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Convergence analysis on time scales for HOBAM neural networks in the Stepanov-like weighted pseudo almost automorphic space

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Abstract

We start by presenting the concept of Stepanov-like weighted pseudo almost automorphic on time-space scales. Besides, we introduce a novel model of high-order BAM neural networks with mixed delays. To the best of our knowledge, this is the first time to study the convergence analysis for one system modeling the recurrent neural networks. Since it is a \(\Delta \)-dynamic system on time-space scales, the results obtained in our work are new and attractive. The main difficulty in our theoretical work is the construction of Lyapunov–Krasovskii functional and the application of the Banach’s fixed-point theorem. By fabricating an appropriate Lyapunov–Krasovskii Functional (LKF), some new sufficient conditions are obtained in terms of linear algebraic equations to guarantee the convergence to the Stepanov-like WPAA on time-space scales solution for the labeled neural networks solutions. The obtained conditions are expressed in terms of algebraic equations whose feasibility can be checked easily by a simple calculus. Furthermore, we have collated our effort with foregoing one in the available literatures and showed that it is less conserved. Finally, two numerical examples show the feasibility of our theoretical outcomes.

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References

  1. Lizama C, Mesquita JG (2013) Almost automorphic solutions of dynamic equations on time scales. J Funct Anal 265(10):2267–2311

    Article  MathSciNet  Google Scholar 

  2. Arbi A, Alsaedi A, Cao J (2018) Delta-differentiable weighted pseudo-almost automorphicity on time-space scales for a novel class of high-order competitive neural networks with WPAA coefficients and mixed delays. Neural Process Lett 47(1):203–232

    Article  Google Scholar 

  3. Arbi A, Cao J, Alsaedi A (2018) Improved synchronization analysis of competitive neural networks with time-varying delays. Nonlinear Anal Model Control 23(1):82–102

    Article  MathSciNet  Google Scholar 

  4. Zhu H, Zhu Q, Sun X, Zhou H (2016) Existence and exponential stability of pseudo almost automorphic solutions for Cohen-Grossberg neural networks with mixed delays. Adv Differ Equ 2016:120

    Article  MathSciNet  Google Scholar 

  5. Li Y, 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 242:679–693

  6. Guo Y (2009) Mean square global asymptotic stability of stochastic recurrent neural networks with distributed delays. Appl Math Comput 215:791–795

    MathSciNet  MATH  Google Scholar 

  7. Sowmiya C, Raja R, Cao J, Rajchakit G, Alsaedi A (2017) Enhanced robust finite-time passivity for Markovian jumping discrete-time BAM neural networks with leakage delay. Adv Differ Equ 1:318

    Article  MathSciNet  Google Scholar 

  8. Bao H, Cao J (2012) Exponential stability for stochastic BAM networks with discrete and distributed delays. Appl Math Comput 218(11):6188–6199

    MathSciNet  MATH  Google Scholar 

  9. Cao J, Rakkiyappan R, Maheswari K (2016) Chandrasekar A (2016) Exponential \(H_{\infty }\) filtering analysis for discrete-time switched neural networks with random delays using sojourn probabilities. Sci China Technol Sci 59(3):387–402

    Article  Google Scholar 

  10. Maharajan C, Raja R, Cao J, Rajchakit G, Tu Z (2018) LMI-based results on exponential stability of BAM-type neural networks with leakage and both time-varying delays: a non-fragile state estimation approach. Appl Math Comput 326:33–55

    MathSciNet  MATH  Google Scholar 

  11. Zhang A, Qiu J, She J (2014) Existence and global exponential stability of periodic solution for high-order discrete-time BAM neural networks. Neural Netw 50:98–109

    Article  Google Scholar 

  12. Ren F, Cao J (2007) Periodic oscillation of higher-order bidirectional associative memory neural networks with periodic coefficients and delays. Nonlinearity 20:605–629

    Article  MathSciNet  Google Scholar 

  13. Kosko B (1988) Bi-directional associative memories. IEEE Trans Syst Man Cybern 18(1):49–60

    Article  MathSciNet  Google Scholar 

  14. Arbi A (2018) Dynamics of BAM neural networks with mixed delays and leakage time-varying delays in the weighted pseudo-almost periodic on time-space scales. Math Methods Appl Sci 41(3):1230–1255. https://doi.org/10.1002/mma.4661

    Article  MathSciNet  MATH  Google Scholar 

  15. 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):177–193

    MathSciNet  MATH  Google Scholar 

  16. Cao J (2003) Global asymptotic stability of delayed bi-directional associative memory neural networks. Appl Math Comput 142:333–339

    MathSciNet  MATH  Google Scholar 

  17. Arik S, Tavsanoglu V (2005) Global asymptotic stability analysis of bidirectional associative memory neural networks with constant time delays. Neurocomputing 68:161–176

    Article  Google Scholar 

  18. Ozcan N, Arik S (2009) A new sufficient condition for global robust stability of bidirectional associative memory neural networks with multiple time delays. Nonlinear Anal Real World Appl 10(5):3312–3320

    Article  MathSciNet  Google Scholar 

  19. Huo HF, Li WT, Tang S (2009) Dynamics of high-order BAM neural networks with and without impulses. Appl Math Comput 215(6):2120–2133

    MathSciNet  MATH  Google Scholar 

  20. Cao J, Liang J, Lam J (2004) Exponential stability of high-order bidirectional associative memory neural networks with time delays. Phys D 199(3):425–436

    Article  MathSciNet  Google Scholar 

  21. Chandran S, Ramachandran R, Cao J, Agarwal Ravi P, Rajchakit G (2019) Passivity analysis for uncertain BAM neural networks with leakage, discrete and distributed delays using novel summation inequality. Int J Control Autom Syst 17(8):2114–2124

    Article  Google Scholar 

  22. Rajchakit G, Pratap A, Raja R, Cao J, Alzabut J, Huang C (2019) Hybrid control scheme for projective lag synchronization of Riemann-Liouville sense fractional order memristive BAM neuralnetworks with mixed delays. Mathematics 7(8):759

    Article  Google Scholar 

  23. Saravanakumar R, Rajchakit G, Syed Ali M, Hoon Joo Y (2019) Exponential dissipativity criteria for generalized BAM neural networks with variable delays. Neural Comput Appl 31(7):2717–2726

    Article  Google Scholar 

  24. Pratap A, Raja R, Rajchakit G, Cao J, Bagdasar O (2019) Mittag-Leffler state estimator design and synchronization analysis for fractional-order BAM neural networks with time delays. Int J Adapt Control Signal Process 33(5):855–874

    Article  MathSciNet  Google Scholar 

  25. Sowmiya C, Raja R, Cao J, Rajchakit G (2018) Enhanced result on stability analysis of randomly occurring uncertain parameters, leakage, and impulsive BAM neural networks with time-varying delays: discrete-time case. Int J Adapt Control Signal Process 32(7):1010–1039

    Article  MathSciNet  Google Scholar 

  26. Sowmiya C, Raja R, Cao J, Li X, Rajchakit G (2018) Discrete-time stochastic impulsive BAM neural networks with leakage and mixed time delays: an exponential stability problem. J Franklin Inst 355(10):4404–4435

    Article  MathSciNet  Google Scholar 

  27. Sowmiya C, Raja R, Cao J, Rajchakit G (2018) Impulsive discrete-time BAM neural networks with random parameter uncertainties and time-varying leakage delays: an asymptotic stability analysis. Nonlinear Dyn 91(4):2571–2592

    Article  Google Scholar 

  28. 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. Neurocomputing 275:2588–2602

    Article  Google Scholar 

  29. Gopalsamy K (2007) Leakage delays in BAM. J Math Anal Appl 325(2):1117–1132

    Article  MathSciNet  Google Scholar 

  30. 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):2808–2825

    Article  MathSciNet  Google Scholar 

  31. Arbi A, Cao J (2017) Pseudo-almost periodic solution on time-space scales for a novel class of competitive neutral-type neural networks with mixed time-varying delays and leakage delays. Neural Process Lett 46(2):719–745

    Article  Google Scholar 

  32. 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):559–566

    Article  MathSciNet  Google Scholar 

  33. Maharajan C, Raja R, Cao J, Rajchakit G (2018) Novel global robust exponential stability criterion for uncertain inertial-type BAM neural networks with discrete and distributed time-varying delays via Lagrange sense. J Franklin Inst 355(11):4727–4754

  34. Bohner M, Peterson A (2003) Advances in dynamic equations on time scales. Birkhäuser, Boston

    Book  Google Scholar 

  35. Guseinov GS (2003) Integration on time scales. J Math Anal Appl 285:107–127

    Article  MathSciNet  Google Scholar 

  36. Wang C, Ravi PA (2014) Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive \(\nabla \)-dynamic equations on time scales. Adv Differ Equ 1:1–29

    MathSciNet  MATH  Google Scholar 

  37. Diagana T (2013) Almost automorphic type and almost periodic type functions in abstract spaces. Spring, Cham

    Book  Google Scholar 

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Acknowledgements

We would like to thank Prof. Dr. Haydar Akca and both anonymous reviewers and the editor for their insightful comments on the paper, as these comments led us to an improvement of the work.

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Authors and Affiliations

  1. Laboratory of Engineering Mathematics (LR01ES13), Tunisia Polytechnic School, University of Carthage, Tunis, Tunisia

    Adnène Arbi

  2. Department of Advanced Sciences and Technologies at National School of Advanced Sciences and Technologies of Borj Cedria, University of Carthage, Tunis, Tunisia

    Adnène Arbi

  3. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, China

    Yingxin Guo

  4. Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing, China

    Jinde Cao

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  1. Adnène Arbi
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  2. Yingxin Guo
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  3. Jinde Cao
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Correspondence to Adnène Arbi.

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Arbi, A., Guo, Y. & Cao, J. Convergence analysis on time scales for HOBAM neural networks in the Stepanov-like weighted pseudo almost automorphic space. Neural Comput & Applic 33, 3567–3581 (2021). https://doi.org/10.1007/s00521-020-05183-0

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  • DOI: https://doi.org/10.1007/s00521-020-05183-0

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