Skip to main content
SpringerLink
Log in

Quantum exponential functions in a Banach algebra

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

Abstract

In this paper, we define and investigate some properties of the quantum exponential functions \(e_{A,\beta }(t)\) and \(E_{A,\beta }(t)\) in a Banach algebra \({\mathbb {E}}\) with a unit \({\mathfrak {e}}\), where \(A:I\rightarrow {\mathbb {E}}\) is a continuous mapping at the unique fixed point \(s_0\) of the strictly increasing continuous function \(\beta \) defined on an interval \(I\subseteq {{\mathbb {R}}}\). Moreover, we define the \(\beta \)-regressive mappings in \({\mathbb {E}}\) and study some of their properties.

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. Annaby, M.H., Hamza, A.E., Aldowah, K.A.: Hahn difference operator and associated Jackson–Nörlund integerals. J. Optim. Theory Appl. 154, 133–153 (2012)

    Article  MathSciNet  Google Scholar 

  2. Annaby, M.H., Mansour, Z.S.: \(q\)-Fractional Calculus and Equations. Springer, Berlin (2012)

    Google Scholar 

  3. Auch, T.J.: Development and application of difference and fractional calculus on discrete time scales. Ph.D. Thesis, University of Nebraska-Lincoln (2013)

  4. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhaüser, Basel (2001)

    Book  Google Scholar 

  5. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhaüser, Basel (2003)

    Book  Google Scholar 

  6. Cresson, J.: Non-differentiable variational principles. J. Math. Anal. Appl. 1(307), 48–64 (2005)

    Article  MathSciNet  Google Scholar 

  7. Faried, N., Shehata, E.M., El Zafarani, R.M.: On homogenous second order linear general quantum difference equations. J. Inequal. Appl. (Springer) 2017, 198 (2017). https://doi.org/10.1186/s13660-017-1471-3

    Article  MATH  Google Scholar 

  8. Faried, N., Shehata, E.M., El Zafarani, R.M.: Theory of nth-order linear general quantum difference equations. Adv. Differ. Equ. (Springer) 2018, 264 (2018). https://doi.org/10.1186/s13662-018-1715-7

    Article  MATH  Google Scholar 

  9. Hamza, A.E., Abdelkhaliq, M.M.: Hahn difference equations in Banach algebras. Adv. Differ. Equ. 2016, 161 (2016). https://doi.org/10.1186/s13662-016-0886-3

    Article  MathSciNet  MATH  Google Scholar 

  10. Hamza, A.E., Sarhan, A.M., Shehata, E.M., Aldowah, K.A.: A general quantum difference calculus. Adv. Differ. Equ. (Springer) 2015, 182 (2015). https://doi.org/10.1186/s13660-015-0518-3

    Article  MATH  Google Scholar 

  11. Hamza, A.E., Sarhan, A.M., Shehata, E.M.: Exponential, trigonometric and hyperbolic functions associated with a general quantum difference operator. Adv. Dyn Syst. Appl. 12(1), 25–38 (2017)

    MathSciNet  Google Scholar 

  12. Hamza, A.E., Shehata, E.M.: Existence and uniqueness of solutions of general quantum difference equations. Adv. Dyn Syst. Appl. 11, 45–58 (2016)

    MathSciNet  Google Scholar 

  13. Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)

    Book  Google Scholar 

  14. Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978)

    MATH  Google Scholar 

  15. Lavagno, A., Swamy, P.N.: \(q\)-Deformed structures and nonextensive statistics, a comparative study. Phys. A 305(1), 310–315 (2002)

    Article  MathSciNet  Google Scholar 

  16. Malinowska, A.B., Torres, D.F.M.: Quantum Variational Calculus, Briefs in Electrical and Computer Engineering-Control, Automation and Robotics. Springer, Berlin (2014)

    Book  Google Scholar 

  17. Page, D.N.: Information in black hole radiation. Phys. Rev. Lett. 71(23), 3743–3746 (1993)

    Article  MathSciNet  Google Scholar 

  18. Sarhan, A.M., Shehata, E.M.: On the fixed points of certain types of functions for constructing associated calculi. J. Fixed Point Theory Appl. 20, 124 (2018). https://doi.org/10.1007/s11784-018-0602-x

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors sincerely thank the referees for their valuable suggestions and comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt

    Nashat Faried & Rasha M. El Zafarani

  2. Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shibin El-Koom, Egypt

    Enas M. Shehata

Authors
  1. Nashat Faried
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Enas M. Shehata
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rasha M. El Zafarani
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Rasha M. El Zafarani.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Additional information

Publisher's Note

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

This work was completed with the support of our TeX-pert.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Faried, N., Shehata, E.M. & El Zafarani, R.M. Quantum exponential functions in a Banach algebra. J. Fixed Point Theory Appl. 22, 22 (2020). https://doi.org/10.1007/s11784-020-0758-z

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s11784-020-0758-z

Keywords

Mathematics Subject Classification

Search

Navigation