Neural Processing Letters

, Volume 49, Issue 1, pp 347–356 | Cite as

Global Exponential Convergence of HCNNs with Neutral Type Proportional Delays and D Operator

  • Songlin XiaoEmail author


This paper deals with a class of high-order cellular neural networks with neutral type proportional delays and D operator. By applying differential inequality techniques, we show that all solutions of the addressed system converge exponentially to zero vector. In addition, we provide an example and its numerical simulations to demonstrate the effectiveness of the proposed results.


Global exponential convergence High-order cellular neural network D operator Neutral type proportional delay 

Mathematics Subject Classification

34C25 34K13 34K25 



The author would like to express the gratitude to the editors and anonymous reviewers for their valuable suggestions, which improved the presentation of this paper. This work was supported by the Natural Scientific Research Fund of Zhejiang Province of China (Grant No. LY18A010019), a Key Project Supported by Scientific Research Fund of Hunan Provincial Education Department (15A038) and Natural Scientific Research Fund of Hunan Provincial of China (Grant Nos. 2016JJ6103, 2016JJ6104).

Compliance with Ethical Standards

Conflicts of interest

The Author declares that he have no conflict of interests.


  1. 1.
    Dembo A, Farotimi O, Kailath T (1991) High-order absolutely stable neural networks. IEEE Trans. Circuits Syst. 38:57–65CrossRefzbMATHGoogle Scholar
  2. 2.
    Karayiannis NB, Venetsanopoulos AN (1995) On the training and performance of high-order neural networks. Math. Biosci 129(2):143–168CrossRefzbMATHGoogle Scholar
  3. 3.
    Tunc C (2015) Pseudo almost periodic solutions for HCNNs with time-varying leakage delays, Moroccan. J Pure Appl Anal 1:51–69Google Scholar
  4. 4.
    Kwon OM, Park JH, Lee SM, Cha EJ (2013) Analysis on delay-dependent stability for neural networks with time-varying delays. Neurocomputing. 103:114–120CrossRefGoogle Scholar
  5. 5.
    Fang M, Park JH (2013) Non-fragile synchronization of neural networks with time-varying delay and randomly occurring controller gain fluctuation. Appl Math Comput 219:8009–8017MathSciNetzbMATHGoogle Scholar
  6. 6.
    Rakkiyappan R, Balasubramaniam P (2008) Delay-dependent asymptotic stability for stochastic delayed recurrent neural networks with time varying delays. Appl Math Comput 198:526–533MathSciNetzbMATHGoogle Scholar
  7. 7.
    Yao L (2016) Global convergence of CNNs with neutral type delays and \(D\) operator. Neural Comput Appl. Google Scholar
  8. 8.
    Yao L (2017) Global exponential convergence of neutral type shunting inhibitory cellular neural networks with \(D\) operator. Neural Process Lett 45:401–409CrossRefGoogle Scholar
  9. 9.
    Zhang A (2017) Pseudo almost periodic solutions for neutral type SICNNs with \(D\) operator. J Exp Theor Artif Intell 29(4):795–807CrossRefGoogle Scholar
  10. 10.
    Zhang A (2017) Almost periodic solutions for SICNNs with neutral type proportional delays and \(D\) operators. Neural Process Lett. Google Scholar
  11. 11.
    Xu Y (2017) Exponential stability of pseudo almost periodic solutions for neutral type cellular neural networks with \(D\) operator. Neural Process Lett 46:329–342CrossRefGoogle Scholar
  12. 12.
    Chen Z (2017) Global exponential stability of anti-periodic solutions for neutral type CNNs with \(D\) operator. Int J Mach Learn Cyber. Google Scholar
  13. 13.
    Ockendon JR, Tayler AB (1971) The dynamics of a current collection systemfor an electric locomotive. Proc R Soc A 322:447–468CrossRefGoogle Scholar
  14. 14.
    Fox L, Mayers DF, Ockendon JR, Tayler AB (1971) On a functional-differential equation. J Inst Math Appl 8(3):271–307MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Derfel GA (1982) On the behaviour of the solutions of functional and functional-differential equations with several deviating arguments. Ukr Math J 34:286–291CrossRefGoogle Scholar
  16. 16.
    Song X, Zhao P, Xing Z, Peng J (2016) Global asymptotic stability of CNNs with impulses and multi-proportional delays. Math Methods Appl Sci 39(4):722–733MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Derfel GA (1990) Kato problem for functional-differential equations and difference Schr\(\ddot{o}\)dinger operators. Oper Theory 46:319–321Google Scholar
  18. 18.
    Zhou L (2011) On the global dissipativity of a class of cellular neural networks with multi-pantograph delays. Adv Artif Neural Syst 941426:1–7Google Scholar
  19. 19.
    Liu B (2017) Finite-time stability of CNNs with neutral proportional delays and time-varying leakage delays. Math Methods Appl Sci 40:167–174MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yu Y (2016) Global exponential convergence for a class of neutral functional differential equations with proportional delays. Math Methods Appl Sci 39:4520–4525MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yu Y (2016) Global exponential convergence for a class of HCNNs with neutral time-proportional delays. Appl Math Comput 285:1–7MathSciNetzbMATHGoogle Scholar
  22. 22.
    Yang G, Wang W (2018) New results on convergence of CNNs with neutral type proportional delays and \(D\) operator. Neural Process Lett.
  23. 23.
    Arthi G, Park JH, Jung HY, Yoo JH (2015) Exponential stability criteria for a neutral type stochastic single neuron system with time-varying delays. Neurocomputing 154(22):317–324CrossRefGoogle Scholar
  24. 24.
    Park Ju H (2009) Synchronization of cellular neural networks of neutral type via dynamic feedback controller. Chaos Solitons Fractals 42(3):1299–1304MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Park JH, Kwon OM (2009) Further results on state estimation for neural networks of neutral-type with time-varying delay. Appl Math Comput 208(1):69–75MathSciNetzbMATHGoogle Scholar
  26. 26.
    Park JH (2004) Design of a dynamic output feedback controller for a class of neutral systems with discrete and distributed delays. IEE Proc Control Theory Appl 151(5):610–614CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Academy of Intelligent SoftwareGuangzhou University, Guangdong Provincial Engineering Technology Research Center for Mathematical Educational SoftwareGuangzhouPeople’s Republic of China

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