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ASYMPTOTIC INTEGRATION OF SECOND-ORDER NONLINEAR DIFFERENCE EQUATIONS

Published online by Cambridge University Press:  08 December 2010

MATS EHRNSTRÖM ,
CHRISTOPHER C. TISDELL  and
ERIK WAHLÉN
MATS EHRNSTRÖM
Affiliation:
Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany. email: ehrnstrom@ifam.uni-hannover.de
CHRISTOPHER C. TISDELL
Affiliation:
School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia. email: cct@unsw.edu.au
ERIK WAHLÉN
Affiliation:
Centre for Mathematical Sciences, Lund University, PO Box 118, 221 00 Lund, Sweden. email: erik.wahlen@math.lu.se
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Abstract

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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.

Keywords

39A1234D05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 2 , May 2011 , pp. 223 - 243
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

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