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

Abstract

Let S be a nonunital commutative semigroup, σ: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 fx+y+fx+σy=2f(x)g(y) and fx+y+fx+σy=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.