Integral Van Vleck's and Kannappan's functional equations on semigroups

Abstract

In this paper we study the solutions of the integral Van Vleck's functional equation for the sine $$\int_{S}f(x\tau(y)t)d\mu(t)-\int_{S}f(xyt)d\mu(t) =2f(x)f(y),\; x,y\in S$$ and the integral Kannappan's functional equation $$\int_{S}f(xyt)d\mu(t)+\int_{S}f(x\tau(y)t)d\mu(t) =2f(x)f(y),\; x,y\in S,$$ where $S$ is a semigroup, $\tau$ is an involution of $S$ and $\mu$ is a measure that is linear combinations of point measures $(\delta_{z_{i}})_{i\in I}$, such that for all $i\in I$, $z_{i}$ is contained in the center of $S$. \\ We express the solutions of the first equation by means of multiplicative functions on $S$, and we prove that the solutions of the second equation are closely related to the solutions of the classic d'Alembert's functional equation with involution.