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Periodic Solution for \(\nabla \)-Stochastic High-Order Hopfield Neural Networks with Time Delays on Time Scales

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Abstract

In this paper, the comparison theorem and Gronwall’s inequality with \(\nabla \)-derivative on time scales are constructed. Based on \(\nabla \)-stochastic integration, the \(\nabla \)-stochastic high-order Hopfield neural networks with time delays on time scales is introduced and studied. By using contraction mapping principal and differential inequality technique on time scales, some sufficient conditions for the existence and exponential stability of periodic solutions for a class of \(\nabla \)-stochastic high-order Hopfield neural networks with time delays on time scales are established. Our results show that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors for periodicity. Finally, a numerical example is provided to illustrate the feasibility of our results. The results of this paper are completely new even if the time scale \(\mathbb {T}=\mathbb {R}\) or \(\mathbb {Z}\).

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References

  1. Li XD, Bohner M, Wang CK (2015) Impulsive differential equations: periodic solutions and applications. Automatica 52:173–178

    Article  MathSciNet  MATH  Google Scholar 

  2. Li XD, Song SJ (2013) Impulsive control for existence, uniqueness, and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays. IEEE Trans Neural Netw Learn Syst 24(6):868–877

    Article  Google Scholar 

  3. Li XD, Rakkiyappan R (2013) Impulsive controller design for exponential synchronization of chaotic neural networks with mixed delays. Commun Nonlinear Sci Numer Simulat 18:1515–1523

    Article  MATH  Google Scholar 

  4. Xiang H, Yan KM, Wang BY (2006) Existence and global exponential stability of periodic solution for delayed high-order Hopfield-type neural networks. Phys Lett A 352:4–5

    Article  MATH  Google Scholar 

  5. Ou CX (2008) Anti-periodic solutions for high-order Hopfield neural networks. Comput Math Appl 56:1838–1844

    Article  MathSciNet  MATH  Google Scholar 

  6. Yua YH, Cai MS (2018) Existence and exponential stability of almost-periodic solutions for high-order Hopfield neural networks. Math Comput Model 47:943–951

    Article  MathSciNet  Google Scholar 

  7. Li YK, Yang L (2014) Almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales. Appl Math Comput 242:679–693

    MathSciNet  MATH  Google Scholar 

  8. Blythe S, Mao X, Liao X (2001) Stability of stochastic delay neural networks. J Frankl Inst 338(5):481–495

    Article  MathSciNet  MATH  Google Scholar 

  9. Balasubramaniam P, Rakkiyappan R (2008) Global asymptotic stability of stochastic recurrent neural networks with multiple discrete delays and unbounded distributed delays. Appl Math Comput 204:680–686

    MathSciNet  MATH  Google Scholar 

  10. Hu SG, Liao XX, Mao XR (2003) Stochastic Hopfield neural networks. J Phys A Math Gen 36:1–15

    Article  MathSciNet  MATH  Google Scholar 

  11. Li XD (2010) Existence and global exponential stability of periodic solution for delayed neural networks with impulsive and stochastic effects. Neurocomputing 73:749–758

    Article  Google Scholar 

  12. Wang CH, Kao YG, Yang GW (2012) Exponential stability of impulsive stochastic fuzzy reaction–diffusion Cohen–Grossberg neural networks with mixed delays. Neurocomputing 89:55–63

    Article  Google Scholar 

  13. Li DS, Wang XH, Xu DY (2012) Existence and global \(p\)-exponential stability of periodic solution for impulsive stochastic neural networks with delays. Nonlinear Anal Hybrid Syst 6:847–858

    Article  MathSciNet  MATH  Google Scholar 

  14. Hilger S (1990) Analysis on measure chains—a unified approach to continuous and discrete calculus. Result Math 18:18–56

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhang G, Dong WL, Li QL, Liang HY (2009) Positive solutions for higher oder nonlinear neutral dynamic equations on time scales. Appl Math Model 33:2455–2463

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang C, Agarwal R (2014) Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive \(\nabla \)-dynamic equations on time scales. Adv Differ Equ 2014:153

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang C, Agarwal R (2014) Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations. Adv Differ Equ 2015:312

    Article  MATH  Google Scholar 

  18. Li YK, Yang L, Wu WQ (2015) Square-mean almost periodic solution for stochastic Hopfield neural networks with time-varying delays on times cales. Neural Comput Appl 26:1073–1084

    Article  Google Scholar 

  19. Wang C (2014) Existence and exponential stability of piecewise mean-square almost periodic solutions for impulsive stochastic Nicholson’s blowflies model on time scales. Appl Math Comput 248:101–112

    MathSciNet  MATH  Google Scholar 

  20. Bohner M, Peterson A (2001) Dynamic equations on time scales, an introduction with applications. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  21. Bohner M, Peterson A (2003) Advances in dynamic equations on time scales. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  22. Lungan C, Lupulescu V (2012) Random dynamical systems on time scales. Electron J Differ Equ 2012(86):1–14

    MathSciNet  MATH  Google Scholar 

  23. Yang L, Li YK (2015) Existence and exponential stability of periodic solution for stochastic Hopfield neural networks on time scales. Neurocomputing 167:543–550

    Article  Google Scholar 

  24. Bohner M, Stanzhytskyi OM, Bratochkina AO (2013) Stochastic dynamic equations on general time scales. Electron J Differ Equ 2013(57):1–15

    MathSciNet  MATH  Google Scholar 

  25. Wu F, Hu S, Liu Y (2010) Positive solution and its asymptotic behavior of stochastic functional Kolmogorov-type system. J Math Anal Appl 364:104–118

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

All authors would like to express their sincere thanks to the editor for handling this paper during reviewing process and to the referees for suggesting some corrections to help making the content of the paper more accurate.

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Authors and Affiliations

  1. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, Yunnan, People’s Republic of China

    Li Yang & Yu Fei

  2. Department of Organization, Yunnan Nationalities University, Kunming, 650091, Yunnan, People’s Republic of China

    Wanqin Wu

Authors
  1. Li Yang
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  2. Yu Fei
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  3. Wanqin Wu
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All authors contributed to the manuscript and typed, read and approved the final manuscript.

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Correspondence to Li Yang.

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This work is supported by Tian Yuan Fund of NSFC (No. 11526180), the NSFC under Grant No. 11561071 and Yunnan Province Education Department Scientific Research Fund Project (Nos. 2018JS315, 2018JS309).

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Yang, L., Fei, Y. & Wu, W. Periodic Solution for \(\nabla \)-Stochastic High-Order Hopfield Neural Networks with Time Delays on Time Scales. Neural Process Lett 49, 1681–1696 (2019). https://doi.org/10.1007/s11063-018-9896-3

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