Solutions and stability of generalized Kannappan's and Van Vleck's functional equations

TL;DR

This paper investigates solutions to generalized integral functional equations on semigroups involving involutive automorphisms, establishing their relation to d'Alembert's equation and proving superstability results.

Contribution

It introduces new solutions to integral Kannappan's and Van Vleck's equations on semigroups with involutive automorphisms, linking them to classical d'Alembert's equation and demonstrating superstability.

Findings

Solutions are related to d'Alembert's functional equation.

Functional equations are shown to be superstable.

Results extend to equations with involutive morphisms.

Abstract

We study the solutions of the integral Kannappan's and Van Vleck's functional equations $$\int_{S}f(xyt)d\mu(t)+\int_{S}f(x\sigma(y)t)d\mu(t) = 2f(x)f(y), \;x,y\in S;$$ $$\int_{S}f( x\sigma(y)t)d\mu(t)-\int_{S}f(xyt)d\mu(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a semigroup, $\sigma$ is an involutive automorphism of $S$ and $\mu$ is a linear combination of Dirac measures $(\delta_{z_{i}})_{i\in I}$, such that for all $i\in I$, $z_{i}$ is contained in the center of $S$. We show that the solutions of these equations are closely related to the solutions of the d'Alembert's classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems that these functional equations are superstable in the general case, where $\sigma$ is an involutive morphism.