On the second part of Hilbert's fifth problem
The following system of functional equations \[ \begin{aligned} \varphi(u+v)\varphi(u-v) &= \varphi(u)^ 2 f(v)^ 2-\varphi(v)^ 2 f(u)^ 2 \tag{1}\\ f(u+v)f(u-v) &= f(u)^ 2 f(v)^ 2-c^ 2\varphi(u)^ 2\varphi(v)^ 2\tag{2}\end{aligned} \] has been solved by N. H. Abel, under suitable differentiability assumptions on \(\varphi:\mathbb{C}\to \mathbb{C}\) and \(f:\mathbb{C}\to\mathbb{C}\). Similarly H. Haruki found the general solution of the first above equation. In this paper the author solves the system (1) and (2) replacing Abel's differentiability assumptions by weaker assumptions. He finds the general solutions \(\varphi: \mathbb{R}^ n\to\mathbb{C}\) and \(f: \mathbb{R}^ n\to\mathbb{C}\) of the system (1) and (2) where only the continuity of \(\varphi\), \(\varphi\not\equiv 0\), is supposed. If \(\varphi=0\) then the system (1)-- (2) reduces to \(f(u+v)f(u-v)=f(u)^ 2f(v)^ 2\), \(u,v\in\mathbb{R}^ n\). The continuous solutions \(f:\mathbb{R}^ n\to\mathbb{C}\) of this functional equation are determined. The paper is connected with recent work of \textit{J. Aczél} [The state of the second part of Hilbert's fifth problem, Bull. Am. Math. Soc., New Ser. 20, No. 2, 153-163 (1989; Zbl 0676.39004)].
scientific article; zbMATH DE number 126394