Piecewise Pseudo Almost-Periodic Solutions of Impulsive Fuzzy Cellular Neural Networks with Mixed Delays

  • Chaouki AouitiEmail author
  • Imen Ben Gharbia


This article examines the existence of the unique piecewise pseudo almost periodic for impulsive fuzzy cellular neural networks by using the contraction mapping principle and piecewise pseudo almost periodic function theory. Further, sufficient certain conditions for their global exponential stability are produced through the use of differential inequality and generalized Gronwall–Bellman inequality. Our results are new and complement some previously known ones. Two examples and their numerical simulations are performed to ensure our theoretical results.


Impulses Fuzzy cellular neural networks Piecewise pseudo almost-periodic function Generalized Gronwall–Bellman inequality 

Mathematics Subject Classification

34C27 37B25 92C20 



  1. 1.
    Aouiti C (2016) Neutral impulsive shunting inhibitory cellular neural networks with time-varying coefficients and leakage delays. Cogn Neurodyn 10(6):573–591MathSciNetCrossRefGoogle Scholar
  2. 2.
    Duan L, Fang X, Huang C (2018) Global exponential convergence in a delayed almost periodic Nicholson’s blowflies model with discontinuous harvesting. Math Methods Appl Sci 41(5):1954–1965MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Aouiti C (2018) Oscillation of impulsive neutral delay generalized high-order hopfield neural networks. Neural Comput Appl 29(9):477–495CrossRefGoogle Scholar
  4. 4.
    Yang W (2014) Periodic solution for fuzzy Cohen–Grossberg BAM neural networks with both time-varying and distributed delays and variable coefficients. Neural Process Lett 40(1):51–73CrossRefGoogle Scholar
  5. 5.
    Cai Z, Huang J, Huang L (2018) Periodic orbit analysis for the delayed Filippov system. Proc Am Math Soc 146(11):4667–4682MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chua LO, Yang L (1988) Cellular neural networks: applications. IEEE Trans Circuits Syst 35(10):1273–1290MathSciNetCrossRefGoogle Scholar
  7. 7.
    Baldi P, Sadowski P (2018) Learning in the machine: recirculation is random backpropagation. Neural Netw 108:479–494CrossRefGoogle Scholar
  8. 8.
    Costarelli D, Vinti G (2016) Pointwise and uniform approximation by multivariate neural network operators of the max-product type. Neural Netw 81:81–90zbMATHCrossRefGoogle Scholar
  9. 9.
    Costarelli D, Vinti G (2017) Convergence for a family of neural network operators in orlicz spaces. Math Nachr 290(2–3):226–235MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Schmidhuber J (2015) Deep learning in neural networks: an overview. Neural Netw 61:85–117CrossRefGoogle Scholar
  11. 11.
    Ahmad S, Stamova IM (2008) Global exponential stability for impulsive cellular neural networks with time-varying delays. Nonlinear Anal Theory Methods Appl 69(3):786–795MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    De Vries B, Principe JC (1992) The gamma model—a new neural model for temporal processing. Neural Netw 5(4):565–576CrossRefGoogle Scholar
  13. 13.
    Huang C, Zhang H, Huang L (2019) Almost periodicity analysis for a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality term. Commun Pure Appl Anal 18(6):3337–3349MathSciNetCrossRefGoogle Scholar
  14. 14.
    Huang C, Zhang H (2019) Periodicity of non-autonomous inertial neural networks involving proportional delays and non-reduced order method. Int J Biomath 12(02):1950016MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Aouiti C, abed Assali E, Cao J, Alsaedi A (2018) Global exponential convergence of neutral-type competitive neural networks with multi-proportional delays, distributed delays and time-varying delay in leakage delays. Int J Syst Sci 49(10):2202–2214MathSciNetCrossRefGoogle Scholar
  16. 16.
    Huang C, Cao J, Wen F, Yang X (2016) Stability analysis of SIR model with distributed delay on complex networks. PLoS ONE 11(8):e0158813CrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    Aouiti C, Dridi F (2018) \((\mu,\nu )\)-Pseudo-almost automorphic solutions for high-order Hopfield bidirectional associative memory neural networks. Neural Comput Appl. CrossRefGoogle Scholar
  19. 19.
    Meng F, Li K, Song Q, Liu Y, Alsaadi FE (2019) Periodicity of Cohen–Grossberg-type fuzzy neural networks with impulses and time-varying delays. Neurocomputing 325:254–259CrossRefGoogle Scholar
  20. 20.
    Aouiti C, Ben Gharbia I, 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 107:111–127MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Yang T, Yang LB (1996) The global stability of fuzzy cellular neural network. IEEE Trans Circuits Syst I Fundam Theory Appl 43(10):880–883MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tang Y (2019) Exponential stability of pseudo almost periodic solutions for fuzzy cellular neural networks with time-varying delays. Neural Process Lett 49(2):851–861CrossRefGoogle Scholar
  23. 23.
    Tang R, Yang X, Wan X, Zou Y, Cheng Z, Fardoun HM (2019) Finite-time synchronization of nonidentical BAM discontinuous fuzzy neural networks with delays and impulsive effects via non-chattering quantized control. Commun Nonlinear Sci Numer Simul 78:104893MathSciNetCrossRefGoogle Scholar
  24. 24.
    Duan L, Huang L, Guo Z, Fang X (2017) Periodic attractor for reaction–diffusion high-order Hopfield neural networks with time-varying delays. Comput Math Appl 73(2):233–245MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Bao H (2018) Existence and stability of anti-periodic solutions for FCNNs with time-varying delays and impulsive effects on time scales. Int J Comput Sci Math 9(5):474–483MathSciNetCrossRefGoogle Scholar
  26. 26.
    ZZhang Q, Yang L, Liu J (2014) Existence and stability of anti-periodic solutions for impulsive fuzzy Cohen–Grossberge neural networks on time scales. Math Slovaca 64(1):119–138MathSciNetzbMATHGoogle Scholar
  27. 27.
    Aouiti C, Gharbia IB, Cao J, Alsaedi A (2019) Dynamics of impulsive neutral-type BAM neural networks. J Frankl Inst 356(4):2294–2324MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Liu B (2013) Global exponential stability for BAM neural networks with time-varying delays in the leakage terms. Nonlinear Anal Real World Appl 14(1):559–566MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Xia Z (2016) Pseudo almost periodic mild solution of nonautonomous impulsive integro-differential equations. Mediterr J Math 13(3):1065–1086MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Li Y, 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(7–9):1621–1630CrossRefGoogle Scholar
  31. 31.
    Li Yt, Yang Cb (2006) Global exponential stability analysis on impulsive BAM neural networks with distributed delays. J Math Anal Appl 324(2):1125–1139MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Samidurai R, Sakthivel R, Anthoni SM (2009) Global asymptotic stability of BAM neural networks with mixed delays and impulses. Appl Math Comput 212(1):113–119MathSciNetzbMATHGoogle Scholar
  33. 33.
    Xia Y, Huang Z, Han M (2008) Existence and globally exponential stability of equilibrium for BAM neural networks with impulses. Chaos Solitons Fractals 37(2):588–597MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Wang W, Liu B (2014) Global exponential stability of pseudo almost periodic solutions for SICNNs with time-varying leakage delays. Abstr Appl Anal 2014:967328MathSciNetzbMATHGoogle Scholar
  35. 35.
    Liu X, Ballinger G (2003) Boundedness for impulsive delay differential equations and applications to population growth models. Nonlinear Anal Theory Methods Appl 53(7–8):1041–1062MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Fullér R (1995) Neural fuzzy systems, Lecture Notes. Abo Akademi University Google Scholar
  37. 37.
    Lakshmikantham V, Simeonov PS (1989) Theory of impulsive differential equations, vol 6. World Scientific, SingaporezbMATHCrossRefGoogle Scholar
  38. 38.
    Cao J, Wang J (2005) Global asymptotic and robust stability of recurrent neural networks with time delays. IEEE Trans Circuits Syst I Regul Pap 52(2):417–426MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Song Q, Zhang J, Maharajan C, Raja R, Cao J, Rajchakit G, Alsaedi A (2018) Impulsive Cohen–Grossberg BAM neural networks with mixed time-delays: an exponential stability analysis issue. Neurocomputing 275:2588–2602CrossRefGoogle Scholar
  40. 40.
    Li Y, Zhao L, Zhang T (2011) Global exponential stability and existence of periodic solution of impulsive Cohen–Grossberg neural networks with distributed delays on time scales. Neural Process Lett 33(1):61–81CrossRefGoogle Scholar
  41. 41.
    Hu M, Wang L (2010) Existence and stability of anti-periodic solutions for an impulsive Cohen–Grossberg sicnns on time scales. Int J Math Comput Sci 6(3):159–165Google Scholar
  42. 42.
    Şaylı M, Yılmaz E (2015) Periodic solution for state-dependent impulsive shunting inhibitory CNNs with time-varying delays. Neural Netw 68:1–11zbMATHCrossRefGoogle Scholar
  43. 43.
    Huang C, Liu B, Tian X, Yang L, Zhang X (2019) Global convergence on asymptotically almost periodic SICNNs with nonlinear decay functions. Neural Process Lett 49(2):625–641CrossRefGoogle Scholar
  44. 44.
    Liang J, Qian H, Liu B (2018) Pseudo almost periodic solutions for fuzzy cellular neural networks with multi-proportional delays. Neural Process Lett 48(2):1201–1212CrossRefGoogle Scholar
  45. 45.
    Huang Z (2017) Almost periodic solutions for fuzzy cellular neural networks with time-varying delays. Neural Comput Appl 28(8):2313–2320CrossRefGoogle Scholar
  46. 46.
    Liu Y, Huang Z, Chen L (2012) Almost periodic solution of impulsive Hopfield neural networks with finite distributed delays. Neural Comput Appl 21(5):821–831CrossRefGoogle Scholar
  47. 47.
    Xu CJ (2016) Existence and exponential stability of anti-periodic solution in cellular neural networks with time-varying delays and impulsive effects. Electron J Differ Equ 2016(02):1–14MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Research Units of Mathematics and Applications UR13ES47, Department of Mathematics, Faculty of Sciences of BizertaUniversity of CarthageZarzouna, BizertaTunisia

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