On necessary and sufficient conditions for the asymptotic stability of higher order linear difference equations

Two types of stability conditions for linear delay difference equations

“…As we have already noted in the introductory section, the case k even is covered (after some simple rearrangements) by the relevant results of [2]. As we have already noted in the introductory section, the case k even is covered (after some simple rearrangements) by the relevant results of [2].…”

“…The basic theory of linear difference equations implies the equivalent formulation of this stability problem for (1), namely whether the associated characteristic polynomial (2) p(λ) ≡ λ k+2 + α λ k + β λ 2 + γ is of a Schur type, i.e. Stability issues traditionally belong among the key and frequently discussed topics of qualitative analysis of difference equations.…”

“…Note that (5) was considered in the purely delayed case (β = 0), the advanced case (α = 0), as well as in the mixed case (αβ = 0), and its corresponding asymptotic stability properties were reported in [13], [6] and [2], respectively. Note that (5) was considered in the purely delayed case (β = 0), the advanced case (α = 0), as well as in the mixed case (αβ = 0), and its corresponding asymptotic stability properties were reported in [13], [6] and [2], respectively.…”

On a three term linear difference equation with complex coefficients

“…Recently, an alternative system of conditions appeared in [1] (the case α, β are real numbers) and in [2] (the case α, β are complex numbers). Although both these systems are different from those of Theorem 1.1 and Theorem 1.2, some comparisons of the relevant results lead us to the hypothesis that condition (3), (4) of Theorem 1.2 is superfluous and can be in the corresponding system replaced simply by |β| < 1.…”

“…. (1) with either real or complex α, β and a positive integer k ≥ 2. If α, β are real constants, then the system of necessary and sufficient conditions for the asymptotic stability of (1) was first derived by K u r u k l i s [10].…”