Reducing the polynomial-like iterative equations order and a generalized Zoltán Boros' problem (Q329385)

The authors consider the functional equation \[ a_Ng^N(x)+\cdots +a_1g(x)+a_0x=0, \leqno(1) \] where \(g:I \to I\) (\(I\) a non-degenerate interval) is continuous, \(N\) is a positive integer, the coefficients \(a_0,a_1,\dots,a_N\) are real and \(a_0 \neq 0\); \(g^n\) is the \(n\)-th iterate of \(g\). In the literature there are many results aimed to find the solutions of the previous equation which are based on lowering the order of the equation. The main goal of the paper under review consists in proving a general procedure for reducing the order of the equation. To Equation (1), its characteristic equation \[ a_Nr^N+\cdots +a_1r+a_0=0; \] is associated, moreover, for a given polynomial \(c_Kr^K+\cdots +c_1r+c_0\), \(\mathcal{R}(c_K,\dots,c_0)\) denotes the set \(\{(r_1,k_1),\dots,(r_p,k_p)\}\) of all pairwise distinct roots \(r_1,\dots,r_p\) and their multiplicities \(k_1,\dots, k_p\), respectively. One of the main results of the paper is given by the following theorem. Theorem. Assume that \[ \mathcal{R}(a_N,\dots,a_0)=\{(\lambda_1,\ell_1),\dots,(\lambda_p,\ell_p),(\mu_1,k_1),(\bar{\mu}_1,k_1),\cdots,(\mu_q,k_q),(\bar{\mu}_q,k_q)\}. \] If \[ |\lambda_1|\leq\cdots \leq |\lambda_p|<|\mu_1|\leq\cdots \leq |\mu_q|, \leqno(*) \] then a continuous function \(g:I \to I\) satisfies Equation (1) if and only if it satisfies the equation \[ b_Mg^M(x)+\cdots +b_1g(x)+b_0x=0 \] with \[ \mathcal{R}(b_M,\dots,b_0)=\{(\lambda_1,\ell_1),\dots,(\lambda_p,\ell_p)\}. \] The same is true when in (\(*\)) all inequalities are reversed. Analogous results are proved for other inequalities similar to (\(*\)). As an application the authors solve the functional equation \[ f^n(x)=\frac{[f(x)]^n}{x^{n-1}} \] proposed by Z. Boros during the 50th International Symposium on Functional Equations.