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

TL;DR

This paper investigates solutions to integral Van Vleck's and Kannappan's functional equations on semigroups, revealing their connections to multiplicative functions and classical d'Alembert's equation, with solutions expressed via measures and involutions.

Contribution

It provides a characterization of solutions to these integral functional equations on semigroups, linking them to multiplicative functions and classical equations, extending previous results.

Findings

Solutions to the Van Vleck's equation are expressed via multiplicative functions.

Solutions to the Kannappan's equation relate closely to d'Alembert's functional equation.

The equations are studied on semigroups with measures supported on central elements.

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.