New results about quadratic functional equation on semigroups
Abstract
Let S be a semigroup, let (H, +) be a uniquely 2-divisible, abelian group and let ϕ, ψ be two endomorphisms of S that need not be involutive. In this paper, we express the solutions f : S→H of the following quadratic functional equation
f(xϕ(y)) + f(ψ(y)x) = 2f(x) + 2f(y), x,y ∈ S,
in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation. Some applications of this result are presented.
Keywords
quadratic functional equation; symmetrized additive Cauchy equation; additive function; bi-additive function; semigroup; endomorphism
References
Y. Aissi, D. Zeglami, and A. Mouzoun, A quadratic functional equation with involutive automorphisms on semigroups, Bol. Soc. Mat. Mex. (3) 28 (2022), no. 1, Paper No. 19, 13 pp.
A. Akkaoui, Jensen’s functional equation on semigroups, Acta Math. Hungar. 170 (2023), no. 1, 261–268.
A. Akkaoui, B. Fadli, and M. El Fatini, The Drygas functional equation on abelian semigroups with endomorphisms, Results Math. 76 (2021), no. 1, Paper No. 42, 13 pp.
A. Akkaoui, M. El Fatini, and B. Fadli, A variant of d’Alembert’s functional equation on semigroups with endomorphisms, Ann. Math. Sil. 36 (2022), no. 1, 1–14.
B.R. Ebanks, Some generalized quadratic functional equations on monoids, Aequationes Math. 94 (2020), no. 4, 737–747.
B.R. Ebanks, Pl. Kannappan, and P.K. Sahoo, A common generalization of functional equations characterizing normed and quasi-inner-product spaces, Canad. Math. Bull. 35 (1992), no. 3, 321–327.
H.H. Elfen, T. Riedel, and P.K. Sahoo, A variant of the quadratic functional equation on groups and an application, Bull. Korean Math. Soc. 54 (2017), no. 6, 2165–2182.
B. Fadli, A. Chahbi, Iz. El-Fassi, and S. Kabbaj, On Jensen’s and the quadratic functional equations with involutions, Proyecciones 35 (2016), no. 2, 213–223.
B. Fadli, D. Zeglami, and S. Kabbaj, A variant of the quadratic functional equation on semigroups, Proyecciones 37 (2018), no. 1, 45–55.
V.A. Faĭziev and P.K. Sahoo, Solution of Whitehead equation on groups, Math. Bohem. 138 (2013), no. 2, 171–180.
P. de Place Friis and H. Stetkær, On the quadratic functional equation on groups, Publ. Math. Debrecen 69 (2006), no. 1–2, 65–93.
P. Jordan and J. Von Neumann, On inner products in linear, metric spaces, Ann. of Math. (2) 36 (1935), no. 3, 719–723.
KH. Sabour and S. Kabbaj, Jensen’s and the quadratic functional equations with an endomorphism, Proyecciones 36 (2017), no. 1, 187–194.
P. Sinopoulos, Functional equations on semigroups, Aequationes Math. 59 (2000), no. 3, 255–261.
H. Stetkær, Functional equations on abelian groups with involution, Aequationes Math. 54 (1997), no. 1–2, 144–172.
H. Stetkær, Functional Equations on Groups, World Scientific Publishing Co. Pte. Ltd., Singapore, 2013.
H. Stetkær, The kernel of the second order Cauchy difference on semigroups, Aequationes Math. 91 (2017), no. 2, 279–288.
Department of Mathematics, Faculty of Sciences, IBN TOFAIL University Morocco
https://orcid.org/0000-0001-8468-3408
Department of Mathematics, Faculty of Sciences, Chouaib Doukkali University 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.
