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  • Xia, Yong-Hui  (4)
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  • 1
    Online Resource
    Online Resource
    Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy
    Hindawi Limited ; 2014
    In:  Abstract and Applied Analysis Vol. 2014 ( 2014), p. 1-10
    Details
    In: Abstract and Applied Analysis, Hindawi Limited, Vol. 2014 ( 2014), p. 1-10
    Abstract: Nonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies | f ( t , x ) | ≤ μ e − ε | t | . However, this condition is very restricted. There are few functions satisfying | f ( t , x ) | ≤ μ e − ε | t | . In some sense, this assumption is not reasonable enough. More suitable assumption should be | f ( t , x ) | ≤ μ . To the best of the authors' knowledge, there is no paper considering the existence and uniqueness of solution to the perturbed nonautonomous system with a relatively conservative assumption | f ( t , x ) | ≤ μ . In this paper, we prove that if the nonlinear term is bounded, the perturbed nonautonomous system with nonuniform exponential dichotomy has a unique solution. The technique employed to prove Theorem 4 is the highlight of this paper.
    Type of Medium: Online Resource
    ISSN: 1085-3375 , 1687-0409
    URL: Article
    DOI: 10.1155/2014/725098
    Language: English
    Publisher: Hindawi Limited
    Publication Date: 2014
    detail.hit.zdb_id: 2064801-7
    SSG: 17,1
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    Online Resource
  • 2
    Online Resource
    Online Resource
    Solve the linear quaternion-valued differential equations having multiple eigenvalues
    AIP Publishing ; 2019
    In:  Journal of Mathematical Physics Vol. 60, No. 2 ( 2019-02-01)
    Details
    In: Journal of Mathematical Physics, AIP Publishing, Vol. 60, No. 2 ( 2019-02-01)
    Abstract: The theory of two-dimensional linear quaternion-valued differential equations (QDEs) was recently established {see the work of Kou and Xia [Stud. Appl. Math. 141(1), 3–45 (2018)]}. They observed some profound differences between QDEs and ordinary differential equations. Also, an algorithm to evaluate the fundamental matrix by employing the eigenvalues and eigenvectors was presented in the work of Kou and Xia [Stud. Appl. Math. 141(1), 3–45 (2018)] . However, the fundamental matrix can be constructed providing that the eigenvalues are simple. If the linear system has multiple eigenvalues, how to construct the fundamental matrix? In particular, if the number of independent eigenvectors might be less than the dimension of the system, that is, the numbers of the eigenvectors are not enough to construct a fundamental matrix, how to find the “missing solutions”? The main purpose of this paper is to answer this question.
    Type of Medium: Online Resource
    ISSN: 0022-2488 , 1089-7658
    URL: Article
    DOI: 10.1063/1.5040237
    RVK:
    UA 4660
    Language: English
    Publisher: AIP Publishing
    Publication Date: 2019
    detail.hit.zdb_id: 1472481-9
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  • 3
    Online Resource
    Online Resource
    On the linearization theorem for nonautonomous differential equations
    Elsevier BV ; 2015
    In:  Bulletin des Sciences Mathématiques Vol. 139, No. 7 ( 2015-10), p. 829-846
    Details
    In: Bulletin des Sciences Mathématiques, Elsevier BV, Vol. 139, No. 7 ( 2015-10), p. 829-846
    Type of Medium: Online Resource
    ISSN: 0007-4497
    URL: Article
    DOI: 10.1016/j.bulsci.2014.12.005
    Language: English
    Publisher: Elsevier BV
    Publication Date: 2015
    detail.hit.zdb_id: 1491945-X
    SSG: 17,1
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    Online Resource
  • 4
    Online Resource
    Online Resource
    Zeros of a Class of Transcendental Equation with Application to Bifurcation of DDE
    World Scientific Pub Co Pte Ltd ; 2016
    In:  International Journal of Bifurcation and Chaos Vol. 26, No. 04 ( 2016-04), p. 1650062-
    Details
    In: International Journal of Bifurcation and Chaos, World Scientific Pub Co Pte Ltd, Vol. 26, No. 04 ( 2016-04), p. 1650062-
    Abstract: Zeros of a class of transcendental equation with small parameter [Formula: see text] are considered in this paper. There have been many works in the literature considering the distribution of zeros of the transcendental equation by choosing the delay [Formula: see text] as bifurcation parameter. Different from standard consideration, we choose [Formula: see text] as bifurcation parameter (not the delay [Formula: see text] ) to discuss the distribution of zeros of such transcendental equation. After mathematical analysis, the obtained results are successfully applied to the bifurcation analysis in a biological model in the real word phenomenon. In the real world model, the effect of climate changes can be seen as the small parameter perturbation, which can induce bifurcations and instability. We present two methods to analyze the stability and bifurcations.
    Type of Medium: Online Resource
    ISSN: 0218-1274 , 1793-6551
    URL: Article
    DOI: 10.1142/S0218127416500620
    Language: English
    Publisher: World Scientific Pub Co Pte Ltd
    Publication Date: 2016
    SSG: 11
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