Solution of Several Functional Equations on Nonunital Semigroups Using Wilson’s Functional Equations with Involution

Abstract

Let S be a nonunital commutative semigroup, $\sigma :S\to S$ an involution, and $\mathbb{C}$ the set of complex numbers. In this paper, first we determine the general solutions $f,g:S\to \mathbb{C}$ of Wilson’s generalizations of d’Alembert’s functional equations $f\left(x+y\right)+f\left(x+\sigma y\right)=2f(x)g(y)$ and $f\left(x+y\right)+f\left(x+\sigma y\right)=2g(x)f(y)$ on nonunital commutative semigroups, and then using the solutions of these equations we solve a number of other functional equations on more general domains.