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June  2019, 24(6): 2719-2743. doi: 10.3934/dcdsb.2018272

Long time behavior of fractional impulsive stochastic differential equations with infinite delay

1. 

School of Mathematics and Statistics, Xi’an Jiaotong University Xi’an 710049, China

2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain

Received  March 2018 Revised  May 2018 Published  October 2018

Fund Project: This work has been supported by grant MTM2015-63723-P (MINECO/FEDER, EU) and Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under grant 2010/FQM314, and Proyecto de Excelencia P12-FQM-1492

This paper is first devoted to the local and global existence of mild solutions for a class of fractional impulsive stochastic differential equations with infinite delay driven by both $\mathbb{K}$-valued Q-cylindrical Brownian motion and fractional Brownian motion with Hurst parameter $H∈(1/2,1)$. A general framework which provides an effective way to prove the continuous dependence of mild solutions on initial value is established under some appropriate assumptions. Furthermore, it is also proved the exponential decay to zero of solutions to fractional stochastic impulsive differential equations with infinite delay.

Citation: Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272
References:
[1]

D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal., 69 (2008), 3692-3705. doi: 10.1016/j.na.2007.10.004. Google Scholar

[2]

D. BahugunaR. Sakthivel and A. Chadha, Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with infinite delay, Stoch. Anal. Appl., 35 (2017), 63-88. doi: 10.1080/07362994.2016.1249285. Google Scholar

[3]

E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, University Press Facilities, Eindhoven University of Technology, 2001. Google Scholar

[4]

M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, 2006. doi: 10.1155/9789775945501. Google Scholar

[5]

T. BlouhiT. Caraballo and A. Ouahab, Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion, Stoch. Anal. Appl., 34 (2016), 792-834. doi: 10.1080/07362994.2016.1180994. Google Scholar

[6]

E.M. BonottoM.C. BortolanT. Caraballo and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differential Equations, 262 (2017), 3524-3550. doi: 10.1016/j.jde.2016.11.036. Google Scholar

[7]

A. BoudaouiT. Caraballo and A. Ouahab, Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay, Math. Methods Appl. Sci., 39 (2016), 1435-1451. doi: 10.1002/mma.3580. Google Scholar

[8]

T. Caraballo and P.E. Kloeden, Non-autonomous attractors for integral-differential evolution equations, Discrete Contin. Dyn. Syst. Ser. S., 2 (2009), 17-36. doi: 10.3934/dcdss.2009.2.17. Google Scholar

[9]

T. CaraballoM.J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684. doi: 10.1016/j.na.2011.02.047. Google Scholar

[10]

A. Chauhan and J. Dabas, Local and global existence of mild solution to an impulsive fractional functional integro-differential equations with nonlocal condition, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 821-829. doi: 10.1016/j.cnsns.2013.07.025. Google Scholar

[11]

R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430. doi: 10.1016/0022-0396(71)90004-0. Google Scholar

[12]

J. Dabas and A. Chauhan, Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay, Math. Comput. Modelling, 57 (2013), 754-763. Google Scholar

[13]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. Google Scholar

[14]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, 2003. Google Scholar

[15]

Z. B. Fan, Characterization of compactness for resolvents and its applications, Appl. Math. Comput., 232 (2014), 60-67. doi: 10.1016/j.amc.2014.01.051. Google Scholar

[16]

M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser Basel, 2006. doi: 10.1007/3-7643-7698-8. Google Scholar

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Spring-Verlag, 1981. Google Scholar

[18]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006. Google Scholar

[19]

P. E. Kloeden, T. Lorenz and M. H. Yang, Forward attractors in discrete time nonautonomous dynamical systems, Springer International Publishing, 2016.Google Scholar

[20]

B. Øksendal, Stochastic Differential Equations, Springer-Verlag, 1985. doi: 10.1007/978-3-662-13050-6. Google Scholar

[21]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, 1989. doi: 10.1142/0906. Google Scholar

[22]

Y. J. Li and Y. J. Wang, Uniform asymptotic stability of solutions of fractional functional differential equations, Abstr. Appl. Anal., 2013 (2013), Art. ID 532589, 8 pp. Google Scholar

[23]

Y. J. Li and Y. J. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, In press.Google Scholar

[24]

A. H. LinY. Ren and N. M. Xia, On neutral impulsive stochastic integro-differential equations with infinite delays via fractional operators, Math. Comput. Modelling, 51 (2010), 413-424. doi: 10.1016/j.mcm.2009.12.006. Google Scholar

[25]

S. Y. Lin, Generalized Gronwall inequalities and their applications to fractional differential equations, J. Inequal. Appl., 2013 (2013), 9pp. doi: 10.1186/1029-242X-2013-549. Google Scholar

[26]

R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Printed in the United States of America, 1976. Google Scholar

[27]

V. D. Mil'man and A. D. Myskis, On the stability of motion in the prensence of impulses, Sibirsk. Mat. Zh., 1 (1960), 233-237. Google Scholar

[28]

Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-verlag, 2008. doi: 10.1007/978-3-540-75873-0. Google Scholar

[29]

X. R. Mao, Stochastic Differential Equations and Applications, Horwood Publication, Chichester, 1997. doi: 10.1533/9780857099402. Google Scholar

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[31]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. Google Scholar

[32]

J. Prüss, Evolutionary Integral Equations and Applications, Brikhauster, Springer Basel, 1993. Google Scholar

[33]

Y. RenX. Cheng and R. Sakthivel, Impulsive neutral stochastic integral-differential equations with infinite delay fBm, Appl. Math. Comput., 247 (2014), 205-212. doi: 10.1016/j.amc.2014.08.095. Google Scholar

[34]

R. SakthivelP. Revathi and Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Anal., 81 (2013), 70-86. doi: 10.1016/j.na.2012.10.009. Google Scholar

[35]

J. H. Shen and X. Z. Liu, Global existence results of impulsive differential equation, J. Math. Anal. Appl., 314 (2006), 546-557. doi: 10.1016/j.jmaa.2005.04.009. Google Scholar

[36]

X. B. ShuY. Z. Lai and Y. M. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal., 74 (2011), 2003-2011. doi: 10.1016/j.na.2010.11.007. Google Scholar

[37]

S. TindelC. A. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab. Theory Related Fields, 127 (2003), 186-204. doi: 10.1007/s00440-003-0282-2. Google Scholar

[38]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235. doi: 10.1016/j.jde.2011.08.048. Google Scholar

[39]

Y. J. WangF. S. Gao and P. E. Kloeden, Impulsive fractional functional differential equations with a weakly continuous nonlinearity, Electron. J. Differ. Equ., 285 (2017), 1-18. Google Scholar

show all references

References:
[1]

D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal., 69 (2008), 3692-3705. doi: 10.1016/j.na.2007.10.004. Google Scholar

[2]

D. BahugunaR. Sakthivel and A. Chadha, Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with infinite delay, Stoch. Anal. Appl., 35 (2017), 63-88. doi: 10.1080/07362994.2016.1249285. Google Scholar

[3]

E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, University Press Facilities, Eindhoven University of Technology, 2001. Google Scholar

[4]

M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, 2006. doi: 10.1155/9789775945501. Google Scholar

[5]

T. BlouhiT. Caraballo and A. Ouahab, Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion, Stoch. Anal. Appl., 34 (2016), 792-834. doi: 10.1080/07362994.2016.1180994. Google Scholar

[6]

E.M. BonottoM.C. BortolanT. Caraballo and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differential Equations, 262 (2017), 3524-3550. doi: 10.1016/j.jde.2016.11.036. Google Scholar

[7]

A. BoudaouiT. Caraballo and A. Ouahab, Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay, Math. Methods Appl. Sci., 39 (2016), 1435-1451. doi: 10.1002/mma.3580. Google Scholar

[8]

T. Caraballo and P.E. Kloeden, Non-autonomous attractors for integral-differential evolution equations, Discrete Contin. Dyn. Syst. Ser. S., 2 (2009), 17-36. doi: 10.3934/dcdss.2009.2.17. Google Scholar

[9]

T. CaraballoM.J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684. doi: 10.1016/j.na.2011.02.047. Google Scholar

[10]

A. Chauhan and J. Dabas, Local and global existence of mild solution to an impulsive fractional functional integro-differential equations with nonlocal condition, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 821-829. doi: 10.1016/j.cnsns.2013.07.025. Google Scholar

[11]

R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430. doi: 10.1016/0022-0396(71)90004-0. Google Scholar

[12]

J. Dabas and A. Chauhan, Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay, Math. Comput. Modelling, 57 (2013), 754-763. Google Scholar

[13]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223. Google Scholar

[14]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, 2003. Google Scholar

[15]

Z. B. Fan, Characterization of compactness for resolvents and its applications, Appl. Math. Comput., 232 (2014), 60-67. doi: 10.1016/j.amc.2014.01.051. Google Scholar

[16]

M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser Basel, 2006. doi: 10.1007/3-7643-7698-8. Google Scholar

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Spring-Verlag, 1981. Google Scholar

[18]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006. Google Scholar

[19]

P. E. Kloeden, T. Lorenz and M. H. Yang, Forward attractors in discrete time nonautonomous dynamical systems, Springer International Publishing, 2016.Google Scholar

[20]

B. Øksendal, Stochastic Differential Equations, Springer-Verlag, 1985. doi: 10.1007/978-3-662-13050-6. Google Scholar

[21]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, 1989. doi: 10.1142/0906. Google Scholar

[22]

Y. J. Li and Y. J. Wang, Uniform asymptotic stability of solutions of fractional functional differential equations, Abstr. Appl. Anal., 2013 (2013), Art. ID 532589, 8 pp. Google Scholar

[23]

Y. J. Li and Y. J. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, In press.Google Scholar

[24]

A. H. LinY. Ren and N. M. Xia, On neutral impulsive stochastic integro-differential equations with infinite delays via fractional operators, Math. Comput. Modelling, 51 (2010), 413-424. doi: 10.1016/j.mcm.2009.12.006. Google Scholar

[25]

S. Y. Lin, Generalized Gronwall inequalities and their applications to fractional differential equations, J. Inequal. Appl., 2013 (2013), 9pp. doi: 10.1186/1029-242X-2013-549. Google Scholar

[26]

R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Printed in the United States of America, 1976. Google Scholar

[27]

V. D. Mil'man and A. D. Myskis, On the stability of motion in the prensence of impulses, Sibirsk. Mat. Zh., 1 (1960), 233-237. Google Scholar

[28]

Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-verlag, 2008. doi: 10.1007/978-3-540-75873-0. Google Scholar

[29]

X. R. Mao, Stochastic Differential Equations and Applications, Horwood Publication, Chichester, 1997. doi: 10.1533/9780857099402. Google Scholar

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[31]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. Google Scholar

[32]

J. Prüss, Evolutionary Integral Equations and Applications, Brikhauster, Springer Basel, 1993. Google Scholar

[33]

Y. RenX. Cheng and R. Sakthivel, Impulsive neutral stochastic integral-differential equations with infinite delay fBm, Appl. Math. Comput., 247 (2014), 205-212. doi: 10.1016/j.amc.2014.08.095. Google Scholar

[34]

R. SakthivelP. Revathi and Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Anal., 81 (2013), 70-86. doi: 10.1016/j.na.2012.10.009. Google Scholar

[35]

J. H. Shen and X. Z. Liu, Global existence results of impulsive differential equation, J. Math. Anal. Appl., 314 (2006), 546-557. doi: 10.1016/j.jmaa.2005.04.009. Google Scholar

[36]

X. B. ShuY. Z. Lai and Y. M. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal., 74 (2011), 2003-2011. doi: 10.1016/j.na.2010.11.007. Google Scholar

[37]

S. TindelC. A. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab. Theory Related Fields, 127 (2003), 186-204. doi: 10.1007/s00440-003-0282-2. Google Scholar

[38]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235. doi: 10.1016/j.jde.2011.08.048. Google Scholar

[39]

Y. J. WangF. S. Gao and P. E. Kloeden, Impulsive fractional functional differential equations with a weakly continuous nonlinearity, Electron. J. Differ. Equ., 285 (2017), 1-18. Google Scholar

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