Sharp algebraic periodicity conditions for linear higher order difference equations

Sharp algebraic periodicity conditions for linear higher order difference equations

In this paper we give easily verifiable, but sharp (in most cases necessary and sufficient) algebraic conditions for the solutions of systems of higher order linear difference equations to be periodic. The main tool in our investigation is a transformation, recently introduced by the authors, which formulates a given higher order recursion as a first order difference equation in the phase space...

General Permanence Conditions for Nonlinear Difference Equations of Higher Order

w . m w . w . where f : 0, ` a 0, ` and the initial values x , . . . , x g 0, ` . 0 1ym Ž . In studying the global behavior of the solutions of Eq. 1 , we often need Ž . to establish that 1 is permanent, i.e., it has the property that every one of its solutions is eventually confined within a fixed compact interval regardless of the initial values chosen. Permanence is needed directly or indire...

On explicit conditions for the asymptotic stability of linear higher order difference equations

We derive some explicit sufficient conditions for the asymptotic stability of the zero solution in a general linear higher order difference equation, and compare our estimations with other related results in the literature. Our main result also applies to some nonlinear perturbations satisfying a kind of sublinearity condition.  2004 Elsevier Inc. All rights reserved.

Reduction of order, periodicity and boundedness in a class of nonlinear, higher order difference equations

Asymptotic Periodicity of a Higher-Order Difference Equation

We give a complete picture regarding the asymptotic periodicity of positive solutions of the following difference equation: xn = f (xn−p1 , . . . ,xn−pk ,xn−q1 , . . . ,xn−qm), n∈N0, where pi, i ∈ {1, . . . ,k}, and qj , j ∈ {1, . . . ,m}, are natural numbers such that p1 < p2 < ··· < pk, q1 < q2 < ··· < qm and gcd(p1, . . . , pk,q1, . . . ,qm) = 1, the function f ∈ C[(0,∞), (α,∞)], α > 0, is i...