Neural Computing and Applications

, Volume 29, Issue 7, pp 513–527 | Cite as

Weighted pseudo-almost automorphic solutions of high-order Hopfield neural networks with neutral distributed delays

  • Lili Zhao
  • Yongkun Li
  • Bing Li
Original Article


In this paper, a class of high-order Hopfield neural networks with neutral distributed delays is considered. Some sufficient conditions are obtained for the existence, uniqueness and global exponential stability of weighted pseudo-almost automorphic solutions for this class of networks by employing the Banach fixed-point theorem and differential inequality techniques. The results of this paper are completely new. An example is given to show the effectiveness of the proposed method and results.


High-order Hopfield neural networks Neutral distributed delays Weighted pseudo-almost automorphic solution Global exponential stability Fixed-point theorem 


  1. 1.
    Wang Z, Fang J, Liu X (2008) Global stability of stochastic high-oreder neural networks with discrete and distributed delay. Chaos Solitons Fractals 36(2):388–396MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Song B, Zhang Y, Shu Z, Hu FN (2016) Stability analysis of Hopfield neural networks perturbed by Poisson noises. Neurocomputing 196:53–58CrossRefGoogle Scholar
  3. 3.
    Duan L, Huang L, Guo Z (2014) Stability and almost periodicity for delayed high-order Hopfield neural networks with discontinuous activations. Nonlinear Dyn 77:1469–1484MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Wang Q, Fang Y, Li H, Su L, Dai B (2014) Anti-periodic solutions for high-order Hopfield neural networks with impulses. Neurocomputing 138:339–346CrossRefGoogle Scholar
  5. 5.
    Wang Y, Lu C, Ji G, Wang L (2011) Global exponential stability of high-order Hopfield-type neural networks with S-type distributed time delays. Commun Nonlinear Sci Numer Simul 16:3319–3325MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Zheng CD, Zhang H, Wang Z (2011) Novel exponential stability criteria of high-order neural networks with time-varying delays. IEEE Trans Syst Man Cybern B Cybern 41(2):486–496CrossRefGoogle Scholar
  7. 7.
    Qiu J (2010) Dynamics of high-order Hopfield neural networks with time delays. Neurocomputing 73:820–826CrossRefGoogle Scholar
  8. 8.
    Xu BJ, Liu XZ (2003) Global asymptotic stability of high-order Hopfield type neurual networks with time delays. Comput Math Appl 45(10–11):1279–1737Google Scholar
  9. 9.
    Kosmatopoulos EB, Christodoulou MA (1995) Structural properties of gradient recurrent high-order neural networks. IEEE Trans Circuits Syst II(42):592–603CrossRefzbMATHGoogle Scholar
  10. 10.
    Zheng CD, Zhang H, Wang Z (2011) Novel exponential stability criteria of high-order neural networks with time-varying delays. IEEE Trans Syst Man Cybern B Cybern 41:486–496CrossRefGoogle Scholar
  11. 11.
    Xu Y (2014) Anti-periodic solutions for HCNNs with time-varying delays in the leakage terms. Neural Comput Appl 24:1047–1058CrossRefGoogle Scholar
  12. 12.
    Gao S, Ning B, Dong H (2015) Adaptive neural control with intercepted adaptation for time-delay saturated nonlinear systems. Neural Comput Appl 26(8):1849–1857CrossRefGoogle Scholar
  13. 13.
    Zhang H, Wang Z, Liu D (2014) A comprehensive review of stability analysis of continuous-time recurrent neural networks. IEEE Trans Neural Netw Learn Syst 25:1229–1264CrossRefGoogle Scholar
  14. 14.
    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 242:679–693MathSciNetzbMATHGoogle Scholar
  15. 15.
    Park JH, Park CH, Kwon OM, Lee SM (2008) A new stability criterion for bidirectional associative memory neural networks of neutral-type. Appl Math Comput 199:716–722MathSciNetzbMATHGoogle Scholar
  16. 16.
    Rakkiyappan R, Balasubramaniam P, Cao J (2010) Global exponential stability results for neutral-type impulsive neural networks. Nonlinear Anal Real World Appl 11:122–130MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rakkiyappan R, Balasubramaniam P (2008) New global exponential stability results for neutral type neural networks with distributed time delays. Neurocomputing 71:1039–1045CrossRefzbMATHGoogle Scholar
  18. 18.
    Rakkiyappan R, Balasubramaniam P (2008) LMI conditions for global asymptotic stability results for neutral-type neural networks with distributed time delays. Appl Math Comput 204:317–324MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lee SM, Kwon OM, Park JH (2010) A novel delay-dependent criterion for delayed neural networks of neutral type. Phys Lett A 374:1843–1848MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhang H, Liu Z, Huang GB (2010) Novel delay-dependent robust stability analysis for switched neutral-type neural networks with time-varying delays via SC technique. IEEE Trans Syst Man Cybern B Cybern 40:1480–1491CrossRefGoogle Scholar
  21. 21.
    Li YK, Zhao L, Chen XR (2012) Existence of periodic solutions for neutral type cellular neural networks with delays. Appl Math Model 36:1173–1183MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Li Y, Li Y (2013) Existence and exponential stability of almost periodic solution for neutral delay BAM neural networks with time-varying delays in leakage terms. J Franklin Inst 350(9):2808–2825MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Du B, Liu Y, Batarfi HA, Alsaadi FE (2016) Almost periodic solution for a neutral-type neural networks with distributed leakage delays on time scales. Neurocomputing 173:921–929CrossRefGoogle Scholar
  24. 24.
    Zhou Y, Li C, Huang T, Wang X (2015) Impulsive stabilization and synchronization of Hopfield-type neural networks with impulse time window. Neural Comput Appl. doi: 10.1007/s00521-015-2105-7 Google Scholar
  25. 25.
    Blot J, Mophou GM, N’Guérékata GM, Pennequin D (2009) Wighted pseudo-almost automorphic functions and applications to abstract differential equations. Nonlinear Anal 71:903–909MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Diagana T, Mophou GM, N’Guérékata GM (2010) Existence of weighted pseudo-almost periodic solutions to some classes of differential equations with \(S^{P}\)-weighted pseudo-almost periodic coefficients. Nonlinear Anal 72:430–438MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Digana T (2011) Doubly-weighted pseudo-almost periodic functions. Afr Diaspora J Math 12(1):121–136MathSciNetGoogle Scholar
  28. 28.
    Digana T (2011) Existence of doubly-weighted pseudo almost periodic solutions to nonautonomous differential equations. Electron J Differential Equ 2011(28):1–15Google Scholar
  29. 29.
    Liang J, Xiao TJ, Zhang J (2010) Decomposition of weighted pseudo-almost periodic functions. Nonlinear Anal 73:3456–3461MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Fink AM (1974) In: Almost periodic differential equations, vol 377. Lecture notes in mathematics. Springer, BerlinGoogle Scholar
  31. 31.
    He CY (1992) Almost periodic differential equations. Higher Education Publishing House, Beijing (in Chinese)Google Scholar
  32. 32.
    Diagana T (2013) Almost automorphic type and almost periodic type functions in abstract spaces. Springer, BerlinCrossRefzbMATHGoogle Scholar
  33. 33.
    Ji D, Zhang C (2012) Translation invariance of weighted pseudo-almost periodic functions and related problems. J Math Anal Appl 391(2):350–362MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  1. 1.Department of MathematicsYunnan UniversityKunmingPeople’s Republic of China

Personalised recommendations