The composition of solutions of the multidimensional translation equation (Q686701)

For an integer \(n\geq 2\), let \(\rho,\sigma,\tau\) be integers from \(\{0,\dots,n\}\). Let \(J_ \tau=[I_ \tau,0]\) be a \(\tau\times n\)-block- matrix, where \(I_ \tau\) is the \(\tau\)-dimensional unit matrix and 0 the \(\tau\times(n-\tau)\)-dimensional zero matrix. The paper deals with the solution of the functional equation \((*)\) \(h\{wJ_ \sigma+h^{-1}[uJ_ \rho+h(p)]\}=uJ_ \rho+h(wJ_ \sigma+p)\) with respect to a continuously differentiable function \(h:\mathbb{R}^ n\to\mathbb{R}^ n\), where \(u\in\mathbb{R}^ \rho\), \(w\in\mathbb{R}^ \sigma\) and \(p\in\mathbb{R}^ n\) are variable vectors, and \(h^{-1}\) is the inverse of \(h\). Equation \((*)\) appears in connection with the multidimensional translation equation, cf. \textit{J. Aczél}, \textit{L. Berg} and \textit{Z. Moszner} [ibid. 19, No. 3/4, 195-210 (1991; Zbl 0724.39008)].