International Journal of Nonlinear Analysis and Applications

A trigonometric functional equation with an automorphism

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco

2 Department of Mathematics, National Technical University of Athens, Athens, Greece

Abstract

Let $S$ be a semigroup. In the present paper, we determine the complex-valued solutions $(f,g)$ of the functional equation
$$g(x\sigma (y)) = g(x)g(y)-f(x)f(y)+\alpha f(x\sigma(y)),\  x,y\in S,$$
where $\sigma :S\rightarrow S$ is an automorphism that need not be involutive, and $\alpha \in \mathbb{C}$ is a fixed constant. Our results generalize and extend the ones by Stetk\ae r in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented.

Keywords

[1] J. Aczel, Lectures on Functional Equations and their Applications, Academic Press, New York, 1966.
[2] Y. Aserrar and E. Elqorachi, Five Trigonometric addition laws on semigroups, https://doi.org/10.48550/arXiv.2210.06181
[3] Y. Aserrar and E. Elqorachi, Cosine and Sine addition and subtraction law with an automorphism, https://doi.org/10.48550/arXiv.2302.10263
[4] Aserrar, Y., Elqorachi, E., A generalization of the cosine addition law on semigroups. Aequat Math. 97  (2023), 787–804.
[5] S. Butler, Problem no. 11030, Amer. Math. Month. 110 (2003), no. 7, 637–639.
[6] B. Ebanks, Some trigonometric functional equations on monoids generated by their squares, Aequat. Math. 95 (2021), no. 2, 383–391.
[7] S.-M. Jung, M.Th. Rassias, and C. Mortici, On a functional equation of trigonometric type, Appl. Math. Comput. 252 (2015), 294–303.
[8] M.Th. Rassias, Solution of a functional equation of Steven Butler, Octogon Math. Mag. 12 (2004), 152–153.
[9] H. Stetkar, Functional Equations on Groups, World Scientific Publishing CO, Singapore, 2013.
[10] H. Stetkar, The cosine addition law with an additional term, Aequat Math. 90 (2016), no. 6, 1147–1168.
[11] H. Stetkar, A Levi-Civita functional equation on semigroups, Aequat Math. 96 (2022), 115–127.
[12] S.-E. Takahasi, T. Miura, and H. Takagi, Exponential type functional equation and its Hyers–Ulam stability, J. Math. Anal. Appl. 329 (2007), 1191–1203.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 11
November 2024
Pages 403-415
  • Receive Date: 05 September 2023
  • Revise Date: 01 November 2023
  • Accept Date: 09 November 2023