Hyers–Ulam stability and discrete dichotomy for difference periodic systems

Buşe, Constantin; O’Regan, Donal; Saierli, Olivia; Tabassum, Afshan

doi:10.1016/j.bulsci.2016.03.010

Cited by 40 publications

(30 citation statements)

References 23 publications

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“…Combining this result with ([3], Theorem 3.1) we get the following Corollary that completes an open problem, raised in [1] for the particular case n = 3. Motivated by the applications suggested in [15], we are also interested in studying the Hyers-Ulam stability of the linear recurrence in (18), but with Z instead of Z + .…”

Section: Background and The Main Resultssupporting

confidence: 61%

“…possesses a discrete dichotomy on Z + ; see ( [1], Proposition 1.2, Theorem 2.1) for a more general framework of this result. A new challenge for us is to see if the following three statements (presented in formal terms) are equivalent.…”

Section: Background and The Main Resultsmentioning

confidence: 93%

“…Here, we continue the analysis started in [3] and, finally, we complete the discussion raised in [1] for periodic linear recurrences of order three. Thus, this article can be seen as a new link in the chain of articles [1][2][3][4][5] which address the Hyers-Ulam stability of linear scalar recurrences. The connections of this topic to those existing in the literature was already presented in [3], so we do not present them again here.…”

Section: Introductionmentioning

confidence: 94%

“…An open problem, arising naturally in [1], is a problem referring to the relationship between the Hyers-Ulam stability of a certain linear recurrence of order n with periodic coefficients and the exponential dichotomy of the monodromy matrix associated to the recurrence. The corresponding problem for second-order recurrences was completed in [2], where second-order linear differential equations were also analyzed.…”

Section: Introductionmentioning

confidence: 99%

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Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients

2019

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Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) , and ( c j ) (with j a non-negative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is as is given below. Assuming that the “monodromy matrix” A ( q ) has at least one multiple eigenvalue, we prove that the linear scalar recurrence x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n ∈ Z + is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } . Connecting this result with a recently obtained one it follows that the above linear recurrence is Hyers-Ulam stable if and only if the spectrum of A ( q ) does not intersect the unit circle.

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Section: Background and The Main Resultssupporting

confidence: 61%

Section: Background and The Main Resultsmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients

2019

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“…The history of the Ulam problem (concerning the stability of a functional equation) and of stability in the sense of Hyers-Ulam is well known. In particular, Hyers-Ulam stability for linear recurrences and for systems of linear recurrences is considered in [3][4][5][6][7][8][9][10][11][12][13][14][15][16], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues

2019

Self Cite

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Let q ≥ 2 be a positive integer and let ( a j ) , ( b j ) and ( c j ) (with j nonnegative integer) be three given C -valued and q-periodic sequences. Let A ( q ) : = A q − 1 ⋯ A 0 , where A j is defined below. Assume that the eigenvalues x , y , z of the “monodromy matrix” A ( q ) verify the condition ( x − y ) ( y − z ) ( z − x ) ≠ 0 . We prove that the linear recurrence in C x n + 3 = a n x n + 2 + b n x n + 1 + c n x n , n ∈ Z + is Hyers–Ulam stable if and only if ( | x | − 1 ) ( | y | − 1 ) ( | z | − 1 ) ≠ 0 , i.e., the spectrum of A ( q ) does not intersect the unit circle Γ : = { w ∈ C : | w | = 1 } .

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Best Ulam constants for two‐dimensional nonautonomous linear differential systems

Anderson,

Onitsuka,

O'Regan

2024

Mathematische Nachrichten

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This study deals with the Ulam stability of nonautonomous linear differential systems without assuming the condition that they admit an exponential dichotomy. In particular, the best (minimal) Ulam constants for two‐dimensional nonautonomous linear differential systems with generalized Jordan normal forms are derived. The obtained results are applicable not only to systems with solutions that exist globally on , but also to systems with solutions that blow up in finite time. New results are included even for constant coefficients. A wealth of examples are presented, and approximations of node, saddle, and focus are proposed. In addition, this is the first study to derive the best Ulam constants for nonautonomous systems other than periodic systems.

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Hyers–Ulam stability and discrete dichotomy for difference periodic systems

Cited by 40 publications

References 23 publications

Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients

Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients

Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues

Best Ulam constants for two‐dimensional nonautonomous linear differential systems

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