A Fully Pexiderized Variant of d’Alembert’s Functional Equations on Monoids

Abstract

We solve the functional equation $$f(xy) + g(\sigma (y)x) = h(x)k(y)$$ for complex-valued functions f, g, h, k on groups or monoids generated by their squares, where $$\sigma $$ is an involutive automorphism. This contains both classical d’Alembert equations $$g(x + y) + g(x - y) = 2g(x)g(y)$$ and $$f(x + y) - f(x - y) = g(x)h(y)$$ in the abelian case, but we do not suppose our groups or monoids are abelian. We also find the continuous solutions on topological groups and monoids.