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Existence and stability of Stepanov-almost periodic solution in distribution for quaternion-valued memristor-based stochastic neural networks with delays

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Abstract

This paper is concerned with the existence and stability of almost periodic solution in Stepanov-like sense for quaternion-valued stochastic delayed neural networks with memristor by a direct method. Firstly, the existence and uniqueness of the Stepanov-almost periodic solution in the distribution sense of quaternion-valued stochastic delayed neural networks with memristor is considered by utilizing contraction mapping principle. Secondly, the stability for a class of neural network is analyzed via the proof by contradiction, and the exponential stability condition of the network is derived. Finally, the feasibility of the obtained theoretical results is illustrated by a numerical example.

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References

  1. Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85(2), 199–225 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. Parcollet, T.: Quaternion neural networks. Artif. Intell. Rev. 53(4), 2957–2982 (2020)

    Article  Google Scholar 

  3. Luo, L. C., Feng, H., Ding, L.J.: Color image compression based on quaternion neural network principal component analysis. In: 2010 International Conference on Multimedia Technology. IEEE (2010)

  4. Xu, H.Y., Kong, J., Jiang, M.: Human action recognition based on quaternion 3D skeleton representation. Laser Optoelectron. Prog. 2, 168–175 (2018)

    Google Scholar 

  5. Li, R.X., Gao, X.B., Cao, J.D., Zhang, K.: Stability analysis of quaternion-valued Cohen–Grossberg neural networks. Math. Methods Appl. Sci. 42(10), 3721–3738 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Li, Y.K., Qin, J.L., Li, B.: Existence and global exponential stability of anti-periodic solutions for delayed quaternion-valued cellular neural networks with impulsive effects. Math. Methods Appl. Sci. 42(1), 5–23 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhang, D.D., Kou, K.I., Liu, Y., Cao, J.D.: Decomposition approach to the stability of recurrent neural networks with asynchronous time delays in quaternion field. Neural Netw. 94, 55–66 (2017)

    Article  MATH  Google Scholar 

  8. Li, Y.K., Lv, G., Meng, X.F.: Weighted pseudo-almost periodic solutions and global exponential synchronization for delayed QVCNNs. J. Inequal. Appl. 2019(1), 1–23 (2019)

    Article  MathSciNet  Google Scholar 

  9. Wang, Z.R., Joshi, S., Savel’ev, S., et al.: Fully memristive neural networks for pattern classification with unsupervised learning. Nat. Electron. 1(2), 137–145 (2018)

    Article  Google Scholar 

  10. Cai, F., Correll, J.M., Lee, S.H., et al.: A fully integrated reprogrammable memristor-CMOS system for efficient multiply-accumulate operations. Nat. Electron. 2(7), 290–299 (2019)

    Article  Google Scholar 

  11. Sun, G.K., Ji, S.H., Kim, H., et al.: Recent advances in memristive materials for artificial synapses. Adv. Mater. Technol. 3(12), 1800457 (2018)

    Article  Google Scholar 

  12. Hsinyu, T., Stefano, A., Pritish, N., et al.: Recent progress in analog memory-based accelerators for deep learning. J. Phys. D Appl. Phys. 51, 283001 (2018)

    Article  Google Scholar 

  13. Sung, C., Hwang, H., Yoo, I.K.: Perspective: a review on memristive hardware for neuromorphic computation. J. Appl. Phys. 124(15), 151903 (2018)

    Article  Google Scholar 

  14. Li, L.L., Sun, Y.F., Wang, M.M., Huang, W.: Synchronization of coupled memristor neural networks with time delay: positive effects of stochastic delayed impulses. Neural Process. Lett. 53, 4349–4364 (2021)

    Article  Google Scholar 

  15. Sheng, Y., Huang, T.W., Zeng, Z.G., Miao, X.S.: Global exponential stability of memristive neural networks with mixed time-varying delays. IEEE Trans. Neural Netw. Learn. Syst. 32(8), 3690–3699 (2021)

    Article  MathSciNet  Google Scholar 

  16. Jiang, P., Zeng, Z.G., Chen, J.J.: Almost periodic solutions for a memristor-based neural networks with leakage, time-varying and distributed delays. Neural Netw. 68, 34–45 (2015)

    Article  MATH  Google Scholar 

  17. Meng, Z.D., Xiang, Z.R.: Stability analysis of stochastic memristor-based recurrent neural networks with mixed time-varying delays. Neural Comput. Appl. 28(7), 1787–1799 (2017)

    Article  Google Scholar 

  18. Tian, Y.F., Wang, Z.S.: Stochastic stability of Markovian neural networks with generally hybrid transition rates. IEEE Trans. Neural Netw. Learn. Syst. https://doi.org/10.1109/TNNLS.2021.3084925

  19. Hou, Y.Y., Dai, L.H.: Square-mean pseudo almost periodic solutions for quaternion-valued stochastic neural networks with time-varying delays. Math. Probl. Eng. 2021, 6679326 (2021)

    Article  MathSciNet  Google Scholar 

  20. Li, Y.K., Meng, X.F.: Almost automorphic solutions in distribution sense of quaternion-valued stochastic recurrent neural networks with mixed time-varying delays. Neural Process. Lett. 51(4), 1353–1377 (2020)

    Article  Google Scholar 

  21. Yang, T.Q., Xiong, Z.L., Yang, C.P.: Analysis of exponential stability for neutral stochastic Cohen–Grossberg neural networks with mixed delays. Discrete Dyn. Nat. Soc. 2019, 4813103 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bohr, H.: Zur Theorie der fast periodischen Funktionen I. Acta Math. 45, 29–127 (1925)

    Article  MathSciNet  Google Scholar 

  23. Bohr, H.: Zur Theorie der fast periodischen Funktionen II. Acta Math. 46, 101–214 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  24. Andres, J., Pennequin, D.: On Stepanov almost-periodic oscillations and their discretizations. J. Differ. Equ. Appl. 18(10), 1665–1682 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Andres, J., Pennequin, D.: On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations. Proc. Am. Math. Soc. 140(8), 2825–2834 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Maqbul, Md., Bahuguna, D.: Almost periodic solutions for Stepanov-almost periodic differential equations. Differ. Equ. Dyn. Syst. 22, 251–264 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Henríquez, H.R.: On Stepanov-almost periodic semigroups and cosine functions of operators. J. Math. Anal. Appl. 146(2), 420–433 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jiang, Q.D., Wang, Q.R.: Almost periodic solutions for quaternion-valued neural networks with mixed delays on time scales. Neurocomputing 439, 363–373 (2021)

    Article  Google Scholar 

  29. Wang, T.Y., Zhu, Q.X., Cai, W.: Mean-square exponential input-to-state stability of stochastic fuzzy recurrent neural networks with multi-proportional delays and distributed delays. Math. Probl. Eng. 2018, 6289019 (2018)

    MATH  Google Scholar 

  30. Wang, P., Li, B., Li, Y.K.: Square-mean almost periodic solutions for impulsive stochastic shunting inhibitory cellular neural networks with delays. Neurocomputing 167, 76–82 (2015)

    Article  Google Scholar 

  31. Liu, W.D., Huang, J.L., Yao, Q.H.: Stability analysis for quaternion-valued inertial memristor-based neural networks with time delays. Neurocomputing 448, 67–81 (2021)

    Article  Google Scholar 

  32. Wang, D.S., Huang, L.H.: Periodicity and global exponential stability of generalized Cohen–Grossberg neural networks with discontinuous activations and mixed delays. Neural Netw. 51, 80–95 (2014)

  33. Fristedt, B., Gray, L.: A Modern Approach to Probability Theory. Birkhäuser, Boston (1997)

    Book  MATH  Google Scholar 

  34. Klenke, A.: Probability Theory: A Comprehensive Course. Springer, Boston (2013)

    MATH  Google Scholar 

  35. Chen, F., Yang, X., Li, Y.: Almost automorphic solutions for stochastic differential equations with state-dependent switching. Mathematics 47, 97–108 (2017)

  36. Corduneanu, C.: Almost Periodic Oscillations and Waves. Springer, New York (2009)

    Book  MATH  Google Scholar 

  37. Kamenskii, M., Mellah, O., Fitte, P.R.D.: Weak averaging of semilinear stochastic differential equations with almost periodic coefficients. J. Math. Anal. Appl. 427(1), 336–364 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Maqbul, Md.: Stepanov-almost periodic solutions of non-autonomous neutral functional differential equations with functional delay. Mediterr. J. Math. 15, 179 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. Li, Y.K., Xiang, J.L.: Existence and global exponential stability of almost periodic solution for quaternion-valued high-order Hopfield neural networks with delays via a direct method. Math. Methods Appl. Sci. 43(10), 6165–6180 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The research is supported by grants from the National Natural Science Foundation of China (Nos.62172188 and 52072130) and the Natural Science Foundation of Guangdong Province in China (No. 2021A1515011753).

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  1. College of Information Science and Technology, Jinan University, Guangzhou, 510632, China

    Jianglian Xiang & Manchun Tan

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  1. Jianglian Xiang
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  2. Manchun Tan
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Correspondence to Manchun Tan.

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Xiang, J., Tan, M. Existence and stability of Stepanov-almost periodic solution in distribution for quaternion-valued memristor-based stochastic neural networks with delays. Nonlinear Dyn 111, 1715–1732 (2023). https://doi.org/10.1007/s11071-022-07877-7

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