A variant of d'Alembert's functional equation on semigroups with endomorphisms
Abstract
Let S be a semigroup, and let ϕ,ψ:S→S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d’Alembert’s functional equation
f(xϕ(y))+f(ψ(y)x) = 2f(x)f(y), x,y∈S,
where f:S→ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented.
Keywords
functional equation; d’Alembert; semigroup; multiplicative function; endomorphism
References
2. M. Ayoubi and D. Zeglami, D’Alembert’s functional equations on monoids with an anti-endomorphism, Results Math. 75 (2020), no. 2, Paper No. 74, 12 pp.
3. A.L. Cauchy, Cours d’Analyse de L’École Royale Polytechnique. Première Partie: Analyse Algébrique, De L’Imprimerie Royale, Paris, 1821.
4. A. Chahbi, B. Fadli, and S. Kabbaj, A generalization of the symmetrized multiplicative Cauchy equation, Acta Math. Hungar. 149 (2016), no. 1, 170–176.
5. J. d’Alembert, Addition au Mémoire sur la courbe que forme une corde tendue mise en vibration, Hist. Acad. Berlin 1750 (1750), 355–360.
6. B. Ebanks and H. Stetkær, d’Alembert’s other functional equation on monoids with an involution, Aequationes Math. 89 (2015), no. 1, 187–206.
7. B. Fadli, S. Kabbaj, K.H. Sabour, and D. Zeglami, Functional equations on semigroups with an endomorphism, Acta Math. Hungar. 150 (2016), no. 2, 363–371.
8. B. Fadli, D. Zeglami, and S. Kabbaj, A joint generalization of Van Vleck’s and Kannappan’s equations on groups, Adv. Pure Appl. Math. 6 (2015), no. 3, 179–188.
9. B. Fadli, D. Zeglami, and S. Kabbaj, A variant of Wilson’s functional equation, Publ. Math. Debrecen 87 (2015), no. 3-4, 415–427.
10. Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York, 2009.
11. K.H. Sabour, A. Charifi, and S. Kabbaj, On a variant of #-Wilson’s functional equation with an endomorphism, in: G.A. Anastassiou, J.M. Rassias (eds.), Frontiers in Functional Equations and Analytic Inequalities, Springer, Cham, 2019, pp. 93–111.
12. H. Stetkær, On multiplicative maps, Semigroup Forum 63 (2001), no. 3, 466–468.
13. H. Stetkær, Functional Equations on Groups, World Scientific Publishing Co., Singapore, 2013.
14. H. Stetkær, A variant of d’Alembert’s functional equation, Aequationes Math. 89 (2015), no. 3, 657–662.
15. D. Zeglami, B. Fadli, and S. Kabbaj, Harmonic analysis and generalized functional equations for the cosine, Adv. Pure Appl. Math. 7 (2016), no. 1, 41–49.
Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Morocco Morocco
https://orcid.org/0000-0001-8468-3408
Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Morocco Morocco
Department of Mathematics, Faculty of Sciences, Chouaib Doukkali University, Morocco Morocco
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
