and quadratic functional equation. Appl. Math. Comput. 228, 13–16 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Park, C., Cui, J.: Generalized stability of \(C^*\)-ternary quadratic mappings. Abst. Appl. Anal. 2007, Article ID 23282 (2007)

  33. Bae, J.H., Park, W.G.: On a cubic and a Jensen-quadratic equations. Abst. Appl. Anal. 2007, Article ID 45179 (2007)

  34. Eshaghi Gordji, M., Ramezani, M., De La Sen, M., Cho, Y.J.: On orthogonal sets and Banach fixed point theorem. Fixed Point Theory 18, 569–578 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research is supported by Islamic Azad University, Central Tehran Branch in Iran. The authors would like to thank the anonymous reviewers for careful reading of the manuscript and also for giving useful comments, which helped to improve the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Islamic Azad University Central Tehran Branch, Tehran, Iran

    A. Bahraini

  2. Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

    G. Askari & M. Eshaghi Gordji

  3. Department of Mathematics, Islamic Azad University Dehloran Branch, Dehloran, Iran

    R. Gholami

Authors
  1. A. Bahraini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Askari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Eshaghi Gordji
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Gholami
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Askari.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bahraini, A., Askari, G., Gordji, M.E. et al. Stability and hyperstability of orthogonally \(*\)-m-homomorphisms in orthogonally Lie \(C^*\)-algebras: a fixed point approach. J. Fixed Point Theory Appl. 20, 89 (2018). https://doi.org/10.1007/s11784-018-0571-0

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s11784-018-0571-0

Keywords

Mathematics Subject Classification

Search

Navigation