A Kannappan-cosine functional equation on semigroups
Abstract
In this paper we determine the complex-valued solutions of the Kannappan-cosine functional equation g(xyz0) = g(x)g(y) − f(x)f(y), x,y∈S, where S is a semigroup and z0 is a fixed element in S.
Keywords
Kannappan; semigroups; multiplicative function; additive function; cosine-sine equation
References
J. Aczél, Lectures on Functional Equations and Their Applications, Mathematics in Science and Engineering, Vol. 19, Academic Press, New York, 1966.
O. Ajebbar and E. Elqorachi, The cosine-sine functional equation on a semigroup with an involutive automorphism, Aequationes Math. 91 (2017), no. 6, 1115–1146.
O. Ajebbar and E. Elqorachi, Solutions and stability of trigonometric functional equations on an amenable group with an involutive automorphism, Commun. Korean Math. Soc. 34 (2019), no. 1, 55–82.
B. Ebanks, The sine addition and subtraction formulas on semigroups, Acta Math. Hungar. 164 (2021), no. 2, 533–555.
B. Ebanks, The cosine and sine addition and subtraction formulas on semigroups, Acta Math. Hungar. 165 (2021), no. 2, 337–354.
B. Ebanks, Around the sine addition law and d’Alembert’s equation on semigroups, Results Math. 77 (2022), no. 1, Paper No. 11, 14 pp.
A. Jafar, O. Ajebbar, and E. Elqorachi, A Kannappan-sine addition law on semigroups, arXiv preprint, 2023. Available at arXiv: 2304.14146.
S. Kabbaj, M. Tial, and D. Zeglami, The integral cosine addition and sine subtraction laws, Results Math. 73 (2018), no. 3, Paper No. 97, 19 pp.
Pl. Kannappan, A functional equation for the cosine, Canad. Math. Bull. 11 (1968), no. 3, 495–498.
T.A. Poulsen and H. Stetkær, The trigonometric subtraction and addition formulas, Aequationes Math. 59 (2000), no. 1–2, 84–92.
H. Stetkær, Functional Equations on Groups, World Scientific Publishing Company, Singapore, 2013.
H. Stetkær, The cosine addition law with an additional term, Aequationes Math. 90 (2016), no. 6, 1147–1168.
H. Stetkær, Kannappan’s functional equation on semigroups with involution, Semigroup Forum 94 (2017), no. 1, 17–30.
Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir Morocco
Department of Mathematics, Multidisciplinary Faculty, Sultan Moulay Slimane University. Beni Mellal Morocco
https://orcid.org/0000-0001-6537-1548
Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir 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.
