On a Variant of μ-Wilson’s Functional Equation with an Endomorphism

The main goal of this chapter is to find the solutions (f, g) of the generalized variant of μ-d’Alembert’s functional equation $$\displaystyle f(xy)+\mu (y)f(\varphi (y)x)=2f(x)f(y), $$ and μ-Wilson’s functional equation $$\displaystyle f(xy)+\mu (y)f(\varphi (y)x)=2f(x)g(y), $$ in the setting of semigroups, monoids, and groups, where φ is an endomorphism not necessarily involutive and μ is a multiplicative function. We prove that their solutions can be expressed in terms of multiplicative and additive functions. Many consequences of these results are presented.