MITTAG-LEFFLER STABILITY OF AN ASYMPTOTICALLY \(\omega\)-PERIODIC SOLUTION TO A DELAYED CELLULAR NEURAL SYSTEM

Mohsen Miraoui     (Laboratory of Chemistry, Materials and Modelling (LR24ES02), Department of Physics, Preparatory Institute for Engineering Studies of Kairouan (IPEIK), University of Kairouan Kairouan 3100, Tunisia)
Sana Mehrez     (Laboratory of Chemistry, Materials and Modelling (LR24ES02), Department of Physics, Preparatory Institute for Engineering Studies of Kairouan (IPEIK), University of Kairouan Kairouan 3100, Tunisia)

Abstract


By using fixed-point methods and the properties of \(S\nu\)-asymptotically \(\omega\)-periodic solutions, we establish the existence, uniqueness, exponential stability, and Mittag-Leffler stability of \(S\nu\)-asymptotically \(\omega\)-periodic solutions for several models of recurrent neural networks with mixed delays and time-varying coefficients. Our main results are illustrated by a numerical example.


Keywords


S-asymptotically ω-periodic, S\(\nu\)-asymptotically ω-periodic, Fixed point theorem, Exponential stability, Mittag-Leffler stability, Neural network

Full Text:

PDF

References


  1. Alimi A.M., Khelifi S., Miraoui M. The measure pseudo almost-periodicity and automorphy in HOHNNs with time varying delays. Int. J. Comput. Math., 2023. Vol. 100, No. 6. P. 1284–1302. DOI: 10.1080/00207160.2023.2178275
  2. Ammar B., Chrif F., Alimi A.M. Existence and uniqueness of pseudo almost-periodic solution recurrent neural networks with time varying coefficient and mixed delays. IEEE Trans. Neural Netw. Learn Syst., 2012. Vol. 23, No. 1. P. 109–118. DOI: 10.1109/TNNLS.2011.2178444
  3. de Andrade B., Cuevas C. \(S\)-asymptotically ω-periodic and asymptotically ω-periodic solutions to semilinear Cauchy problems with non-dense domain. Nonlinear Anal., 2010. Vol. 72. P. 3190–3208. DOI: 10.1016/j.na.2009.12.016
  4. Aouiti C., Ghanmi B., Miraoui M. On the differential equations of recurrent neural networks. Int. J. Comput. Math., 2021. Vol. 98, No. 7. P. 1385–1407. DOI: 10.1080/00207160.2020.1820493
  5. Bai Ch. Existence and stability of almost periodic solutions of Hopfield neural networks with continuously distributed delays. Nonlinear Anal., 2009. Vol. 71, No. 11. P. 5850–5859. DOI: 10.1016/j.na.2009.05.008
  6. Caicedo A., Cuevas C. \(S\)-asymptotically ω-periodic solutions of abstract partial neutral integro-differential equations. Funct. Differ. Equ., 2010. Vol. 17, No. 1–2. P. 59–77.Â
  7. Cao J., Liang J., Lam J. Exponential stability of high-order bidirectional associative memory neural networks with time delays. Physica D., 2004. Vol. 199, No. 3–4. P. 425–436. DOI: 10.1016/j.physd.2004.09.012
  8. Chua L.O., Yang L. Cellular neural networks: theory. IEEE Trans. Circuits Syst., 1988. Vol. 35, No. 10. P. 1257–1272. DOI: 10.1109/31.7600
  9. Cuevas C., Pierri M., Sepulveda A. Weighted \(S\)-asymptotically ω-periodic solutions of a class of fractional differential equations. Adv. Differential Equations, 2011. Vol. 2011, No. 2. Art. no. 584874. DOI: 10.1155/2011/584874
  10. Cuevas C., de Souza J.C. \(S\)-asymptotically ω-periodic solutions of semilinear fractional integro-differential equations. Appl. Math. Lett., 2009. Vol. 22, No. 6. P. 865–870. DOI: 10.1016/j.aml.2008.07.013
  11. Cuevas C., Lizama C. \(S\)-asymptotically ω-periodic solutions for semilinear Volterra equations. Math. Methods Appl. Sci., 2010. Vol. 33, No. 13. P. 1628–1636. DOI: 10.1002/mma.1284
  12. Cuevas C., de Souza J.C. Existence of \(S\)-asymptotically ωω-periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Anal., 2010. Vol. 72, No. 3–4. P. 1683–1689. DOI: 10.1016/j.na.2009.09.007
  13. Duan L. Existence and global exponential stability of pseudo almost periodic solutions of a general delayed BAM neural networks. J. Syst. Sci. Complex., 2018. Vol. 31. P. 608–620. /DOI: 10.1007/s11424-017-6180-y
  14. Eidinejad Z., Saadati R., Repovs D.D. Mittag-Leffler stability and attractiveness of pseudo almost periodic solutions for delayed cellular neural networks. J. Funct. Spaces, 2022. Vol. 2022. Art. no. 3186963. DOI:Â 10.1155/2022/3186963
  15. Ghanmi B., Miraoui M. Stability of unique pseudo almost periodic solutions with measure. Appl. Math., 2020. Vol. 65, No. 4. P. 421–445. DOI: 10.21136/AM.2020.0252-19
  16. Henriquez H.R., Pierri M., Taboas P. On \(S\)-asymptotically ω-periodic functions on Banach spaces and applications. J. Math. Anal. Appl., 2008. Vol. 343, No. 2. P. 1119–1130. DOI: 10.1016/j.jmaa.2008.02.023
  17. Henriquez H.R., Pierri M., Taboas P. Existence of \(S\)-asymptotically ω-periodic solutions for abstract neutral equations. Bull. Aust. Math. Soc., 2008. Vol. 78, No. 3. P. 365–382. DOI: 10.1017/S0004972708000713
  18. Huang H., Cao J., Wang J. Global exponential stability and periodic solutions of recurrent neural networks with delays. Phys. Lett. A, 2002. Vol. 298, No. 5–6. P. 393–404. DOI: 10.1016/S0375-9601(02)00537-6
  19. Liang Z. Asymptotically periodic solutions of a class of second order nonlinear differential equations. Proc. Amer. Math. Soc., 1987. Vol. 99, No. 4. P. 693–699. DOI: 10.2307/2046478
  20. Miraoui M., Alimi A.M., Khelifi S. The measure pseudo almost-periodicity and automorphy in HOHNNs with time-varying delays. Int. J. Comput. Math., 2023. Vol. 100, No. 6. P. 1284–1302. DOI: 10.1080/00207160.2023.2178275
  21. Miraoui M., Yaakoubi N. Measure pseudo almost periodic solutions of shunting inhibitory cellular neural networks with mixed delays. Numer. Funct. Anal. Optim., 2019. Vol. 40, No. 5. P. 571–585. DOI: 10.1080/01630563.2018.1561469
  22. Pierri M. On \(S\)-asymptotically ω-periodic functions and applications. Nonlinear Anal., 2012. Vol. 75, No. 2. P. 651–661. DOI: 10.1016/j.na.2011.08.059
  23. Roska T., Chua L.O. Cellular neural networks with non-linear and delay-type template elements and non-uniform grids. Int. J. Circuit Theory Appl., 1992. Vol. 20, No. 5. P. 469–481.
  24. Utz W.R., Waltman P. Asymptotic almost periodicity of solutions of a system of differential equations. Proc. Amer. Math. Soc., 1967. Vol. 18. P. 597–601.DOI: 10.1090/S0002-9939-1967-0212285-6
  25. Zhang S., Yu Y., Wang H. Mittag-Leffler stability of fractional-order Hopfield neural networks. Nonlin. Anal. Hybrid Syst., 2015. Vol. 16. P. 104–121. DOI: 10.1016/j.nahs.2014.10.001
  26. Zhao H. Existence and global attractivity of almost periodic solution for cellular neural network with distributed delays. Appl. Math. Comput., 2004. Vol. 154, No. 3. P. 683–695. DOI: 10.1016/S0096-3003(03)00743-4




DOI: http://dx.doi.org/10.15826/umj.2026.1.007

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.