Solutions of linear iterative functional equations behaving like the forcing term in the equation
type: Item
instance of: scholarly article
title: Solutions of linear iterative functional equations behaving like the forcing term in the equation
zbMATH Open document ID: 0972.39014
published in: Aequationes Mathematicae
publication date: 2001-11-09T00:00:00Z
review text: The author considers the functional equation \[ \varphi \circ f=g\cdot \varphi+h \tag{*} \] where \(\varphi\) is the unknown function, and studies continuous solutions which are asymptotically comparable at the origin with the function \(h\). The general assumptions are the following. 1) \(I\) is an interval with \(0\) as left-end point (belonging or not to \(I\)), \(f:I \to I\) is continuous, strictly increasing and \(0<f(x)<x\), for \(x\in I\), \(x>0\); 2) \(h,g:I \to \mathbb R\) are continuous, \(h(0)=0\), \(h(x)\) and \(g(x)\) are different from zero for \(x\in I\), \(x>0\). Theorem: If the previous assumptions hold, the limit \[ q:=\lim_{x \to 0^+} \frac{h(f(x))}{h(x)} \] exists and \(0<q<|g(0)|\), then equation (*) has a unique continuous solution \(\varphi\) such that the limit \[ L_{\varphi}:=\lim_{x \to 0^+} \frac{\varphi(x)}{h(x)} \] exists and \(L_{\varphi}=(q-g(0))^{-1}\). The previous theorem is applied to determine a unique solution to the iterative functional inequality \[ \varphi(f^2(x))\leq (p(x)+q(f(x)))\psi(f(x))+p(x)q(x)\psi(x) . \]
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 39B62
zbMATH DE Number: 1556606
zbMATH Keywords: linear iterative functional equations
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: continuous solution
zbMATH Keywords: iterative functional inequality
author: Bogdan A. Choczewski
reviewed by: Gian Luigi Forti
MaRDI profile type: Publication
full work available at URL: s000100050157
OpenAlex ID: W1988715058
DOI: 10.1007/S000100050157
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description: scientific article; zbMATH DE number 1556606