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A trigonometric functional equation with an automorphism | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 30، دوره 15، شماره 11، بهمن 2024، صفحه 403-415 اصل مقاله (384.64 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2023.31988.4745 | ||
| نویسندگان | ||
| Youssef Aserrar* 1؛ Elhoucien Elqorachi1؛ Th.M. Rassias2 | ||
| 1Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco | ||
| 2Department of Mathematics, National Technical University of Athens, Athens, Greece | ||
| تاریخ دریافت: 14 شهریور 1402، تاریخ بازنگری: 10 آبان 1402، تاریخ پذیرش: 18 آبان 1402 | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Functional equation؛ Semigroup؛ Addition law؛ Automorphism | ||
| مراجع | ||
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[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. | ||
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