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Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy

Author

Listed:
  • Yong-Hui Xia
  • Xiaoqing Yuan
  • Kit Ian Kou
  • Patricia J. Y. Wong

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)| ≤ μe−ε|t|. However, this condition is very restricted. There are few functions satisfying |f(t, x)| ≤ μe−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.

Suggested Citation

  • Yong-Hui Xia & Xiaoqing Yuan & Kit Ian Kou & Patricia J. Y. Wong, 2014. "Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:725098
    DOI: 10.1155/2014/725098
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    References listed on IDEAS

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    1. Yongfei Gao & Yonghui Xia & Xiaoqing Yuan & P. J. Y. Wong, 2014. "Linearization of Nonautonomous Impulsive System with Nonuniform Exponential Dichotomy," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    2. Yongfei Gao & Yonghui Xia & Xiaoqing Yuan & P. J. Y. Wong, 2014. "Linearization of Nonautonomous Impulsive System with Nonuniform Exponential Dichotomy," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-7, March.
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    Cited by:

    1. Huicheng Wang & Patricia J. Y. Wong & Yonghui Xia, 2014. "Periodic Solutions of a Stage‐Structured Plant‐Hare Model with Toxin‐Determined Functional Responses," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    2. Xueru Lin, 2014. "A Note on Gronwall’s Inequality on Time Scales," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

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