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, 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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.

[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.

[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.

[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.

[26]

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

[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.

[28]

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

[29]

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

[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.

[31]

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

[32]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.

[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.

[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.

[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.

[26]

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

[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.

[28]

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

[29]

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

[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.

[31]

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

[32]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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