D’Alembert’s Functional Equations on Monoids with an Anti-endomorphism

Let M be a topological monoid. Our main goal is to introduce the functional equation g(xy)+μ(y)g(xψ(y))=2g(x)g(y),x,y∈M,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g(xy)+\mu (y)g(x\psi (y))=2g(x)g(y),\ \ x,y\in M, \end{aligned}$$\end{document}where ψ:M→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi :M\rightarrow M$$\end{document} is a continuous anti-endomorphism that need not be involutive and μ:M→C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu :M\rightarrow \mathbb {C}$$\end{document} is a continuous multiplicative function such that μ(xψ(x))=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (x\psi (x))=1$$\end{document} for all x∈M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in M$$\end{document}. We exploit results on the pre-d’Alembert functional equation by Davison (Publ Math Debrecen 75(1–2):41–66, 2009) and Stetkær (Functional equations on groups. World Scientific Publishing Company, Singapore (2013)) to prove that the continuous solutions g:M→C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ g:M\rightarrow \mathbb {C}$$\end{document} of this equation can be expressed in terms of multiplicative functions and characters of 2-dimensional representations of M. Interesting consequences of this result are presented.