\((\mu ,\nu )\)-Pseudo-almost automorphic solutions for high-order Hopfield bidirectional associative memory neural networks

  • Chaouki AouitiEmail author
  • Farah Dridi
Original Article


This article is concerned with a high-order Hopfield bidirectional associative memory neural networks with time-varying coefficients and mixed delays. Sufficient conditions are derived for the existence, the uniqueness and the exponential stability of \((\mu ,\nu )\)-pseudo-almost automorphic solutions of the considered model. Banach fixed-point theorem is applied for the existence and the uniqueness results. Global exponential stability is derived via differential inequalities. Finally, two examples are provided to support the feasibility of the theoretical results.


\((\mu , \nu )\)-Pseudo-almost automorphic function High-order BAM neural networks Global exponential stability 

Mathematics Subject Classification

34C27 37B25 92C20 


Compliance with ethical standards

Conflict of interest

There is no conflict of interest.


  1. 1.
    Abbas S, Yonghui XIA (2013) Existence and attractivity of \(k\)-almost automorphic sequence solution of a model of cellular neural networks with delay. Acta Math Sci 33(1):290–302MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abbas S, Mahto L, Hafayed M, Alimi AM (2014) Asymptotic almost automorphic solutions of impulsive neural network with almost automorphic coefficients. Neurocomputing 142:326–334CrossRefGoogle Scholar
  3. 3.
    Abbas S, Xia Y (2015) Almost automorphic solutions of impulsive cellular neural networks with piecewise constant argument. Neural Process Lett 42(3):691–702CrossRefGoogle Scholar
  4. 4.
    Ait Dads EH, Ezzinbi K, Miraoui M (2015) \((\mu,\nu )\)-Pseudo almost automorphic solutions for some non-autonomous differential equations. Int J Math 26(11):1550090MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Alimi AM, Aouiti C, Chérif F, Dridi F, M’hamdi MS (2018) Dynamics and oscillations of generalized high-order Hopfield Neural Networks with mixed delays. Neurocomputing.
  6. 6.
    Ammar B, Brahmi H, Chérif F (2017) On the weighted pseudo-almost periodic solution for BAM networks with delays. Neural Process Lett. Google Scholar
  7. 7.
    Aouiti C (2018) Oscillation of impulsive neutral delay generalized high-order Hopfield neural networks. Neural Comput Appl 29(9):477–495CrossRefGoogle Scholar
  8. 8.
    Aouiti C (2016) Neutral impulsive shunting inhibitory cellular neural networks with time-varying coefficients and leakage delays. Cogn Neurodynamics 10(6):573–591MathSciNetCrossRefGoogle Scholar
  9. 9.
    Aouiti C, M’hamdi MS, Chérif F (2017) New results for impulsive recurrent neural networks with time-varying coefficients and mixed delays. Neural Process Lett 46(2):487–506CrossRefGoogle Scholar
  10. 10.
    Aouiti C, Coirault P, Miaadi F, Moulay E (2017) Finite time boundedness of neutral high-order Hopfield neural networks with time delay in the leakage term and mixed time delays. Neurocomputing 260:378–392CrossRefGoogle Scholar
  11. 11.
    Aouiti C, M’hamdi MS, Touati A (2017) Pseudo Almost Automorphic Solutions of Recurrent Neural Networks with Time-Varying Coefficients and Mixed Delays. Neural Process Lett 45(1):121–140CrossRefGoogle Scholar
  12. 12.
    Aouiti C, M’hamdi MS, Cao J, Alsaedi A (2017) Piecewise Pseudo Almost Periodic Solution for Impulsive Generalised High-Order Hopfield Neural Networks with Leakage Delays. Neural Process Lett 45(2):615–648CrossRefGoogle Scholar
  13. 13.
    Aouiti C, Dridi F (2018) Piecewise asymptotically almost automorphic solutions for impulsive non-autonomous high-order Hopfield neural networks with mixed delays. Neural Comput Appl. Google Scholar
  14. 14.
    Aouiti C, Gharbia IB, Cao J, M’hamdi MS, Alsaedi A (2018) Existence and global exponential stability of pseudo almost periodic solution for neutral delay BAM neural networks with time-varying delay in leakage terms. Chaos Solitons Fractals 107:111–127MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Blot J, Cieutat P, Ezzinbi K (2012) Measure theory and pseudo almost automorphic functions: new developments and applications. Nonlinear Anal Theory Methods Appl 75(4):2426–2447MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cao J, Wang L (2002) Exponential stability and periodic oscillatory solution in BAM networks with delays. IEEE Trans Neural Netw 13(2):457–463CrossRefGoogle Scholar
  17. 17.
    Cao J, Liang J, Lam J (2004) Exponential stability of high-order bidirectional associative memory neural networks with time delays. Physica D Nonlinear Phenom 199(3):425–436MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chang CY, Chung PC (2000) Two-layer competitive based Hopfield neural network for medical image edge detection. Opt Eng 39(3):695–704CrossRefGoogle Scholar
  19. 19.
    Chang CA, Angkasith V (2001) Using Hopfield neural networks for operational sequencing for prismatic parts on NC machines. Eng Appl Artif Intell 14(3):357–368CrossRefGoogle Scholar
  20. 20.
    Chang YK, Luo XX (2014) Existence of \(\mu\)-pseudo almost automorphic solutions to a neutral differential equation by interpolation theory. Filomat 28(3):603–614MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen A, Huang L, Cao J (2003) Existence and stability of almost periodic solution for BAM neural networks with delays. Appl Math Comput 137(1):177–193MathSciNetzbMATHGoogle Scholar
  22. 22.
    Choi E, Schuetz A, Stewart WF, Sun J (2016) Using recurrent neural network models for early detection of heart failure onset. J Am Med Inform Assoc 24(2):361–370Google Scholar
  23. 23.
    Diagana T (2013) Almost automorphic type and almost periodic type functions in abstract spaces. Springer, New YorkCrossRefzbMATHGoogle Scholar
  24. 24.
    Ait Dads EH, Drisi N, Ezzinbi K, Ziat M (2016) Exponential dichotomy and \((\mu , \nu )\)-Pseudo almost automorphic solutions for some ordinary differential equations. Commun Optim Theory 2016:6Google Scholar
  25. 25.
    Ezzinbi K, Fatajou S, N’Guérékata GM (2009) Pseudo-almost-automorphic solutions to some neutral partial functional differential equations in Banach spaces. Nonlinear Anal Theory Methods Appl 70(4):1641–1647MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gui R, Yang Z (2006) Application of Hopfield neural network for extracting Doppler spectrum from ocean echo. Radio Sci 41(4):RS4S90-1–RS4S90-6MathSciNetCrossRefGoogle Scholar
  27. 27.
    Huo HF, Li WT, Tang S (2009) Dynamics of high-order BAM neural networks with and without impulses. Appl Math Comput 215(6):2120–2133MathSciNetzbMATHGoogle Scholar
  28. 28.
    Jagannatha AN, Yu H (2016, June). Bidirectional RNN for medical event detection in electronic health records. In: Proceedings of the conference. Association for Computational Linguistics. North American Chapter. Meeting (Vol 2016, p 473). NIH Public AccessGoogle Scholar
  29. 29.
    Kavitha V, Wang PZ, Murugesu R (2013) Existence of weighted pseudo almost automorphic mild solutions to fractional integro-differential equations. J Fract Calc Appl 4(1):37–55Google Scholar
  30. 30.
    Liang J, Zhang J, Xiao TJ (2008) Composition of pseudo almost automorphic and asymptotically almost automorphic functions. J Math Anal Appl 340(2):1493–1499MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Li Y, Zhao L, Yang L (2015) \(C^1\)-Almost periodic solutions of BAM neural networks with time-varying delays on time scales. Sci World J. Google Scholar
  32. 32.
    Ma F, Chitta R, Zhou J, You Q, Sun T, Gao J (2017, August). Dipole: Diagnosis prediction in healthcare via attention-based bidirectional recurrent neural networks. In: Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining (pp 1903-1911). ACMGoogle Scholar
  33. 33.
    Marcus CM, Westervelt RM (1989) Stability of analog neural networks with delay. Phys Rev A 39(1):347MathSciNetCrossRefGoogle Scholar
  34. 34.
    Miraoui M (2017) \(\mu\)-Pseudo-Almost Automorphic Solutions for some Differential Equations with Reflection of the Argument. Numer Funct Anal Optim 38(3):376–394MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Pu Z, Rao R (2018) Exponential stability criterion of high-order BAM neural networks with delays and impulse via fixed point approach. Neurocomputing 292:63–71CrossRefGoogle Scholar
  36. 36.
    Ren F, Cao J (2007) Periodic oscillation of higher-order bidirectional associative memory neural networks with periodic coefficients and delays. Nonlinearity 20(3):605MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sammouda R, Mathkour HB (2015) Lung region segmentation using artificial neural network hopfield model for cancer diagnosis in thorax CT images. Autom Control Intell Syst 3(2):19–25CrossRefGoogle Scholar
  38. 38.
    Sammouda RS, Wang X, Basilion JP (2015) Hopfield Neural Network for the segmentation of Near Infrared Fluorescent images for diagnosing prostate cancer. In: 2015 6th International conference on information and communication systems (ICICS), pp 111-118. IEEEGoogle Scholar
  39. 39.
    Singh YP, Yadav SV, Gupta A, Khare A (2009) Bi directional associative memory neural network method in the character recognition. J Theor Appl Inf Technol 5(4):382–386Google Scholar
  40. 40.
    Wang J, Jiang H, Hu C (2014) Existence and stability of periodic solutions of discrete-time Cohen-Grossberg neural networks with delays and impulses. Neurocomputing 142:542–550CrossRefGoogle Scholar
  41. 41.
    Xu C, Chen L, Guo T (2018) Anti-periodic oscillations of bidirectional associative memory (BAM) neural networks with leakage delays. J Inequal Appl. MathSciNetGoogle Scholar
  42. 42.
    Yang W, Yu W, Cao J, Alsaadi FE, Hayat T (2017) Almost automorphic solution for neutral type high-order Hopfield BAM neural networks with time-varying leakage delays on time scales. Neurocomputing 267:241–260CrossRefGoogle Scholar
  43. 43.
    Zhang L, Si L (2007) Existence and exponential stability of almost periodic solution for BAM neural networks with variable coefficients and delays. Appl Math Comput 194(1):215–22MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Zhou H, Zhou Z, Jiang W (2015) Almost periodic solutions for neutral type BAM neural networks with distributed leakage delays on time scales. Neurocomputing 157:223–230CrossRefGoogle Scholar
  45. 45.
    Zhou H, Alzabut J (2017) Existence and stability of neutraltype BAM neural networks with time delay in the neutral and leakage terms on time scales. Glob J Pure Appl Math 13(2):589–616Google Scholar

Copyright information

© The Natural Computing Applications Forum 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences of Bizerta, Research Units of Mathematics and Applications UR13ES47University of CarthageZarzouna, BizertaTunisia

Personalised recommendations