%0 Journal Article
%T Solutions and stability of variant of Van Vleck's and D'Alembert's functional equations
%J International Journal of Nonlinear Analysis and Applications
%I Semnan University
%Z 2008-6822
%A Rassias, Th.M.
%A Elqorachi, Elhoucien
%A Redouani, Ahmed
%D 2016
%\ 12/20/2016
%V 7
%N 2
%P 279-301
%! Solutions and stability of variant of Van Vleck's and D'Alembert's functional equations
%K semigroup
%K d'Alembert's equation
%K Van Vleck's equation, sine function
%K involution
%K multiplicative function, homomorphism, superstability
%R 10.22075/ijnaa.2017.1803.1472
%X 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.
%U https://ijnaa.semnan.ac.ir/article_774_ac5ba88e6d8ed3f180cc2ff75a074111.pdf