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Rassias, Th.M., Elqorachi, Elhoucien, Redouani, Ahmed. (1395). Solutions and stability of variant of Van Vleck's and D'Alembert's functional equations. هنرهای کاربردی, 7(2), 279-301. doi: 10.22075/ijnaa.2017.1803.1472
Th.M. Rassias; Elhoucien Elqorachi; Ahmed Redouani. "Solutions and stability of variant of Van Vleck's and D'Alembert's functional equations". هنرهای کاربردی, 7, 2, 1395, 279-301. doi: 10.22075/ijnaa.2017.1803.1472
Rassias, Th.M., Elqorachi, Elhoucien, Redouani, Ahmed. (1395). 'Solutions and stability of variant of Van Vleck's and D'Alembert's functional equations', هنرهای کاربردی, 7(2), pp. 279-301. doi: 10.22075/ijnaa.2017.1803.1472
Rassias, Th.M., Elqorachi, Elhoucien, Redouani, Ahmed. Solutions and stability of variant of Van Vleck's and D'Alembert's functional equations. هنرهای کاربردی, 1395; 7(2): 279-301. doi: 10.22075/ijnaa.2017.1803.1472

Solutions and stability of variant of Van Vleck's and D'Alembert's functional equations

International Journal of Nonlinear Analysis and Applications
مقاله 24، دوره 7، شماره 2، اسفند 2016، صفحه 279-301    اصل مقاله (456.41 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22075/ijnaa.2017.1803.1472
نویسندگان
Th.M. Rassias* 1؛ Elhoucien Elqorachi2؛ Ahmed Redouani2
1Department of Mathematics, National Technical University of Athens, Zofrafou Campus, 15780 Athens, Greece
2Ibn Zohr University, Faculty of Sciences Department of Mathematic, Agadir, Morocco
تاریخ دریافت: 29 آذر 1394،  تاریخ بازنگری: 26 مرداد 1395،  تاریخ پذیرش: 19 آبان 1395 
چکیده
In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation
$$\int_{S}f(\sigma(y)xt)d\mu(t)-\int_{S}f(xyt)d\mu(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a semigroup, $\sigma$ is an involutive morphism of $S$, and $\mu$ is a complex measure that is linear combinations of Dirac measures $(\delta_{z_{i}})_{i\in I}$, such that for all $i\in I$, $z_{i}$ is contained in the center of $S$. (2) We determine the complex-valued continuous solutions of the following variant of d'Alembert's functional equation
$$\int_{S}f(xty)d\upsilon(t)+\int_{S}f(\sigma(y)tx)d\upsilon(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a topological semigroup, $\sigma$ is a continuous involutive automorphism of $S$, and $\upsilon$ is a complex measure with compact support and which is $\sigma$-invariant. (3) We prove the superstability theorems of the first functional equation.
کلیدواژه‌ها
semigroup؛ d'Alembert's equation؛ Van Vleck's equation, sine function؛ involution؛ multiplicative function, homomorphism, superstability
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