Advertisement

Piecewise Pseudo Almost Periodic Solutions of Generalized Neutral-Type Neural Networks with Impulses and Delays

  • Fanchao Kong
  • Zhiguo Luo
  • Xiaoping Wang
Article

Abstract

This paper is concerned with the generalized neutral-type neural networks with impulses and delays. By applying the contraction mapping principle and generalized Gronwall–Bellman’s inequality, we employ a novel argument to establish new results on the existence, uniqueness and exponential stability of piecewise pseudo almost periodic solutions. Some corresponding results in the literature can be enriched and extended. Moreover, a numerical example is given to illustrate the effectiveness of our results.

Keywords

Neutral-type neural networks Contraction mapping principle Gronwall–Bellman’s inequality Exponential stability Impulses Delays 

Mathematics Subject Classification

34C37 

Notes

Acknowledgements

The authors thank the anonymous reviewers for their insightful suggestions which improved this work significantly. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11471109, 11471278).

Funding was provided by Scientific Research Fund of Hunan Provincial Education Department (Grant No. 13A093).

References

  1. 1.
    Alonso AI, Hong J, Rojo J (1998) A class of ergodic solutions of differential equations with piecewise constant arguments. Dyn Syst Appl 7:561–574MathSciNetMATHGoogle Scholar
  2. 2.
    Gui ZJ, Ge WG, Yang XS (2007) Periodic oscillation for a Hopfield neural networks with neutral delays. Phys Lett A 364:267–273CrossRefMATHGoogle Scholar
  3. 3.
    Hale J (1977) Theory of functional differential equations. Springer, New YorkCrossRefMATHGoogle Scholar
  4. 4.
    Kong FC (2017) Positive piecewise pseudo-almost periodic solutions of first-order singular differential equations with impulses. J Fixed Point Theory Appl.  https://doi.org/10.1007/s11784-017-0438-9.
  5. 5.
    Lakshmikantham V, Bainov DD, Simeonov PS (1989) Theory of impulse differential equations. World Scientific, SingaporeCrossRefGoogle Scholar
  6. 6.
    Liu YR, Wang ZD, Liu XH (2012) Stability analysis for a class of neutral-type neural networks with Markovian jumping parameters and mode-dependent mixed delays. Neurocomputing 94:46–53CrossRefGoogle Scholar
  7. 7.
    Liu J, Zhang C (2013) Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations. Adv Differ Equ 11:1–21MathSciNetGoogle Scholar
  8. 8.
    Liu DY, Du Y (2015) New results of stability analysis for a class of neutral-type neural network with mixed time delays. Int J Mach Learn Cybern 6(4):555–566CrossRefGoogle Scholar
  9. 9.
    Li XD, O’Regan D, Akca H (2015) Global exponential stabilization of impulsive neural networks with unbounded continuously distributed delays. IMA J Appl Math 80(1):85–99MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Liu BW (2015) Pseudo almost periodic solutions for neutral type CNNs with continuously distributed leakage delays. Neurocomputing 148:445–454CrossRefGoogle Scholar
  11. 11.
    Liu BW (2015) Pseudo almost periodic solutions for CNNs with continuously distributed leakage delays. Neural Process Lett 42:233–256CrossRefGoogle Scholar
  12. 12.
    Liu BW, Tunç C (2015) Pseudo almost periodic solutions for CNNs with leakage delays and complex deviating arguments. Neural Comput Appl 26:429–435CrossRefGoogle Scholar
  13. 13.
    Liu C, Liu WP, Yang Z (2016) Stability of neural networks with delay and variable-time impulses. Neurocomputing 171:1644–1654CrossRefGoogle Scholar
  14. 14.
    Li YK, Yang L, Li B (2016) Existence and stability of pseudo almost periodic solution for neutral type high-order hopfield neural networks with delays in leakage terms on time scales. Neural Process Lett 44(3):603–623MathSciNetCrossRefGoogle Scholar
  15. 15.
    Rakkiyappan R, Balasubramaniam P (2008) New global exponential stability results for neutral type neural networks with distributed time delays. Neurocomputing 71:1039–1045CrossRefMATHGoogle Scholar
  16. 16.
    Rakkiyappan R, Balasubramaniam P (2008) LMI conditions for global asymptotic stability results for neutral-type neural networks with distributed time delays. Appl Math Comput 204(1):317–324MathSciNetMATHGoogle Scholar
  17. 17.
    Samoilenko AM, Perestyuk NA (1995) Impulsive differential equations. World Scientific Publishing Corporation, SingaporeCrossRefMATHGoogle Scholar
  18. 18.
    Samli R, Arik S (2009) New results for global stability of a class of neutral-type neural systems with time delays. Appl Math Comput 210(2):564–570MathSciNetMATHGoogle Scholar
  19. 19.
    Stamov GT (2012) Almost periodic solutions of impulsive differential equations. Springer, BerlinCrossRefMATHGoogle Scholar
  20. 20.
    Wang K, Zhu YL (2010) Stability of almost periodic solution for a generalized neutral-type neural networks with delays. Neurocomputing 73(16):3300–3307CrossRefGoogle Scholar
  21. 21.
    Wang XH, Li SY, Xu DY (2011) Globally exponential stability of periodic solutions for impulsive neutral-type neural networks with delays. Nonlinear Dyn 64(1):65–75MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Wang C, Agarwal RP (2014) Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive \(\nabla \)-dynamic equations on time scales. Adv Differ Equ 2014:153MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wang C (2016) Piecewise pseudo almost periodic solution for impulsive non-autonomous high-order Hopfield neural networks with variable delays. Neurocomputing 171:1291–1301CrossRefGoogle Scholar
  24. 24.
    Xu CJ, Zhang QM, Wu YS (2014) Existence and stability of pseudo almost periodic solutions for shunting inhibitory cellular neural networks with neutral type delays and time-varying leakage delays. Netw Comput Neural 25(4):168–192Google Scholar
  25. 25.
    Xie D, Jiang YP (2016) Global exponential stability of periodic solution for delayed complex-valued neural networks with impulses. Neurocomputing 207:528–538CrossRefGoogle Scholar
  26. 26.
    Yang CB, Huang TZ (2013) New results on stability for a class of neural networks with distributed delays and impulses. Neurocomputing 111:115–121CrossRefGoogle Scholar
  27. 27.
    Zhang J, Gui ZJ (2009) Periodic solutions of nonautonomous cellular neural networks with impulses and delays. Nonlinear Anal Real World Appl 10(3):1891–1903MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Zhao DL, Han D (2011) Stability of linear neutral differential equations with delays and impulses established by the fixed points method. Nonlinear Anal 74(18):7240–7251MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Zheng CD, Wang Y, Wang ZS (2014) New stability results of neutral-type neural networks with continuously distributed delays and impulses. Int J Comput Math 91(9):1880–1896MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zhang CY (2003) Almost periodic type functions and ergodicity. Kluwer Academic/Science Press, BeijingCrossRefMATHGoogle Scholar
  31. 31.
    Zhang CY (1995) Pseudo almost periodic solutions of some differential equations II. J Math Anal Appl 192:543–561MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Zhang CY (1994) Pseudo almost periodic solutions of some differential equations. J Math Anal Appl 151:62–76MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Zhou QY (2016) Pseudo almost periodic solutions for SICNNs with leakage delays and complex deviating arguments. Neural Process Lett 44(2):375–386CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer ScienceHunan Normal UniversityChangshaPeople’s Republic of China
  2. 2.Mathematics and Computer Science CollegeChangsha UniversityChangshaPeople’s Republic of China
  3. 3.College of Mathematics and FinanceXiangnan UniversityChenzhouPeople’s Republic of China

Personalised recommendations