AIMS Mathematics, 2020, 5(6): 6169-6182. doi: 10.3934/math.2020396

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Boundedness analysis of non-autonomous stochastic differential systems with Lévy noise and mixed delays

1 Department of Mathematics, Zhejiang International Studies University, Hangzhou 310023, PR China
2 Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, PR China

The present research studies the boundedness issue of Lévy driven non-autonomous stochastic differential systems with mixed discrete and distributed delays. A set of sufficient conditions of the $p$th moment globally asymptotical boundedness is obtained by combining the Lyapunov function method with the inequality technique. The proposed results reveal that the convergence rate $\lambda$ and the coefficients of the estimates for Lyapunov function $W$ and Itô operator $\mathcal {L}W$ can determine the upper bound for the solution. The presented results are demonstrated by an illustrative example.
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1. X. Mao, Stochastic Differential Equations and Applications, Chichester: Horwood, 1997.

2. T. Taniguchi, K. Liu, A. Truman, Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differ. Equations, 181 (2002), 72-91.

3. D. Xu, Z. Yang, Y. Huang, Existence-uniqueness and continuation theorems for stochastic functional differential equations, J. Differ. Equations, 245 (2008), 1681-1703.    

4. R. Z. Has'minskii, Stochastic Stability of Differential Equations, Heidelberg: Springer-Verlag, 2012.

5. H. Hu, L. Xu, Existence and uniqueness theorems for periodic Markov process and applications to stochastic functional differential equations, J. Math. Anal. Appl., 466 (2018), 896-926.    

6. D. Xu, Y. Huang, Z. Yang, Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Contin. Dyn. Syst., 24 (2009), 1005-1023.    

7. R. Sakthivel, J. Luo, Asymptotic stability of nonlinear impulsive stochastic differential equations, Statist. Probab. Lett., 79 (2009), 1219-1223.    

8. R. Sakthivel, J. Luo, Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math. Anal. Appl., 356 (2009), 1-6.    

9. R. Sakthivel, Y. Ren, H. Kim, Asymptotic stability of second-order neutral stochastic differential equations, J. Math. Phys., 51 (2010), 052701.

10. L. Xu, S. S. Ge, The pth moment exponential ultimate boundedness of impulsive stochastic differential systems, Appl. Math. Lett., 42 (2015), 22-29.    

11. L. Xu, Z. Dai, D. He, Exponential ultimate boundedness of impulsive stochastic delay differential equations, Appl. Math. Lett., 85 (2018), 70-76.    

12. L. Xu, Z. Dai, H. Hu, Almost sure and moment asymptotic boundedness of stochastic delay differential systems, Appl. Math. Comput., 361 (2019), 157-168.    

13. L. Xu, S. S. Ge, H. Hu, Boundedness and stability analysis for impulsive stochastic differential equations driven by G-Brownian motion, Int. J. Control, 92 (2019), 642-652.    

14. L. Xu, H. Hu, Boundedness analysis of stochastic pantograph differential systems, Appl. Math. Lett., 111 (2021), 106630.

15. B. Tojtovska, S. Jankovic, On a general decay stability of stochastic Cohen-Grossberg neural networks with time-varying delays, Appl. Math. Comput., 219 (2012), 2289-2302.

16. A. Rathinasamy, J. Narayanasamy, Mean square stability and almost sure exponential stability of two step Maruyama methods of stochastic delay Hopfield neural networks, Appl. Math. Comput., 348 (2019), 126-152.

17. O. M. Otunuga, Closed-form probability distribution of number of infections at a given time in a stochastic SIS epidemic model, Heliyon, 5 (2019), e02499.

18. A. Raza, M. Rafiq, D. Baleanu, et al. Competitive numerical analysis for stochastic HIV/AIDS epidemic model in a two-sex population, IET syst. biol., 13 (2019), 305-315.    

19. D. He, L. Xu, Globally impulsive asymptotical synchronization of delayed chaotic systems with stochastic perturbation, Rocky MT. J. Math., 42 (2012), 617-632.    

20. J. Zhao, Adaptive Q-S synchronization between coupled chaotic systems with stochastic perturbation and delay, Appl. Math. Model., 36 (2012), 3312-3319.    

21. R. Sakthivel, T. Saravanakumar, B. Kaviarasan, et al., Dissipativity based repetitive control for switched stochastic dynamical systems, Appl. Math. Comput., 291 (2016), 340-353.

22. L. Xu, D. He, Mean square exponential stability analysis of impulsive stochastic switched systems with mixed delays, Comput. Math. Appl., 62 (2011), 109-117.    

23. L. Liu, F. Deng, pth moment exponential stability of highly nonlinear neutral pantograph stochastic differential equations driven by Lévy noise, Appl. Math. Lett., 86 (2018), 313-319.    

24. Q. Zhu, Stability analysis of stochastic delay differential equations with Lévy noise, Syst. Control Let., 118 (2018), 62-68.    

25. Y. Xu, B. Pei, G. Guo, Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise, Appl. Math. Comput., 263 (2015), 398-409.

26. J. Yang, X. Liu, X. Liu, Stability of stochastic functional differential systems with semi-Markovian switching and Lévy noise by functional Itô's formula and its applications, J. Frankl. Inst., 357 (2020), 4458-4485.    

27. D. Applebaum, M. Siakalli, Asymptotic stability of stochastic differential equations driven by Lévy noise, J. Appl. Probab. 46 (2009), 1116-1129.    

28. D. He, L. Xu, Boundedness analysis of stochastic integrodifferential systems with Lévy noise, J. Taibah Univ. Sci., 14 (2020), 87-93.    

29. E. Beckenbach, R. Bellman, Inequalities, New York: Springer-Verlag, 1961.

30. X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, London: Imperial College Press, 2006.

31. Y. Miyahara, Ultimate boundedness of the systems governed by stochastic differential equations, Nagoya Math. J., 47 (1972), 111-144.    

32. H. Ahmad, A. R. Seadawy, T. A. Khan, et al. Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations, J. Taibah. Univ. Sci., 14 (2020), 346-358.    

33. S. Peszat, J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise-Evolution Equation Approach, Cambridge: Cambridge Univ Press, 2009.

34. L. Xu, S. S. Ge, Asymptotic behavior analysis of complex-valued impulsive differential systems with time-varying delays, Nonlinear Anal.: Hybrid Syst., 27 (2018), 13-28.    

35. L. Xu, X. Chu, H. Hu, Exponential ultimate boundedness of non-autonomous fractional differential systems with time delay and impulses, Appl. Math. Lett., 99 (2020), 106000.

36. S. Ma, Y. Kang, Periodic averaging method for impulsive stochastic differential equations with Lévy noise, Appl. Math. Lett., 93 (2019), 91-97.    

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