Wilson’s functional equation with an anti-endomorphism

Let M be a topological monoid, let $$\psi :M\rightarrow M$$ be a continuous anti-homomorphism of M, and let $$\mu :M\rightarrow {\mathbb {C}}$$ be a continuous multiplicative function such that $$\mu (x\psi (x))=1$$ for all $$ x\in M$$ . We describe, in terms of multiplicative functions, additive functions and characters of 2-dimensional representations of M, the solutions (w, g), where $$w,g:M\rightarrow {\mathbb {C}}$$ , such that w is central and g is continuous, of the new functional equation $$\begin{aligned} w(xy)+\mu (y)w(x\psi (y))=2w(x)g(y),\quad x,y\in M. \end{aligned}$$ We also treat the equation on compact groups and the special equation (when $$ \mu =g=1$$ ): Jensen’s functional equation.