D'Alembert's functional equation on monoids with both an endomorphism and an anti-endomorphism

Let $M$ be a topological monoid, $e$ its neutral element, let $\mathbb{K}$ denote a division ring of characteristic $\neq 2$, and let $\psi:M\rightarrow M$ (resp. $\varphi:M\rightarrow M$) be a continuous anti-endomorphism (resp. endomorphism) of $M$ as a semigroup. We solve the functional equation $$g(xy)-g(x\psi(y))=cg(x)g(y),\quad x,y\in M,$$ where $g:M\rightarrow\mathbb{K}$ is the unknown function and $c\in\mathbb{K}\backslash\{0\}$. This enables us to find, when $\psi(e)=\varphi(e)=e$, the solutions $g:M\rightarrow\mathbb{C}$ of each of the new functional equations $$g(\varphi(x)y)\pm g(x\psi(y))=2g(x)g(y),\quad x,y\in M.$$ We also find the continuous, complex valued solutions of a Van Vleck's functional equation with an anti-endomorphism on $M$.