Details

Autor(en) / Beteiligte

• El Rabih, Abir
• Schäfke, Reinhard

Titel

Overstable analytic solutions for non-linear systems of difference equations with small step size containing an additional parameter

Ist Teil von

• Journal of difference equations and applications, 2005-03, Vol.11 (3), p.183-213

Ort / Verlag

Taylor & Francis GroupAbingdon, UK

Erscheinungsjahr

2005

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• We study the existence of analytic solutions of the system of difference equations given by where F is an analytic function of ϵ, x, a and y on a certain neighborhood of a point with We assume the existence of a "slow curve" φ 0 (x) satisfying in a neighborhood of x 0 and Under the assumption that the Jacobian is invertible except at we first show, under some transversality condition, the existence of a unique formal solution and establish its Gevrey-1 character. Then, we construct a quasi-solution, in the sense that it solves our system equation up to an exponentially small remainder. Next, we show the existence of an actual analytic solution (a(ϵ), y(ϵ, x)), defined for ϵ in some sector with vertex at and in some neighborhood of (to be constructed), that is exponentially close to the above quasi-solution. We also show the exponential closeness of any two quasi-solutions (and hence also of actual solutions) tending to the same slow curve. Finally, as an application of this theory, we give two numerical examples with some figures of discrete bounded solutions for two difference equations of second order. Most of this work was done when the author was a scholarship holder of the National Council of Scientific research in Lebanon. schaefke@math.u-strasbg.fr