×

Attracting and quasi-invariant sets of neutral stochastic integro-differential equations with impulses driven by fractional Brownian motion. (English) Zbl 1444.60069

Summary: The paper is devoted to investigating a class of neutral stochastic integro-differential equations with impulses driven by fractional Brownian motion. By establishing two new impulsive integral inequalities which improve the inequalities established by Z. Li [“Global attractiveness and quasi-invariant sets of impulsive neutral stochastic functional differential equations driven by fBm”, Neurocomputing 177, 620–627 (2016; doi:10.1016/j.neucom.2015.11.070)] and S. Long et al. [Stat. Probab. Lett. 82, No. 9, 1699–1709 (2012; Zbl 1250.93124)], attracting and quasi-invariant sets of the system are obtained. Moreover, exponential stability of the mild solution is established with sufficient conditions.

MSC:

60H20 Stochastic integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
47G20 Integro-differential operators
60G22 Fractional processes, including fractional Brownian motion
45K05 Integro-partial differential equations

Citations:

Zbl 1250.93124
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Li, Z: Global attractiveness and quasi-invariant sets of impulsive neutral stochastic functional differential equations driven by fBm. Neurocomputing 177, 620-627 (2016) · doi:10.1016/j.neucom.2015.11.070
[2] Long, S, Teng, L, Xu, D: Global attracting set and stability of stochastic neutral partial functional differential equations with impulses. Stat. Probab. Lett. 82(9), 1699-1709 (2012) · Zbl 1250.93124 · doi:10.1016/j.spl.2012.05.018
[3] Balasubramaniam, P, Park, JY, Muthukumar, P: Approximate controllability of neutral stochastic functional differential systems with infinite delay. Stochastic Anal. Appl. 28(2), 389-400 (2010) · Zbl 1186.93014 · doi:10.1080/07362990802405695
[4] Chen, H: Integral inequality and exponential stability for neutral stochastic partial differential equations with delays. J. Inequal. Appl. 2009, Article ID 297478 (2009). doi:10.1155/2009/297478 · Zbl 1188.60034 · doi:10.1155/2009/297478
[5] Cui, J, Yan, L, Sun, X: Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps. Stat. Probab. Lett. 81(12), 1970-1977 (2011) · Zbl 1227.60085 · doi:10.1016/j.spl.2011.08.010
[6] Govindan, TE: Sample path exponential stability of stochastic neutral partial functional differential equations. J. Numer. Math. Stoch. 3(1), 1-12 (2011) · Zbl 1360.60130
[7] Luo, J: Exponential stability for stochastic neutral partial functional differential equations. J. Math. Anal. Appl. 355(1), 414-425 (2009) · Zbl 1165.60024 · doi:10.1016/j.jmaa.2009.02.001
[8] Randjelovi, J, Jankovi, S: On the pth moment exponential stability criteria of neutral stochastic functional differential equations. J. Math. Anal. Appl. 326(1), 266-280 (2007) · Zbl 1115.60065 · doi:10.1016/j.jmaa.2006.02.030
[9] Taniguchi, T, Liu, K, Truman, A: Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces. J. Differential Equ. 181(1), 72-91 (2002) · Zbl 1009.34074 · doi:10.1006/jdeq.2001.4073
[10] Yang, H, Jiang, F: Exponential stability of mild solutions to impulsive stochastic neutral partial differential equations with memory. Adv. Differ. Equ. 2013, 148 (2013) · Zbl 1390.60250 · doi:10.1186/1687-1847-2013-148
[11] Chen, H: Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays. Stat. Probab. Lett. 80(1), 50-56 (2010) · Zbl 1177.93075 · doi:10.1016/j.spl.2009.09.011
[12] Chen, H, Zhu, C, Zhang, Y: A note on exponential stability for impulsive neutral stochastic partial functional differential equations. Appl. Math. Comput. 227, 139-147 (2014) · Zbl 1364.35448 · doi:10.1016/j.cam.2013.11.002
[13] Jiang, F, Shen, Y: Stability of impulsive stochastic neutral partial differential equations with infinite delays. Asian J. Control 14(6), 1706-1709 (2012) · Zbl 1303.93181 · doi:10.1002/asjc.491
[14] Sakthivel, R, Luo, J: Asymptotic stability of impulsive stochastic partial differential equations with infinite delays. J. Math. Anal. Appl. 356(1), 1-6 (2009) · Zbl 1166.60037 · doi:10.1016/j.jmaa.2009.02.002
[15] Huan, D, Agarwal, A: Global attracting and quasi-invariant sets for stochastic Volterra-Levin equations with jumps. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 21, 343-353 (2014) · Zbl 1304.34133
[16] Xu, D, Long, S: Attracting and quasi-invariant sets of non-autonomous neural networks with delays. Neurocomputing 77(1), 222-228 (2012) · doi:10.1016/j.neucom.2011.09.004
[17] Xu, L, Xu, D: P-Attracting and p-invariant sets for a class of impulsive stochastic functional differential equations. J. Math. Anal. Appl. 57(1), 54-61 (2009) · Zbl 1165.60329
[18] Li, D, Xu, D: Attracting and quasi-invariant sets of stochastic neutral partial functional differential equations. Acta. Math. Scientia B 33(2), 578-588 (2013) · Zbl 1289.35341 · doi:10.1016/S0252-9602(13)60021-1
[19] Wang, L, Li, D: Impulsive-integral inequality for attracting and quasi-invariant sets of impulsive stochastic partial functional differential equations with infinite delays. J. Ineq. Appl. 2013, 238 (2013) · Zbl 1282.05032 · doi:10.1186/1029-242X-2013-238
[20] Boufoussi, B, Hajji, S: Functional differential equations driven by a fractional Brownian motion. Comput. Math. Appl. 62(2), 746-754 (2011) · Zbl 1228.60064 · doi:10.1016/j.camwa.2011.05.055
[21] Duncan, TE, Maslowski, B, Pasik-Duncan, B: Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stoch. Process Appl. 115(8), 1357-1383 (2005) · Zbl 1076.60054 · doi:10.1016/j.spa.2005.03.011
[22] Ferrante, M, Rovira, C: Convergence of delay differential equations driven by fractional Brownian motion. J. Evol. Equ. 10(4), 761-783 (2009) · Zbl 1239.60040 · doi:10.1007/s00028-010-0069-8
[23] Ferrante, M, Rovira, C: Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter \(H>12H>\frac{1}{2} \). Bernouilli 12(1), 85-100 (2006) · Zbl 1102.60054
[24] Mishura, Y: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics (Morel, JM, Takens, F, Teissier, B (eds.)), vol. 1929. Springer, Berlin (2008) · Zbl 1138.60006
[25] Nualart, D, Saussereau, B: Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Stoch. Process. Appl. 119(2), 391-409 (2009) · Zbl 1169.60013 · doi:10.1016/j.spa.2008.02.016
[26] Tindel, S, Tudor, C, Viens, F: Stochastic evolution equations with fractional Brownian motion. Probab. Theory Relat. Fields 127(4), 186-204 (2003) · Zbl 1036.60056 · doi:10.1007/s00440-003-0282-2
[27] Boufoussi, B, Hajji, S: Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space. Stat. Probab. Lett. 82(8), 1549-1558 (2012) · Zbl 1248.60069 · doi:10.1016/j.spl.2012.04.013
[28] Tien, ND: Neutral stochastic differential equations driven by a fractional Brownian motion with impulsive effects and varying-time delays. J. Korean Stat. Soc. 43(4), 599-608 (2014) · Zbl 1304.60074 · doi:10.1016/j.jkss.2014.02.003
[29] Arthi, G, Park, JH, Jung, HY: Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion. Commun. Nonlinear Sci. Numer. Simulat. 32, 145-157 (2016) · Zbl 1510.60045 · doi:10.1016/j.cnsns.2015.08.014
[30] Caraballo, T, Garrido-Atienza, MJ, Taniguchi, T: The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal. 74(11), 3671-3684 (2011) · Zbl 1218.60053 · doi:10.1016/j.na.2011.02.047
[31] Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences (Marsden, JE, Sirovich, L, John, F (eds.)), vol. 44. Springer, New York (1983) · Zbl 0516.47023
[32] Wu, F, Hu, S, Liu, Y: Positive solution and its asymptotic behavior of stochastic functional Kolmogorov type system. J. Math. Anal. Appl. 364(4), 104-118 (2010) · Zbl 1198.34181 · doi:10.1016/j.jmaa.2009.10.072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.