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

Let S be a nonunital commutative semigroup, sigma : S -> S an involution, and C the set of complex numbers. In this paper, first we determine the general solutions f, g : S -> C of Wilson's generalizations of d'Alembert's functional equations f(x + y) + f(x + sigma y) = 2 f(x)g(y) and f(x + y) + f(x + sigma y) = 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.