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Stability in mean for uncertain delay differential equations based on new Lipschitz conditions

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  • Gao, Yin
  • Jia, Lifen

Abstract

Uncertain delay differential equations (UDDEs) involving a current state and a certain past state have been proposed to model an uncertain system with a delay time, such as uncertain delay logistic model. Stability of the UDDEs is a vital problem for its applications. Based on the strong Lipschitz condition, the stability in mean for UDDEs have been investigated. Actually, the strong Lipschitz condition is assumed that it only relates to the current state, it is difficult to be employed to determine the stability in mean for the UDDEs. In this paper, we propose two kinds of new Lipschitz conditions containing the current state and the past state, which are more weaker than the strong Lipschitz condition. Meanwhile, two sufficient theorems based on these new Lipschitz conditions as the tools to judge the stability in mean for the UDDEs are verified. For a special class of the UDDEs, which are proved to be stable in mean without any limited condition. Besides, some examples are discussed in this paper.

Suggested Citation

  • Gao, Yin & Jia, Lifen, 2021. "Stability in mean for uncertain delay differential equations based on new Lipschitz conditions," Applied Mathematics and Computation, Elsevier, vol. 399(C).
    • Handle: RePEc:eee:apmaco:v:399:y:2021:i:c:s0096300321000989
    DOI: 10.1016/j.amc.2021.126050
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    References listed on IDEAS

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    1. Yang, Xiangfeng & Ralescu, Dan A., 2015. "Adams method for solving uncertain differential equations," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 993-1003.
    2. Jia, Lifen & Sheng, Yuhong, 2019. "Stability in distribution for uncertain delay differential equation," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 49-56.
    3. Xiao Wang & Yufu Ning, 2019. "A New Stability Analysis of Uncertain Delay Differential Equations," Mathematical Problems in Engineering, Hindawi, vol. 2019, pages 1-8, January.
    4. Xiangfeng Yang & Kai Yao, 2017. "Uncertain partial differential equation with application to heat conduction," Fuzzy Optimization and Decision Making, Springer, vol. 16(3), pages 379-403, September.
    5. Zhang, Tian & Chen, Huabin, 2019. "The stability with a general decay of stochastic delay differential equations with Markovian switching," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 294-307.
    6. Ali, Ishtiaq & Ullah Khan, Sami, 2020. "Analysis of stochastic delayed SIRS model with exponential birth and saturated incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
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