On continuous and smooth solutions of the Schröder equation in normed spaces
The author considers the Schröder functional equation \[ \varphi(f(x))=A\varphi(x) \tag{S} \] and proves the following existence theorem for smooth solutions. Theorem. Let \(X, Y\) be normed spaces, \(m \in \{0\}\cup \mathbb N \cup \{\infty\}\) and \(D\) be a nonempty, open if \(m>0\), subset of \(X\). Assume the following conditions: {\parindent=6mm \begin{itemize} \item[i)] \(A:Y \to Y\) is a linear bijection such that \(A\) and \(A^{-1}\) are bounded; \item [ii)] \(f:D \to D\) is a bijection such that \(f\) and \(f^{-1}\) are of class \(C^m\); \item [iii)] \(a>0\) and \(l:D \to \mathbb R\) of class \(C^m\) are such that \(l(f(x)) \geq l(x)+a\) for \(x \in D\). \end{itemize}} Then the equation (S) has a nonzero sulution \(\varphi:D \to Y\) of class \(C^m\). Moreover, if the spectral radius of \(A\) is less than \(1\), then \(\varphi\) is bounded on every set of the form \(l^{-1}((t,\infty))\), for \(t \in \mathbb R\) .
scientific article; zbMATH DE number 5294655