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Asymptotic Integration Of Second-Order Nonlinear Difference Equations

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Author

Summary, in English

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.

Department/s

Publishing year

2011

Language

English

Pages

223-243

Publication/Series

Glasgow Mathematical Journal

Volume

53

Document type

Journal article

Publisher

Cambridge University Press

Topic

  • Mathematics

Status

Published

Research group

  • Partial differential equations

ISBN/ISSN/Other

  • ISSN: 0017-0895