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، 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 | ||
| 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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