Asymptotic Integration of Differential and Difference Equations

Bodine, Sigrun; Lutz, Donald A.

doi:10.1007/978-3-319-18248-3

Cited by 24 publications

(11 citation statements)

References 96 publications

(185 reference statements)

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“…whereΛ(n) is a diagonal matrix and R(n) is a certain matrix, playing the role of the perturbation term (in our case R(n) ∈ ℓ 1 ). The problem of construction the asymptotics for L -diagonal systems was discussed, e.g., in paper [4] (see also [6]). The main result that can be applied to construct the asymptotics for solutions of Eq.…”

Section: Theorem 23mentioning

confidence: 99%

On the dynamics of certain higher-order scalar difference equation: asymptotics, oscillation, stability

Nesterov¹

2020

Turk J Math

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We construct the asymptotics for solutions of the higher-order scalar difference equation that is equivalent to the linear delay difference equation ∆y(n) = −g(n)y(n − k). We assume that the coefficient of this equation oscillates at the certain level and the oscillation amplitude decreases as n → ∞. Both the ideas of the centre manifold theory and the averaging method are used to construct the asymptotic formulae. The obtained results are applied to the oscillation and stability problems for the solutions of the considered equation.

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Section: Theorem 23mentioning

confidence: 99%

On the dynamics of certain higher-order scalar difference equation: asymptotics, oscillation, stability

Nesterov¹

2020

Turk J Math

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“…For some alternative definitions of an exponential dichotomy for Equation (3), see, e.g., [17][18][19]. In particular, Palmer gives in [18] a finite-time condition for an exponential dichotomy.…”

Section: Remarkmentioning

confidence: 99%

Note on Limit-Periodic Solutions of the Difference Equation xt + 1 − [h(xt) + λ]xt = rt, λ > 1

Andres

Pennequin

2019

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As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar as well as vector cases. The nonlinearity h is not necessarily globally Lipschitzian. Several simple illustrative examples are supplied.

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“…It ha also been generalized in many different ways by the authors of [3][4][5][6][7]. The exposition in [8] conveys well the historical context and the recent development of the theory as well. We first describe the rudiments of the problem, introduce Levinson's Theorem, and state which aspects of the theorem we still want to generalize.…”

Section: Introductionmentioning

confidence: 99%

“…It is remarkable that they also worked out to give the Lemma showing that under suitable conditions there is an upper triangular factorization that converges to a Jordan factorization in the limit. Theorem 6.6 in [8] and Theorem 4 in [11] (revised in the form of Theorem 6.8 in [8]) are other persistence theorems on a matrix with multiple Jordan blocks. The results in [8,10] are restricted on matrices with Jordan blocks, which makes their results so explicit, but their arguments seems not heavily dependent on having the exact Jordan form.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 6.6 in [8] and Theorem 4 in [11] (revised in the form of Theorem 6.8 in [8]) are other persistence theorems on a matrix with multiple Jordan blocks. The results in [8,10] are restricted on matrices with Jordan blocks, which makes their results so explicit, but their arguments seems not heavily dependent on having the exact Jordan form. Our result can be easier to apply since we do not check if the matrix is in a certain specific form such as Jordan form, but it is still enough to characterize the asymptotic rate of the corresponding block.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Stability of Non-Autonomous Systems and a Generalization of Levinson’s Theorem

Lee

2019

Mathematics

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We study the asymptotic stability of non-autonomous linear systems with time dependent coefficient matrices { A ( t ) } t ∈ R . The classical theorem of Levinson has been an indispensable tool for the study of the asymptotic stability of non-autonomous linear systems. Contrary to constant coefficient system, having all eigenvalues in the left half complex plane does not imply asymptotic stability of the zero solution. Levinson’s theorem assumes that the coefficient matrix is a suitable perturbation of the diagonal matrix. Our objective is to prove a theorem similar to Levinson’s Theorem when the family of matrices merely admits an upper triangular factorization. In fact, in the presence of defective eigenvalues, Levinson’s Theorem does not apply. In our paper, we first investigate the asymptotic behavior of upper triangular systems and use the fixed point theory to draw a few conclusions. Unless stated otherwise, we aim to understand asymptotic behavior dimension by dimension, working with upper triangular with internal blocks adds flexibility to the analysis.

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Asymptotic Integration of Differential and Difference Equations

Cited by 24 publications

References 96 publications

On the dynamics of certain higher-order scalar difference equation: asymptotics, oscillation, stability

On the dynamics of certain higher-order scalar difference equation: asymptotics, oscillation, stability

Note on Limit-Periodic Solutions of the Difference Equation xt + 1 − [h(xt) + λ]xt = rt, λ > 1

Asymptotic Stability of Non-Autonomous Systems and a Generalization of Levinson’s Theorem

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