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The existence and averaging principle for Caputo fractional stochastic delay differential systems

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Abstract

In this paper, we first establish the existence and uniqueness theorem for solutions of Caputo type fractional stochastic delay differential systems by using delayed perturbation of Mittag-Leffler function. Secondly, we obtain an averaging principle for the solution of the considered system under some suitable assumptions. Finally, two simulation examples are given to verify the theoretical results.

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References

  1. Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific, Singapore (2001)

    Google Scholar 

  2. Wang, J., Fečkan, M., Zhou, Y.: A Survey on Impulsive Fractional Differential Equations. Fract. Calc. Appl. Anal. 21(3), 806–831 (2018). https://doi.org/10.1515/fca-2016-0044

    Article  MathSciNet  MATH  Google Scholar 

  3. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  4. Sabatier, J., Agrawal, O.P., Machado, J.A.T.: Advances in Fractional Calculus. Springer, New York (2007)

    Book  MATH  Google Scholar 

  5. Wang, J., Fečkan, M., Zhou, Y.: Center stable manifold for planar fractional damped equations. Appl. Math. Comput. 296, 257–269 (2017)

    MathSciNet  MATH  Google Scholar 

  6. You, Z., Wang, J., O’Regan, D., Zhou, Y.: Relative controllability of delay differential systems with impulses and linear parts defined by permutable matrices. Math. Meth. Appl. Sci. 42(3), 954–968 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wang, X., Wang, J., Fečkan, M.: Controllability of conformable differential systems. Nonlinear Anal. Model. Control 25(4), 658–674 (2020)

    MathSciNet  MATH  Google Scholar 

  8. Oksendal, B.: Stochastic Differential Equations. Springer, New York (2003)

    Book  MATH  Google Scholar 

  9. Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, New York (2013)

    MATH  Google Scholar 

  10. Mao, X.: Stochastic Differential Equations and Applications. Horwood Publishing Limited, Chichester (2008)

    Book  Google Scholar 

  11. Zhu, Q.: Stabilization of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control. IEEE T. Automat. Contr. 64(9), 3764–3771 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhu, Q., Huang, T.: Stability analysis for a class of stochastic delay nonlinear systems driven by G-Brownian motion. Systems Control Lett. 140, 104699 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. **ao, G., Wang, J.: Stability of solutions of Caputo fractional stochastic differential equations. Nonlinear Anal. Model. Control 26(4), 581–596 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Khasminskii, R.Z.: On the principle of averaging the Itô stochastic differential equations. Kibernetika 4(3), 260–279 (1968)

    Google Scholar 

  15. Xu, Y., Duan, J., Xu, W.: An averaging principle for stochastic dynamical systems with Lévy noise. Physica D. 240(17), 1395–1401 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gao, P.: Averaging principle for stochastic Korteweg-de Vries equation. J. Differential Equations 267(12), 6872–6909 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ma, S., Kang, Y.: An averaging principle for stochastic switched systems with Lévy noise. Math. Meth. Appl. Sci. 43(15), 8714–8727 (2020)

    Article  MATH  Google Scholar 

  18. Xu, W., Xu, W., Zhang, S.: The averaging principle for stochastic differential equations with Caputo fractional derivative. Appl. Math. Lett. 93, 79–84 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guo, Z., Hu, J., Yuan, C.: Averaging principle for a type of Caputo fractional stochastic differential equations. Chaos 31, 053123 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Luo, D., Zhu, Q., Luo, Z.: An averaging principle for stochastic fractional differential equations with time-delays. Appl. Math. Lett. 105, 106290 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Luo, D., Zhu, Q., Luo, Z.: A novel result on averaging principle of stochastic Hilfer-type fractional systems involving non-Lipshitz coefficients. Appl. Math. Lett. 122, 107549 (2021)

    Article  MATH  Google Scholar 

  22. Ahmed, H.M., Zhu, Q.: The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jumps. Appl. Math. Lett. 112, 106755 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. **ao, G., Fečkan, M., Wang, J.: On the averaging principle for stochastic differential equations involving Caputo fractional derivative. Chaos 31, 101105 (2022)

    Article  MathSciNet  Google Scholar 

  24. Xu, W., Duan, J., Xu, W.: An averaging principle for fractional stochastic differential equations with Lévy noise. Chaos 30, 083126 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mahmudov, N.I.: Delayed perturbation of Mittag-Leffler functions and their applications to fractional linear delay differential equations. Math. Meth. Appl. Sci. 42(16), 5489–5497 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. You, Z., Fečkan, M., Wang, J.: Relative controllability of fractional delay differential equations via delayed perturbation of Mittag-Leffler functions. J. Comput. Appl. Math. 378, 112939 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Son, D., Huong, P., Kloeden, P., Tuan, H.: Asymptotic separation between solutions of Caputo fractional stochastic differential equations. Stoch. Anal. Appl. 36(4), 654–664 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, New York (2012)

    Book  MATH  Google Scholar 

  29. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2), 1075–1081 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (12161015 and 12201148), Guizhou Provincial Science and Technology Projects (No.QKHJC-ZK[2022]YB069), Qian Ke He ** Tai Ren Cai-YSZ[2022] 002, Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020] 5016), and Major Project of Guizhou Postgraduate Education and Teaching Reform (YJSJGKT[2021]041). The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the help from the editor too.

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Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, People’s Republic of China

    Mengmeng Li & **rong Wang

  2. Supercomputing Algorithm and Application Laboratory of Guizhou University and Gui’an Scientific Innovation Company, Guiyang, 550025, Guizhou, People’s Republic of China

    **rong Wang

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  1. Mengmeng Li
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  2. **rong Wang
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Correspondence to **rong Wang.

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Li, M., Wang, J. The existence and averaging principle for Caputo fractional stochastic delay differential systems. Fract Calc Appl Anal 26, 893–912 (2023). https://doi.org/10.1007/s13540-023-00146-3

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  • DOI: https://doi.org/10.1007/s13540-023-00146-3

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