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A topological approach to the existence of solutions for nonlinear differential equations with piecewise constant argument

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  • Huang, Zhenkun
  • Wang, Xinghua
  • Xia, Yonghui

Abstract

In this paper, we investigate qualitative behavior of nonlinear differential equations with piecewise constant argument (PCA). A topological approach of Ważewski-type which gives sufficient conditions to guarantee that the graph of at least one solution stays in a given domain is formulated. Moreover, our results are also suitable for a class of general system of discrete equations. By using a regular polyfacial set, we apply our developed topological approach to cellular neural networks (CNNs) with PCA. Some new results are attained to reveal dynamic behavior of CNNs with PCA and discrete-time CNNs. Finally, an illustrative example of CNNs with PCA shows usefulness and effectiveness of our results.

Suggested Citation

  • Huang, Zhenkun & Wang, Xinghua & Xia, Yonghui, 2009. "A topological approach to the existence of solutions for nonlinear differential equations with piecewise constant argument," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1121-1131.
    • Handle: RePEc:eee:chsofr:v:39:y:2009:i:3:p:1121-1131
    DOI: 10.1016/j.chaos.2007.04.029
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    1. Huang, Chuangxia & Huang, Lihong & Yuan, Zhaohui, 2005. "Global stability analysis of a class of delayed cellular neural networks," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 70(3), pages 133-148.
    2. Huang, Zhenkun & Xia, Yonghui, 2009. "Exponential periodic attractor of impulsive BAM networks with finite distributed delays," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 373-384.
    3. Huang, Zhenkun & Xia, Yonghui, 2008. "Global exponential stability of BAM neural networks with transmission delays and nonlinear impulses," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 489-498.
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    1. Marat Akhmet & Duygu Aruğaslan Çinçin & Madina Tleubergenova & Zakhira Nugayeva, 2021. "Unpredictable Oscillations for Hopfield-Type Neural Networks with Delayed and Advanced Arguments," Mathematics, MDPI, vol. 9(5), pages 1-19, March.

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