Skip to main content
SpringerLink
Log in

Nice results about quadratic type functional equations on semigroups

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

Let \((S,+)\) be an abelian semigroup, let \(\sigma \) be an involution of S,  let X be a linear space over the field \({\mathbb {K}}\in \{{\mathbb {R}},{\mathbb {C}}\}\) and let \(\mu \),\(\nu \) be linear combinations of Dirac measures. In the present paper, we find the general solution \(f:S\rightarrow X\) of the following functional equation

$$\begin{aligned} \int _{S}f(x+y+t)d\mu (t)+\int _{S}f(x+\sigma (y)+t)d\nu (t)=f(x)+f(y), \ \ \ x,y \in S, \end{aligned}$$

in terms of additive and bi-additive maps. Many consequences of this result are presented.

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. Aczél, J.: The general solution of two functional equations by reduction to functions additive in two variables and with the aid of Hamel bases. Glas. Mat.-Fiz. Astronom. 2, 65–73 (1965)

    MathSciNet  MATH  Google Scholar 

  2. Charifi, A., Łukasik, R., Zeglami, D.: A special class of functional equations. Math. Slovaca 68(2), 397–404 (2018)

    Article  MathSciNet  Google Scholar 

  3. Fadli, B., Zeglami, D., Kabbaj, S.: On a Gajda’s type quadratic equation on a locally compact abelian group. Indag. Math. 26(4), 660–668 (2015)

    Article  MathSciNet  Google Scholar 

  4. Fadli, B., Chahbi, A., EL-Fassi, I., Kabbaj, S.: On Jensen’s and the quadratic functional equations with involutions. Proyecciones (Antofagasta) 35(2), 213–223 (2016)

    Article  MathSciNet  Google Scholar 

  5. Fadli, B.: A note on a generalization of the quadratic functional equation. Math. Recherche Appl. 15, 40–43 (2016)

    Google Scholar 

  6. Fadli, B., Zeglami, D., Kabbaj, S.: A variant of Jensen’s functional equation. Demonstr. Math. 49(4), 413–420 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Fadli, B., Zeglami, D., Kabbaj, S.: A variant of the quadratic functional equation on semigroups. Proyecciones (Antofagasta) 37(1), 45–55 (2018)

    Article  MathSciNet  Google Scholar 

  8. de Place Friis, P., Stetkær, H.: On the quadratic functional equation on groups. Publ. Math. Debr. 69(1–2), 65–93 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Gajda, Z.: A generalization of d’Alembert’s functional equation. Funkcial. Ekvac. 33(1), 69–77 (1990)

    MathSciNet  MATH  Google Scholar 

  10. Jung, S.-M.: Quadratic functional equations of Pexider type. Int. J. Math. Math. Sci. 24(5), 351–359 (2000)

    Article  MathSciNet  Google Scholar 

  11. Kannappan, P.: Quadratic functional equation and inner product spaces. Results Math. 27(3–4), 368–372 (1995)

    Article  MathSciNet  Google Scholar 

  12. Łukasik, R.: Some generalization of Cauchy’s and the quadratic functional equations. Aequationes Math. 83(1), 75–86 (2012)

    Article  MathSciNet  Google Scholar 

  13. Sinopoulos, P.: Functional equations on semigroups. Aequationes Math. 59(3), 255–261 (2000)

    Article  MathSciNet  Google Scholar 

  14. Stetkær, H.: Functional equations on abelian groups with involution. Aequationes Math. 54(1–2), 144–172 (1997)

    Article  MathSciNet  Google Scholar 

  15. Stetkær, H.: Functional Equations on Groups. World Scientific, Singapore (2013)

    Book  Google Scholar 

  16. Székelyhidi, L.: Convolution Type Functional Equations on Topological Abelian Groups. World Scientific, Teaneck (1991)

    Book  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the referee for a number of constructive comments which have led to essential improvement of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Ibn Tofail University, B.P. 14000, Kenitra, Morocco

    A. Akkaoui & M. El Fatini

  2. Department of Mathematics, Faculty of Sciences, Chouaib Doukkali University, B.P. 20, 24000, El Jadida, Morocco

    B. Fadli

  3. Department of Mathematics, E.N.S.A.M, Moulay ISMAIL University, B.P. 15290, Al Mansour, Meknes, Morocco

    D. Zeglami

Authors
  1. A. Akkaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. El Fatini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Fadli
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. Zeglami
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Fadli.

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

Akkaoui, A., Fatini, M.E., Fadli, B. et al. Nice results about quadratic type functional equations on semigroups. Aequat. Math. 94, 83–96 (2020). https://doi.org/10.1007/s00010-019-00653-w

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-019-00653-w

Mathematics Subject Classification

Keywords

Search

Navigation