Skip to main content
Springer Nature Link
Log in

Square-mean almost periodic solution for stochastic Hopfield neural networks with time-varying delays on timescales

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

In this paper, we first propose a concept of square-mean almost periodic function on timescales. Then, by means of the fixed point theory and differential inequality techniques on timescales, we establish some sufficient conditions on the existence and global exponential stability of square-mean almost periodic solutions for a class of stochastic Hopfield neural networks with time-varying delays on timescales. Our results are new even if the timescale \({\mathbb {T}}={\mathbb {R}}\) or \({\mathbb {Z}}\). Finally, we present an example to illustrate our theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Chen SH, Zhang Q, Wang CP (2004) Existence and stability of equilibria of the continuous-time Hopfield neural network. J Comput Appl Math 169:117–125

    Article  MATH  MathSciNet  Google Scholar 

  2. Zhao HY (2004) Global asymptotic stability of Hopfield neural network involving distributed delays. Neural Netw 17:47–53

    Article  MATH  Google Scholar 

  3. Zhou J, Li SY, Yang ZG (2009) Global exponential stability of Hopfield neural networks with distributed delays. Appl Math Model 33:1513–1520

    Article  MATH  MathSciNet  Google Scholar 

  4. Yang XF, Liao XF, Evans DJ, Tang YY (2005) Existence and stability of periodic solution in impulsive Hopfield neural networks with finite distributed delays. Phys Lett A 343:108–116

    Article  MATH  Google Scholar 

  5. 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 352:341–349

    Article  MATH  Google Scholar 

  6. Jiang HJ, Teng ZD (2009) Boundedness, periodic solutions and global stability for cellular neural networks with variable coefficients and infinite delays. Neurocomputing 72:2455–2463

    Article  Google Scholar 

  7. Kaslik E, Sivasundaram S (2011) Multiple periodic solutions in impulsive hybrid neural networks with delays. Appl Math Comput 217:4890–4899

    Article  MATH  MathSciNet  Google Scholar 

  8. Liu BW, Huang LH (2005) Existence and exponential stability of almost periodic solutions for Hopfield neural networks with delays. Neurocomputing 68:196–207

    Article  Google Scholar 

  9. Xiao B (2009) Existence and uniqueness of almost periodic solutions for a class of Hopfield neural networks with neutral delays. Appl Math Lett 22:528–533

    Article  MATH  MathSciNet  Google Scholar 

  10. Bai CZ (2009) Existence and stability of almost periodic solutions of Hopfield neural networks with continuously distributed delays. Nonlinear Anal 71:5850–5859

    Article  MATH  MathSciNet  Google Scholar 

  11. Shi PL, Dong LZ (2010) Existence and exponential stability of anti-periodic solutions of Hopfield neural networks with impulses. Appl Math Comput 216:623–630

    Article  MATH  MathSciNet  Google Scholar 

  12. Pan LJ, Cao Jinde (2011) Anti-periodic solution for delayed cellular neural networks with impulsive effects. Nonlinear Anal Real World Appl 12:3014–3027

    Article  MATH  MathSciNet  Google Scholar 

  13. Haykin S (1994) Neural networks. Prentice-Hall, NJ

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Wang ZD, Shu HS, Fang J, Liu XH (2006) Robust stability for stochastic Hopfield neural networks with time delays. Nonlinear Anal Real World Appl 7:1119–1128

    Article  MATH  MathSciNet  Google Scholar 

  16. Lou XY, Cui BT (2007) Delay-dependent stochastic stability of delayed Hopfield neural networks with Markovian jump parameters. J Math Anal Appl 328:316–326

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhang JH, Shi P, Qiu JQ (2007) Novel robust stability criteria for uncertain stochastic Hopfield neural networks with time-varying delays. Nonlinear Anal Real World Appl 8:1349–1357

    Article  MATH  MathSciNet  Google Scholar 

  18. Zhou QH, Wan L (2008) Exponential stability of stochastic delayed Hopfield neural networks. Appl Math Comput 199:84–89

    Article  MATH  MathSciNet  Google Scholar 

  19. Sheng L, Gao M, Yang HZ (2009) Delay-dependent robust stability for uncertain stochastic fuzzy Hopfield neural networks with time-varying delays. Fuzzy Sets Syst 160:3503–3517

    Article  MATH  MathSciNet  Google Scholar 

  20. Li XD (2010) Global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type. Appl Math Comput 215:4370–4384

    Article  MATH  MathSciNet  Google Scholar 

  21. Fu XL, Li XD (2011) LMI conditions for stability of impulsive stochastic Cohen–Grossberg neural networks with mixed delays. Commun Nonlinear Sci Numer Simul 16:435–454

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. 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  MATH  MathSciNet  Google Scholar 

  24. Li YK, Chen X, Zhao L (2009) Stability and existence of periodic solutions to delayed Cohen–Grossberg BAM neural networks with impulses on time scales. Neurocomputing 72:1621–1630

    Article  Google Scholar 

  25. Zheng FY, Zhou Z, Ma CQ (2010) Periodic solutions for a delayed neural network model on a special time scale. Appl Math Lett 23:571–575

    Article  MATH  MathSciNet  Google Scholar 

  26. Zhang ZQ, Liu KY (2011) Existence and global exponential stability of a periodic solution to interval general bidirectional associative memory (BAM) neural networks with multiple delays on time scales. Neural Netw 24:427–439

    Article  MATH  Google Scholar 

  27. Li YK (2013) Periodic solutions of nonautonomous cellular neural networks with impulses and delays on time scales. IMA J Math Control Inf. doi:10.1093/imamci/dnt012

  28. Li YK, Wang C (2012) Almost periodic solutions of shunting inhibitory cellular neural networks on time scales. Commun Nonlinear Sci Numer Simul 17:3258–3266

    Article  MATH  MathSciNet  Google Scholar 

  29. Li YK, Yang L (2013) Almost periodic solutions for neutral-type BAM neural networks with delays on time scales. J Appl Math. Article ID 942309.

  30. Li YK, Shu JY (2011) Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales. Commun Nonlinear Sci Numer Simul 16:3326–3336

    Article  MATH  MathSciNet  Google Scholar 

  31. Li YK, Yang L, Wu WQ (2011) Anti-periodic solutions for a class of Cohen–Grossberg neural networks with time-varying delays on time scales. Int J Syst Sci 42:1127–1132

    Article  MATH  MathSciNet  Google Scholar 

  32. Grow D, Sanyal S (2011) Brownian motion indexed by a time scale. Stoch Anal Appl 29:457–472

    Article  MATH  MathSciNet  Google Scholar 

  33. Bohner M, Wintz N (2013) The Kalman filter for linear systems on time scales. J Math Anal Appl 406:419–436

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  36. Sanyal S (2008) Stochastic dynamic equations, Ph.D. Dissertation, Missouri University of Science and Technology, Rolla, Missouri, USA.

  37. Grow D, Sanyal S (2012) The quadratic variation of Brownian motion on a time scale. Stat Probab Lett 82:1677–1680

    Article  MATH  MathSciNet  Google Scholar 

  38. Du NH, Dieu NH (2013) Stochastic dynamic equations on time scales. Acta Math 38(2):317–338

    MATH  MathSciNet  Google Scholar 

  39. Du NH, Dieu NH (2011) The first attempt on the stochastic calculus on time scale. Stoch Anal Appl 29(11):1057–1080

    Article  MATH  MathSciNet  Google Scholar 

  40. David G, Suman S (2013) Existence and uniqueness for stochastic dynamic equations. Int J Stat Probab 2(2):77–88

    Google Scholar 

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

    Book  Google Scholar 

  42. Adıvar M, Raffoul YN (2009) Existence of periodic solutions in totally nonlinear delay dynamic equations. Electron J Qual Theory Differ Equ 1:1–20

    Google Scholar 

  43. Li YK, Wang C (2011) Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales. Abs Appl Anal. Article ID 341520.

  44. Lizama C, Mesquita JG, Ponce R (2014) A connection between almost periodic functions defined on time scales and \({\mathbb{R}}\). Appl Anal. doi:10.1080/00036811.2013.875161.

  45. 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  MATH  MathSciNet  Google Scholar 

  46. Fu M, Liu Z (2010) Square-mean almost automorphic solutions for some stochastic differential equations. Proc Am Math Soc 138(10):3689–3701

    Article  MATH  MathSciNet  Google Scholar 

  47. Bezandry PH, Diagana T (2007) Existence of almost periodic solutions to some stochastic differential equations. Appl Anal 86(7):819–827

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11361072.

Author information

Authors and Affiliations

  1. Department of Mathematics, Yunnan University, Kunming, 650091, Yunnan, People’s Republic of China

    Yongkun Li

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

    Li Yang

  3. School of Mathematics and Computer Science, Yunnan Nationalities University, Kunming, 650091, Yunnan, People’s Republic of China

    Wanqin Wu

Authors
  1. Yongkun Li
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Li Yang
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Wanqin Wu
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Yongkun Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Y., Yang, L. & Wu, W. Square-mean almost periodic solution for stochastic Hopfield neural networks with time-varying delays on timescales. Neural Comput & Applic 26, 1073–1084 (2015). https://doi.org/10.1007/s00521-014-1784-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-014-1784-9

Keywords