Extensions of Kannappan’s and Van Vleck’s Functional Equations on Semigroups

Abstract

This paper treats two functional equations, the Kannappan-Van Vleck functional equation

$$\displaystyle \mu (y)f(x\tau (y)z_0)\pm f(xyz_0) =2f(x)f(y), \;x,y\in S $$

and the following variant of it

$$\displaystyle \mu (y)f(\tau (y)xz_0)\pm f(xyz_0) = 2f(x)f(y), \;x,y\in S, $$

in the setting of semigroups S that need not be abelian or unital, τ is an involutive morphism of S, μ : S→C is a multiplicative function such that μ(xτ(x)) = 1 for all x ∈ S and z ₀ is a fixed element in the center of S.

We find the complex-valued solutions of these equations in terms of multiplicative functions and solutions of d’Alembert’s functional equation.