Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-68227220161220Solutions and stability of variant of Van Vleck's and D'Alembert's functional equations27930177410.22075/ijnaa.2017.1803.1472ENTh.M. RassiasDepartment of Mathematics, National Technical University of Athens, Zofrafou Campus, 15780 Athens, GreeceElhoucien ElqorachiIbn Zohr University, Faculty of Sciences
Department of Mathematic, Agadir, MoroccoAhmed RedouaniIbn Zohr University, Faculty of Sciences
Department of Mathematic, Agadir, MoroccoJournal Article20151220In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation<br />$$\int_{S}f(\sigma(y)xt)d\mu(t)-\int_{S}f(xyt)d\mu(t) = 2f(x)f(y), \;x,y\in S,$$ 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<br />$$\int_{S}f(xty)d\upsilon(t)+\int_{S}f(\sigma(y)tx)d\upsilon(t) = 2f(x)f(y), \;x,y\in S,$$ 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.https://ijnaa.semnan.ac.ir/article_774_ac5ba88e6d8ed3f180cc2ff75a074111.pdf