Asymptotics of families of solutions of nonlinear difference equations
type: Item
instance of: scholarly article
title: Asymptotics of families of solutions of nonlinear difference equations
zbMATH Open document ID: 1170.39006
published in: Logic and Analysis
publication date: 2009-02-11T00:00:00Z
full work available at URL: http://hdl.handle.net/10174/1393
review text: This article deals with first-order scalar difference equations. In a conventional notation they read as \[ y_{k+1}=f(k,y_k);\tag{\(\ast\)} \] note that these problems can be nonautonomous and nonlinear, where the right-hand side \(f\) is supposed to be continuously partially differentiable in the second argument. In the framework of Nonstandard Analysis, the author provides conditions for the existence of solutions to \((\ast)\) which have an asymptotic behavior given by \(\hat y_k\). Here, the family \(\hat y_k\) of ``asymptotic fixed points'' satisfies the so-called asymptotic functional equation \[ \lim_{k\to\infty}\frac{f(k,\hat y_k)-\hat y_k}{\hat y_k(|D_2f(k,\hat y_k)|-1)} = 0. \tag{\(\ast\ast\)} \] Under additional regularity conditions on \(f\) such asymptotic fixed points are denoted as approximate solution by the author. Various kinds of local asymptotic behavior of true solutions are presented using the notions of a (strongly, moderately, weakly exponential) attractive/repulsive river; a precise definition involves the notion of telescopes and macroscopes. In Sections 3 and 6 attractive and repulsive rivers are characterized in terms of the right-hand side \(f\) and the sequence \(\hat y_k\). Section 4 contains a convenient lemma which states that although a nonlinear difference equation \((\ast)\) in general lacks the property of uniqueness of solutions, it is possible to move back in time for a subclass of its solutions. Additionally, it is shown that repulsive solutions are unique in their asymptotic direction. Effects of the rescalings on the behavior of the equation and its solutions are treated in Section 5. Proofs of the main results are postponed to Section 6. In the final Section 7, the author considers some examples of quadratic equations, a natural class of drains which admits a simple formula for deviations and finally the question whether the river or drain itself satisfies the asymptotic functional equation \((\ast\ast)\).
Mathematics Subject Classification ID: 39A30
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 26E35
zbMATH DE Number: 5506510
zbMATH Keywords: difference equations
zbMATH Keywords: asymptotics
zbMATH Keywords: stability
zbMATH Keywords: rivers
zbMATH Keywords: drains
zbMATH Keywords: nonstandard analysis
zbMATH Keywords: change of scale
zbMATH Keywords: asymptotic fixed points
zbMATH Keywords: asymptotic functional equation
zbMATH Keywords: attractive and repulsive rivers
zbMATH Keywords: quadratic equations
author: I. P. Van den Berg
reviewed by: Christian Pötzsche
MaRDI profile type: Publication
OpenAlex ID: W2037493434
cites work: Q4734516
cites work: Asymptotic Representation of Solutions of Perturbed Systems of Linear Difference Equations
cites work: On solutions of polynomial growth of ordinary differential equations
cites work: Q4345280
cites work: Extended use of IST
cites work: On the relation between elementary partial difference equations and partial differential equations
cites work: Q3993282
cites work: Nonstandard analysis in practice
cites work: Q4730102
cites work: Q4679023
cites work: Analytic solutions of difference equations with small step size*
cites work: Asymptotic solutions of systems of nonlinear difference equations
cites work: On asymptotic solutions of systems of nonlinear difference equations.
cites work: Asymptotics of analytic difference equations
cites work: Q4821145
cites work: Q4531875
cites work: Internal set theory: A new approach to nonstandard analysis
cites work: Q4870980
DOI: 10.1007/S11813-008-0006-0
description: scientific article; zbMATH DE number 5506510