Differentiability for the high dimensional polynomial-like iterative equation (Q1774247)

The authors study the functional equation \[ \sum_{i=1}^\infty \lambda_i f^{i}(x)=F(x)\quad(x\in B) \] where the function \(F\) and \(\lambda_1>0\), \(\lambda_i\geq 0\) \((i=2,3,\dots)\), \(\sum_{i=1}^\infty \lambda_i=1\) are given, \(f^{k}\) is the \(k\)-th iterate of the unknown function \(f\) and \(B\) is a compact convex subset of \({\mathbb R}^N.\) Improving the methods of \textit{W. Zhang} [Nonlinear Anal., Theory Methods Appl. 15, 387--398 (1990; Zbl 0717.39005)] and \textit{M. Kulczycki} and \textit{J. Tabor} [Aequationes Math. 64, 24--33 (2002; Zbl 1009.39021)] existence, uniqueness and stability of continuously differentiable solutions are given.