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Boundness and Linearisation of a Class of Differential Equations with Piecewise Constant Argument

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Abstract

The differential equations with piecewise constant argument (DEPCAs, for short) is a class of hybrid dynamical systems (combining continuous and discrete). In this paper, under the assumption that the nonlinear term is partially unbounded, we study the bounded solution and global topological linearisation of a class of DEPCAs of general type. One of the purpose of this paper is to obtain a new criterion for the existence of a unique bounded solution, which improved the previous results. The other aim of this paper is to establish a generalized Grobman–Hartman-type theorem for the topological conjugacy between a nonlinear perturbation system and its linear system. The method is based on the new obtained criterion for bounded solution. The obtained results generalized and improved some previous papers. Some novel techniques are employed in the proof.

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

    Changwu Zou & Jinlin Shi

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Yonghui Xia

  3. Departamento de Matematica, Universidad de Chile, Santiago, Chile

    Manuel Pinto

  4. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

    Yuzhen Bai

Authors
  1. Changwu Zou
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  2. Yonghui Xia
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  3. Manuel Pinto
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  4. Jinlin Shi
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  5. Yuzhen Bai
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Correspondence to Yonghui Xia.

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The authors declare that there is no conflict of interests regarding the publication of this article.

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Additional information

Changwu Zou was supported by the National Natural Science Foundation of China under Grant (No. 11471027) and Foundation of Fujiang Province Education Department under Grant (No. JAT160082)

Yonghui Xia was supported by the National Natural Science Foundation of China under Grant (No. 11671176), Natural Science Foundation of Zhejiang Province under Grant (No. LY15A010007), Natural Science Foundation of Fujian Province under Grant (No. 2018J01001)

Manuel Pinto was supported by FONDECYT Grant (Nos. 1120709 and 1170466 ).

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Zou, C., Xia, Y., Pinto, M. et al. Boundness and Linearisation of a Class of Differential Equations with Piecewise Constant Argument. Qual. Theory Dyn. Syst. 18, 495–531 (2019). https://doi.org/10.1007/s12346-018-0297-9

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  • DOI: https://doi.org/10.1007/s12346-018-0297-9

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