Global solutions for leading coefficient problem of a general iterative equation
type: Item
instance of: scholarly article
title: Global solutions for leading coefficient problem of a general iterative equation
zbMATH Open document ID: 1331.39013
author: Wei Song
author: Lin Li
published in: Results in Mathematics
publication date: 2015-09-28T00:00:00Z
review text: The paper deals with the iterative equation \[ G(f(x),f^2(x),\dots,f^n(x))=F(x), \leqno(*) \] where \(x \in I:=[a,b]\) and \(f:I \to J\), \(J\) a real interval. The product of \(n\) intervals \(I\) is denoted by \(I^n\) and \(\mathbf a=(a,a,\dots,a)\), \(\mathbf b=(b,b,\dots,b)\), \(\mathbf x=(x_1,x_2,\dots,x_n)\); \(G_i^{\prime}(\mathbf x):=\frac{\partial{G}}{\partial{x_i}}(\mathbf x)\). The following properties on the given function \(G\) are requested. {\parindent=10mm \begin{itemize}\item[(G1)] \(G\) is increasing with respect to each variable \(x_i\), \(i=1,2,\dots,n-1\), and strictly increasing with respect to \(x_n\); \item[(G2)] for every \(\mathbf x, \mathbf y \in I^n\), it is \[ 0\leq G_i^{\prime}(\mathbf x)\leq G_i^{\prime}(\mathbf a) \quad \text{and} \quad |G_i^{\prime}(\mathbf x)-G_i^{\prime}(\mathbf y)|\leq \sum_{j+1}^n N_{ij}| x_j-y_j|, \] where \(N_{ij}\) are nonnegative constants; \item[(G3)] For every \(\mathbf x, \mathbf y \in I^n\), it is \[ 0\leq G_i^{\prime}(\mathbf x)\leq G_i^{\prime}(\mathbf b) \quad \text{and} \quad |G_i^{\prime}(\mathbf x)-G_i^{\prime}(\mathbf y)|\leq \sum_{j+1}^n M_{ij}| x_j-y_j|, \] where \(M_{ij}\) are nonnegative constants. \end{itemize}} Put \[ \mathcal G_1(I):=\{G\mid G:I^n \to \mathbb R \text{ satisfies (G1) and (G2)} \} \] and \[ \mathcal G_2(I):=\{G\mid G:I^n \to \mathbb R \text{ satisfies (G1) and (G3)} \}. \] Similarly, the following properties on \(F\) are needed. Define \[ \mathcal H_+(I,J):=\{f:I \to J\mid f \text{ is a strictly increasing homeomomorphism} \}; \] \[ \mathcal A_G[I]:=\{F \in \mathcal H_+(I,J)\mid F(x)<G(x,x,\dots,x), \quad x \in (a,b) \}; \] \[ \mathcal B_G[I]:=\{F \in \mathcal H_+(I,J)\mid F(x)>G(x,x,\dots,x), \quad x \in (a,b) \}; \] {\parindent=10mm \begin{itemize}\item[(F1)] \(F \in \mathcal A_G[I]\); \item[(F2)] \(F^{\prime}\) is Lipschitz on \(I\) and \(F^{\prime}(x)\geq F^{\prime}(a)>0\) for all \(x \in I\); \item[(F3)] \(F \in \mathcal B_G[I]\); \item[(F4)] \(F^{\prime}\) is Lipschitz o…
reviewed by: Gian Luigi Forti
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 26A18
zbMATH DE Number: 6485806
zbMATH Keywords: iterative functional equations
zbMATH Keywords: leading coefficient problem
zbMATH Keywords: Schroeder transformation
zbMATH Keywords: fixed point theorem
zbMATH Keywords: continuous solution
MaRDI profile type: Publication
full work available at URL: s00025-014-0432-0
OpenAlex ID: W2155371610
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DOI: 10.1007/S00025-014-0432-0
description: scientific article; zbMATH DE number 6485806