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Asymptotic Integration Of Second-Order Nonlinear Difference Equations

Ehrnstroem, Mats; Tisdell, Christopher C. and Wahlén, Erik LU (2011) In Glasgow Mathematical Journal 53. p.223-243
Abstract
In this work we analyse a nonlinear, second-order difference equation on an unbounded interval. We present new conditions under which the problem admits a unique solution that is of a particular linear asymptotic form. The results concern the general behaviour of solutions to the initial-value problem, as well as solutions with a given asymptote. Our methods involve establishing suitable complete metric spaces and an application of Banach's fixed-point theorem. For the solutions found in our two main theorems-fixed initial data and fixed asymptote, respectively-we establish exact convergence rates to solutions of the differential equation related to our difference equation. It turns out that for the asymptotic case there is uniform... (More)
In this work we analyse a nonlinear, second-order difference equation on an unbounded interval. We present new conditions under which the problem admits a unique solution that is of a particular linear asymptotic form. The results concern the general behaviour of solutions to the initial-value problem, as well as solutions with a given asymptote. Our methods involve establishing suitable complete metric spaces and an application of Banach's fixed-point theorem. For the solutions found in our two main theorems-fixed initial data and fixed asymptote, respectively-we establish exact convergence rates to solutions of the differential equation related to our difference equation. It turns out that for the asymptotic case there is uniform convergence for both the solution and its derivative, while in the other case the convergence is somewhat weaker. Two different techniques are utilized, and for each one has to employ ad-hoc methods for the unbounded interval. Of particular importance is the exact form of the operators and metric spaces formulated in the earlier sections. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Glasgow Mathematical Journal
volume
53
pages
223 - 243
publisher
Cambridge University Press
external identifiers
  • wos:000288615900002
  • scopus:82455175688
ISSN
0017-0895
DOI
10.1017/S0017089510000650
language
English
LU publication?
yes
id
de02ebc2-f8a4-4eaa-86f5-93f25e0c173c (old id 1925950)
date added to LUP
2011-05-10 14:51:25
date last changed
2017-02-09 11:55:29
@article{de02ebc2-f8a4-4eaa-86f5-93f25e0c173c,
  abstract     = {In this work we analyse a nonlinear, second-order difference equation on an unbounded interval. We present new conditions under which the problem admits a unique solution that is of a particular linear asymptotic form. The results concern the general behaviour of solutions to the initial-value problem, as well as solutions with a given asymptote. Our methods involve establishing suitable complete metric spaces and an application of Banach's fixed-point theorem. For the solutions found in our two main theorems-fixed initial data and fixed asymptote, respectively-we establish exact convergence rates to solutions of the differential equation related to our difference equation. It turns out that for the asymptotic case there is uniform convergence for both the solution and its derivative, while in the other case the convergence is somewhat weaker. Two different techniques are utilized, and for each one has to employ ad-hoc methods for the unbounded interval. Of particular importance is the exact form of the operators and metric spaces formulated in the earlier sections.},
  author       = {Ehrnstroem, Mats and Tisdell, Christopher C. and Wahlén, Erik},
  issn         = {0017-0895},
  language     = {eng},
  pages        = {223--243},
  publisher    = {Cambridge University Press},
  series       = {Glasgow Mathematical Journal},
  title        = {Asymptotic Integration Of Second-Order Nonlinear Difference Equations},
  url          = {http://dx.doi.org/10.1017/S0017089510000650},
  volume       = {53},
  year         = {2011},
}