Skip to main content
SpringerLink
Log in

A New Variant of Wilson’s Functional Equation on Monoids

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We find on a monoid M the complex-valued solutions f, g : M → ℂ such that f is central and g is continuous of the functional equation

$$f\left( {x\sigma \left( y \right)} \right) + f\left( {\tau \left( y \right)x} \right) = 2f\left( x \right)g\left( y \right),\,\,\,\,\,\,\,\,x,y \in M,$$

where σ : M → M is an involutive automorphism and τ : M → M is an involutive anti-automorphism. The solutions are described in terms of multiplicative functions, additive functions and characters of 2-dimensional representations of M.

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., Dhombres, J.: Functional Equations in Several Variables with Applications to Mathematics, Information Theory and to the Natural and Social Sciences. Encyclopedia of Mathematics and its Applications, Vol. 31, Cambridge University Press, New York, 1989

    Book  Google Scholar 

  2. Akkouchi, M., Bakali, A., Khalil, I.: A class of functional equations on a locally compact group. J. Lond. Math. Soc., 57(3), 694–705 (1998)

    Article  MathSciNet  Google Scholar 

  3. d’Alembert, J.: Addition au Mémoire sur la courbe que forme une corde tendue mise en vibration. Hist. Acad. Berl., 6, 355–360 (1750)

    Google Scholar 

  4. Ayoubi, M., Zeglami, D., Aissi, Y.: Wilson’s functional equation with an anti-endomorphism. Aequationes Math., 95, 535–549 (2021)

    Article  MathSciNet  Google Scholar 

  5. Chahbi, A., Fadli, B., Kabbaj, S.: A generalization of the symmetrized multiplicative Cauchy equation. Acta Math. Hung., 149, 170–176 (2020)

    Article  MathSciNet  Google Scholar 

  6. Chahbi, A., Elqorachi, E.: A variant of d’Alembert’s functional equation on monoids. Bull. Iran. Math. Soc., (2021). https://doi.org/10.1007/s41980-021-00568-w

  7. Davison, T. M. K.: D’Alembert’s functional equation on topological monoids. Publ. Math. Debrecen, 75(1–2), 41–66 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Dimou, H., Chahbi, A., Kabbaj, S.: A new generalization of Wilson’s functional equation. Proyecciones, 38(5), 943–954 (2019)

    Article  MathSciNet  Google Scholar 

  9. Dimou, H., Chahbi, A., Kabbaj, S.: New idea of generalized of variant of dAlembert functional equation. Palestine Journal of Mathematics, 10(2), 432–439 (2021)

    MathSciNet  MATH  Google Scholar 

  10. Ebanks, B. R.: Some trigonometric functional equations on monoids generated by their squares. Aequationes Math., 95(2), 383–391 (2021)

    Article  MathSciNet  Google Scholar 

  11. Ebanks, B. R., Stetkaer, H.: d’Alembert’s other functional equation on monoids with an involution. Aequationes Math., 89, 187–206 (2015)

    Article  MathSciNet  Google Scholar 

  12. Ebanks, B. R., Stetkaer, H.: On Wilson’s functional equations. Aequationes Math., 89, 339–354 (2015)

    Article  MathSciNet  Google Scholar 

  13. Elqorachi, E., Redouani, A.: Solutions and stability of a variant of Wilsons functional equation. Proyecciones, 37(2), 317–344 (2018)

    Article  MathSciNet  Google Scholar 

  14. El Fassi, I., Chahbi, A., Kabbaj, S.: The solution of a class functional equations on semi-groups. Filomat, 31(14), 4613–4618 (2017)

    Article  MathSciNet  Google Scholar 

  15. Fadli, B., Zeglami, D., Kabbaj, S.: A variant of Wilson’s functional equation. Publ. Math. Debrecen, 87(3–4), 415–427 (2015)

    Article  MathSciNet  Google Scholar 

  16. Kannapan, P. L.: Functional Equations and Inequalities with Applications, Springer, New York, 2009

    Book  Google Scholar 

  17. Ng, C. T., Zhao, H. Y., Lin, X.: A functional equation on groups with involutions. Aequationes Math., 94(3), 511–533 (2020)

    Article  MathSciNet  Google Scholar 

  18. Perkins, A. M., Sahoo, P. K.: On two functional equations with involution on groups related to sine and cosine functions. Aequationes Math., 89, 1251–1263 (2015)

    Article  MathSciNet  Google Scholar 

  19. Sabour, K. H., Fadli, B., Kabbaj, S.: Wilson’s functional equation on monoids with involutive automorphisms. Aequationes Math., 90, 1001–1011 (2016)

    Article  MathSciNet  Google Scholar 

  20. Sinopoulos, P.: Generalized sine equations. Aequationes Math., 48, 171–193 (1994)

    Article  MathSciNet  Google Scholar 

  21. Stetkær, H.: On multiplicative maps. Semigr. Forum, 63, 466–468 (2001)

    Article  MathSciNet  Google Scholar 

  22. Stetkær, H.: Functional Equations on Groups, World Scientific Publishing Co., Singapore, 2013

    Book  Google Scholar 

  23. Stetkær, H.: A variant of d’Alembert’s functional equation, Aequationes Math., 89, 657–662 (2015)

    Article  MathSciNet  Google Scholar 

  24. Stetkær, H.: More about Wilson’s functional equation. Aequationes Math., 94, 429–446 (2020)

    Article  MathSciNet  Google Scholar 

  25. Székelyhidi, L.: Convolution Type Functional Equations on Topological Abelian Groups. World Scientific Publishing Co., Inc., Teaneck, 1991

    Book  Google Scholar 

  26. Wilson, W. H.: On certain related functional equations. Bull. Amer. Math. Soc., 26, 300–312 (1920)

    Article  MathSciNet  Google Scholar 

  27. Wilson, W. H.: Two general functional equations. Bull. Amer. Math. Soc., 31(7), 330–334 (1925)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the referees for their time and comments.

Author information

Authors and Affiliations

  1. Laboratory EEFA, Faculty of Applied Sciences, Ibn Zohr University, Agadir, Morocco

    Hajira Dimou

  2. Laboratory EEFA, Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco

    Elhoucien Elqorachi & Abdellatif Chahbi

Authors
  1. Hajira Dimou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elhoucien Elqorachi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abdellatif Chahbi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hajira Dimou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dimou, H., Elqorachi, E. & Chahbi, A. A New Variant of Wilson’s Functional Equation on Monoids. Acta. Math. Sin.-English Ser. 38, 1303–1316 (2022). https://doi.org/10.1007/s10114-022-1233-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-022-1233-0

Keywords

MR(2010) Subject Classification

Search

Navigation