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

10.22075/ijnaa.2023.31988.4745

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.
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[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.

Articles in Press, Corrected Proof
Available Online from 25 December 2023
  • Receive Date: 05 September 2023
  • Revise Date: 01 November 2023
  • Accept Date: 09 November 2023