Asymptotic Expansions for Higher-Order Scalar Difference Equations

Abstract

We give an asymptotic expansion of the solutions of higher-order Poincaré difference equation in terms of the characteristic solutions of the limiting equation. As a consequence, we obtain an asymptotic description of the solutions approaching a hyperbolic equilibrium of a higher-order nonlinear difference equation with sufficiently smooth nonlinearity. The proof is based on the inversion formula for the z-transform and the residue theorem.

Zeros, growth and Taylor coefficients of entire solutions of linear q-difference equations

“…The proof is based on results of Agarwal and Pituk [3] as well as Bodine and Lutz [9]; see Lemma 2.1 below. In the case that K = 1 and thus j 0 = 0 and j 1 = m we can apply their results directly, but the general case requires some extensions of their arguments; see Lemma 2.2.…”

Exponentially asymptotically constant systems of difference equations with an application to hyperbolic equilibria

“…[1], we first note a scalar equation (4) can be expressed as a system (1) in the standard way and that a solution of the scalar equation is the first component of the corresponding vector-valued solution of the system. First, in the diagonal case, we introduce solutions w i ðnÞ ¼ l n i e i of the unperturbed diagonal system w(n þ 1) ¼ Lw(n) and rewrite (10) as…”

“…We will use more standard methods from matrix analysis to derive our result, which contains that of Agarwal/Pituk as a special case, but also strengthens it in the sense of obtaining more precise estimates for the error term by finding bounds on d. In Ref. [1], Agarwal and Pituk have also applied their result on linear systems to obtain an asymptotic representation for solutions of certain non-linear scalar difference equations in the neighbourhood of a hyperbolic equilibrium (Theorem 3.1). We consider analogous nonlinear systems of difference equations and show that 'linearization' of the system in a neighbourhood of an equilibrium solution leads to a weakly nonlinear system for the perturbation, yðn þ 1Þ ¼ AyðnÞ þ Gðn; yðnÞÞ:…”

“…As mentioned in the Introduction, Agarwal and Pituk [1] applied their result for solutions of (4) to obtain an asymptotic representation for solutions of certain nonlinear autonomous scalar difference equations in the neighbourhood of a hyperbolic equilibrium. Here, we apply Theorem 3 to obtain an analogous result for nonlinear autonomous systems.…”

“…Our results are motivated by a recent paper by Agarwal and Pituk [1], who considered asymptotically constant scalar difference equations of the form uðn þ kÞ þ ½a 1 þ b 1 ðnÞuðn þ k 2 1Þ þ · · · þ ½a k þ b k ðnÞuðnÞ ¼ 0;…”