Asymptotic integration of second-order nonlinear difference equations

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Ehrnström, M.; Tisdell, C.C.; Wahlén, E.: Asymptotic integration of second-order nonlinear difference equations. In: Glasgow Mathematical Journal 53 (2011), Nr. 2, S. 223-243. DOI: https://doi.org/10.1017/S0017089510000650

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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. © 2011 Glasgow Mathematical Journal Trust.
License of this version: Es gilt deutsches Urheberrecht. Das Dokument darf zum eigenen Gebrauch kostenfrei genutzt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. Dieser Beitrag ist aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.
Document Type: article
Publishing status: publishedVersion
Issue Date: 2011
Appears in Collections:Fakultät für Mathematik und Physik

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1 image of flag of Germany Germany 25 78.12%
2 image of flag of United States United States 4 12.50%
3 image of flag of Italy Italy 1 3.12%
4 image of flag of France France 1 3.12%
5 image of flag of Austria Austria 1 3.12%

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