On the dichotomic behavior of discrete dynamical systems on the half-line

Let m be a given positive integer and let A be an m × m complex matrix. We prove that the discrete systemis Hyers-Ulam stable if and only if the matrix A possesses a discrete dichotomy. Also we prove that the scalar difference equation of order m x n+m = a 1 x n+m−1 + a 2 x n+m−2 + · · · + a m x n , n ∈ Z + , is Hyers-Ulam stable if and only if the algebraic equation z m = a 1 z m−1 + · · · + a m−1 z + a m , z ∈ C has no roots on the unit circle. This latter result is essentially known, for further details see for example [24] and [2]. However, our proofs are completely different and moreover, it seems that our approach opens the way to obtain many other results in this topic.

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“…The authors wish to thank the anonymous reviewers who indicated references [18,22,28] which allowed shortening the proof of Proposition 2.1.…”

Section: Acknowledgmentmentioning

confidence: 99%

Hyers–Ulam stability and discrete dichotomy

Barbu

Buşe

Tabassum

2015

Journal of Mathematical Analysis and Applications

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“…We are also able to say what happens with the ranges of projections P 2 n (again we refer to [45]). In other words, the unstable direction at time m 0 can be an image under the action of the cocycle of any closed subspace Z with the property that X can be decomposed as a direct sum of S(0) and Z.…”

Section: Characterization Of Exponential Trichotomies On Z +mentioning

confidence: 99%

“…It turns out that in this case only the ranges of projections P 1 n are uniquely determined (see [45]). …”

Section: Characterization Of Exponential Trichotomies On Z +mentioning

confidence: 99%

“…The major reason for this is the fact that Theorem 2.4 is no longer valid when one passes to the case of dichotomies on the half-line and consequently dichotomies on Z + 0 cannot be characterized in terms of the invertibility (hyperbolicity) of a single operator. For appropriate versions of Theorem 2.4 for dichotomies on the half-line we refer to [45]. In order to overcome this difficulty, we will apply a different strategy of extending dichotomies on Z + 0 to dichotomies on Z and applying already established results.…”

Section: Characterization Of Exponential Trichotomies On Z +mentioning

confidence: 99%

See 1 more Smart Citation

Lyapunov type equation for discrete exponential trichotomies

Dragičević¹

2017

J. Nonlinear Sci. Appl.

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For a nonautonomous dynamics obtained by a sequence of linear operators acting on an arbitrary Hilbert space, we give a complete characterization of the notion of a uniform exponential trichotomy in terms of what can be considered to be a discrete version of the Lyapunov equation. We then use this characterization to study the stability of exponential trichotomies under small linear and nonlinear perturbations.

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“…For more recent results devoted to continuous and discrete evolution families, we refer to [5][6][7][8][9][10][11][12][13][14][15] for those dealing with uniform exponential behaviour and to [16][17][18][19][20] for those that consider various concepts of nonuniform exponential behaviour.…”

Section: Introductionmentioning

confidence: 99%