-
Previous Article
Bifurcation and final patterns of a modified Swift-Hohenberg equation
- DCDS-B Home
- This Issue
- Next Article
Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion
1. | Laboratory of Mathematics, Univ Sidi Bel Abbes, PoBox 89,22000 Sidi-Bel-Abbes, Algeria |
2. | Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160,41080 Sevilla, Spain |
This paper is concerned with the existence and continuous dependence of mild solutions to stochastic differential equations with non-instantaneous impulses driven by fractional Brownian motions. Our approach is based on a Banach fixed point theorem and Krasnoselski-Schaefer type fixed point theorem.
References:
[1] |
H. M. Ahmed,
Semilinear neutral fractional stochastic integro-differential equations with nonlocal conditions, J. Theoret. Probab., 28 (2015), 667-680.
doi: 10.1007/s10959-013-0520-1. |
[2] |
E. Alos, O. Mazet and D. Nualart,
Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29 (2001), 766-801.
doi: 10.1214/aop/1008956692. |
[3] |
C. Avramescu,
Some remarks on a fixed point theorem of Krasnoselskii, Electron. J. Qual. Theory Differ. Equ., 5 (2003), 1-15.
|
[4] |
J. Bao and Z. Hou,
Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl., 59 (2010), 207-214.
doi: 10.1016/j.camwa.2009.08.035. |
[5] |
I. Bihari,
A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar., 7 (1956), 81-94.
doi: 10.1007/BF02022967. |
[6] |
A. Boudaoui, T. Caraballo and A. Ouahab,
Existence of mild solutions to stochastic delay evolution equations with a fractional Brownian motion and impulses, Stoch. Anal. Appl., 33 (2015), 244-258.
doi: 10.1080/07362994.2014.981641. |
[7] |
A. Boudaoui, T. Caraballo and A. Ouahab,
Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay, Math. Meth. Appl. Sci., 39 (2016), 1435-1451.
doi: 10.1002/mma.3580. |
[8] |
A. Boudaoui, T. Caraballo and A. Ouahab,
Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay, Appl. Anal., 95 (2016), 2039-2062.
doi: 10.1080/00036811.2015.1086756. |
[9] |
B. Boufoussi and S. Hajji,
Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statist. Probab. Lett., 82 (2012), 1549-1558.
doi: 10.1016/j.spl.2012.04.013. |
[10] |
G. Cao, K. He and X. Zhang,
Successive approximations of infinite dimensional SDES with jump, Stoch. Dyn., 5 (2005), 609-619.
doi: 10.1142/S0219493705001584. |
[11] |
T. Caraballo, M. 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. |
[12] |
T. Caraballo,
Mamadou A. Diop, Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion, Front. Math. China, 8 (2013), 745-760.
doi: 10.1007/s11464-013-0300-3. |
[13] |
M. M. El-Borai, K. EI-Said EI-Nadi and H. A. Fouad,
On some fractional stochastic delay differential equations, Comput. Math. Appl., 59 (2010), 1165-1170.
doi: 10.1016/j.camwa.2009.05.004. |
[14] |
G. R. Gautam and J. Dabas,
Existence result of fractional functional integrodifferential equation with not instantaneous impulse, Int. J. Adv. Appl. Math. Mech, 1 (2014), 11-21.
|
[15] |
T. E. Govindan,
Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics, 77 (2005), 139-154.
doi: 10.1080/10451120512331335181. |
[16] |
J. R. Graef, J. Henderson and A. Ouahab,
Impulsive Differential Inclusions. A Fixed Point Approach De Gruyter Series in Nonlinear Analysis and Applications, 20. De Gruyter, Berlin, 2013.
doi: 10.1515/9783110295313. |
[17] |
J. K. Hale and J. Kato,
Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.
|
[18] |
E. Hernández and D. O'Regan,
On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641-1649.
doi: 10.1090/S0002-9939-2012-11613-2. |
[19] |
F. Jiang and Y. Shen,
A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl., 61 (2011), 1590-1594.
doi: 10.1016/j.camwa.2011.01.027. |
[20] |
V. Lakshmikantham, D. Bainov and P. Simeonov,
Theory of Impulsive Differential Equations Series in Modern Applied Mathematics, 6. World Scientific Publishing Co. , Inc. , Teaneck, NJ, 1989.
doi: 10.1142/0906. |
[21] |
X. Li and M. Bohner,
An impulsive delay differential inequality and applications, Comput. Math. Appl., 64 (2012), 1875-1881.
doi: 10.1016/j.camwa.2012.03.013. |
[22] |
X. Li and X. Fu,
On the global exponential stability of impulsive functional differential equations with infinite delays or finite delays, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 442-447.
doi: 10.1016/j.cnsns.2013.07.011. |
[23] |
Y. Mishura,
Stochastic Calculus for Fractional Brownian Motion and Related Topics Lecture Notes in Mathematics, 1929. Springer-Verlag, Berlin, 2008.
doi: 10.1007/978-3-540-75873-0. |
[24] |
D. Nualart,
The Malliavin Calculus and Related Topics, 2nd ed. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. |
[25] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[26] |
M. Pierri, D. O'Regan and V. Rolnik,
Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comp., 219 (2013), 6743-6749.
doi: 10.1016/j.amc.2012.12.084. |
[27] |
R. Sakthivel and J. Luo,
Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math. Anal. Appl., 356 (2009), 1-6.
doi: 10.1016/j.jmaa.2009.02.002. |
[28] |
A. M. Samoilenko and N. A. Perestyuk,
Impulsive Differential Equations World Scientific, Singapore 1995.
doi: 10.1142/9789812798664. |
[29] |
G. Shen and Y. Ren,
Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space, J. Korean Statist. Soc., 44 (2015), 123-133.
doi: 10.1016/j.jkss.2014.06.002. |
[30] |
T. Taniguchi,
Successive approximations to solutions of stochastic differential equations, J. Differential Equations, 96 (1992), 152-169.
doi: 10.1016/0022-0396(92)90148-G. |
[31] |
S. Tindel, C. 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. |
[32] |
J. R. Wang, Y. Zhou and Z. Lin,
On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242 (2014), 649-657.
doi: 10.1016/j.amc.2014.06.002. |
[33] |
Z. Yan and X. Yan,
Existence of solutions for impulsive partial stochastic neutral integro-differential equations with state-dependent delay, Collect. Math., 64 (2013), 235-250.
doi: 10.1007/s13348-012-0063-2. |
[34] |
Q. Zhu,
Asymptotic stability in the $p$th moment for stochastic differential equations with Levy noise, J. Math. Anal. Appl., 416 (2014), 126-142.
doi: 10.1016/j.jmaa.2014.02.016. |
show all references
References:
[1] |
H. M. Ahmed,
Semilinear neutral fractional stochastic integro-differential equations with nonlocal conditions, J. Theoret. Probab., 28 (2015), 667-680.
doi: 10.1007/s10959-013-0520-1. |
[2] |
E. Alos, O. Mazet and D. Nualart,
Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29 (2001), 766-801.
doi: 10.1214/aop/1008956692. |
[3] |
C. Avramescu,
Some remarks on a fixed point theorem of Krasnoselskii, Electron. J. Qual. Theory Differ. Equ., 5 (2003), 1-15.
|
[4] |
J. Bao and Z. Hou,
Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl., 59 (2010), 207-214.
doi: 10.1016/j.camwa.2009.08.035. |
[5] |
I. Bihari,
A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar., 7 (1956), 81-94.
doi: 10.1007/BF02022967. |
[6] |
A. Boudaoui, T. Caraballo and A. Ouahab,
Existence of mild solutions to stochastic delay evolution equations with a fractional Brownian motion and impulses, Stoch. Anal. Appl., 33 (2015), 244-258.
doi: 10.1080/07362994.2014.981641. |
[7] |
A. Boudaoui, T. Caraballo and A. Ouahab,
Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay, Math. Meth. Appl. Sci., 39 (2016), 1435-1451.
doi: 10.1002/mma.3580. |
[8] |
A. Boudaoui, T. Caraballo and A. Ouahab,
Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay, Appl. Anal., 95 (2016), 2039-2062.
doi: 10.1080/00036811.2015.1086756. |
[9] |
B. Boufoussi and S. Hajji,
Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statist. Probab. Lett., 82 (2012), 1549-1558.
doi: 10.1016/j.spl.2012.04.013. |
[10] |
G. Cao, K. He and X. Zhang,
Successive approximations of infinite dimensional SDES with jump, Stoch. Dyn., 5 (2005), 609-619.
doi: 10.1142/S0219493705001584. |
[11] |
T. Caraballo, M. 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. |
[12] |
T. Caraballo,
Mamadou A. Diop, Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion, Front. Math. China, 8 (2013), 745-760.
doi: 10.1007/s11464-013-0300-3. |
[13] |
M. M. El-Borai, K. EI-Said EI-Nadi and H. A. Fouad,
On some fractional stochastic delay differential equations, Comput. Math. Appl., 59 (2010), 1165-1170.
doi: 10.1016/j.camwa.2009.05.004. |
[14] |
G. R. Gautam and J. Dabas,
Existence result of fractional functional integrodifferential equation with not instantaneous impulse, Int. J. Adv. Appl. Math. Mech, 1 (2014), 11-21.
|
[15] |
T. E. Govindan,
Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics, 77 (2005), 139-154.
doi: 10.1080/10451120512331335181. |
[16] |
J. R. Graef, J. Henderson and A. Ouahab,
Impulsive Differential Inclusions. A Fixed Point Approach De Gruyter Series in Nonlinear Analysis and Applications, 20. De Gruyter, Berlin, 2013.
doi: 10.1515/9783110295313. |
[17] |
J. K. Hale and J. Kato,
Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.
|
[18] |
E. Hernández and D. O'Regan,
On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641-1649.
doi: 10.1090/S0002-9939-2012-11613-2. |
[19] |
F. Jiang and Y. Shen,
A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl., 61 (2011), 1590-1594.
doi: 10.1016/j.camwa.2011.01.027. |
[20] |
V. Lakshmikantham, D. Bainov and P. Simeonov,
Theory of Impulsive Differential Equations Series in Modern Applied Mathematics, 6. World Scientific Publishing Co. , Inc. , Teaneck, NJ, 1989.
doi: 10.1142/0906. |
[21] |
X. Li and M. Bohner,
An impulsive delay differential inequality and applications, Comput. Math. Appl., 64 (2012), 1875-1881.
doi: 10.1016/j.camwa.2012.03.013. |
[22] |
X. Li and X. Fu,
On the global exponential stability of impulsive functional differential equations with infinite delays or finite delays, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 442-447.
doi: 10.1016/j.cnsns.2013.07.011. |
[23] |
Y. Mishura,
Stochastic Calculus for Fractional Brownian Motion and Related Topics Lecture Notes in Mathematics, 1929. Springer-Verlag, Berlin, 2008.
doi: 10.1007/978-3-540-75873-0. |
[24] |
D. Nualart,
The Malliavin Calculus and Related Topics, 2nd ed. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. |
[25] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[26] |
M. Pierri, D. O'Regan and V. Rolnik,
Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comp., 219 (2013), 6743-6749.
doi: 10.1016/j.amc.2012.12.084. |
[27] |
R. Sakthivel and J. Luo,
Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math. Anal. Appl., 356 (2009), 1-6.
doi: 10.1016/j.jmaa.2009.02.002. |
[28] |
A. M. Samoilenko and N. A. Perestyuk,
Impulsive Differential Equations World Scientific, Singapore 1995.
doi: 10.1142/9789812798664. |
[29] |
G. Shen and Y. Ren,
Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space, J. Korean Statist. Soc., 44 (2015), 123-133.
doi: 10.1016/j.jkss.2014.06.002. |
[30] |
T. Taniguchi,
Successive approximations to solutions of stochastic differential equations, J. Differential Equations, 96 (1992), 152-169.
doi: 10.1016/0022-0396(92)90148-G. |
[31] |
S. Tindel, C. 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. |
[32] |
J. R. Wang, Y. Zhou and Z. Lin,
On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242 (2014), 649-657.
doi: 10.1016/j.amc.2014.06.002. |
[33] |
Z. Yan and X. Yan,
Existence of solutions for impulsive partial stochastic neutral integro-differential equations with state-dependent delay, Collect. Math., 64 (2013), 235-250.
doi: 10.1007/s13348-012-0063-2. |
[34] |
Q. Zhu,
Asymptotic stability in the $p$th moment for stochastic differential equations with Levy noise, J. Math. Anal. Appl., 416 (2014), 126-142.
doi: 10.1016/j.jmaa.2014.02.016. |
[1] |
Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281 |
[2] |
Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199 |
[3] |
Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255 |
[4] |
Brahim Boufoussi, Soufiane Mouchtabih. Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $ 1/2 $. Evolution Equations and Control Theory, 2021, 10 (4) : 921-935. doi: 10.3934/eect.2020096 |
[5] |
Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 |
[6] |
María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473 |
[7] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[8] |
Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 |
[9] |
Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197 |
[10] |
Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257 |
[11] |
Yejuan Wang, Tongtong Liang. Mild solutions to the time fractional Navier-Stokes delay differential inclusions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3713-3740. doi: 10.3934/dcdsb.2018312 |
[12] |
Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4927-4962. doi: 10.3934/dcdsb.2020320 |
[13] |
Jin Li, Jianhua Huang. Dynamics of a 2D Stochastic non-Newtonian fluid driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2483-2508. doi: 10.3934/dcdsb.2012.17.2483 |
[14] |
Xin Meng, Cunchen Gao, Baoping Jiang, Hamid Reza Karimi. Observer-based SMC for stochastic systems with disturbance driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022027 |
[15] |
Yousef Alnafisah, Hamdy M. Ahmed. Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion. Evolution Equations and Control Theory, 2022, 11 (3) : 925-937. doi: 10.3934/eect.2021031 |
[16] |
Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations and Control Theory, 2022, 11 (1) : 1-24. doi: 10.3934/eect.2020100 |
[17] |
Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435 |
[18] |
Qi Yao, Linshan Wang, Yangfan Wang. Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4727-4743. doi: 10.3934/dcdsb.2020310 |
[19] |
Chunhong Li, Jiaowan Luo. Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3299-3318. doi: 10.3934/dcdsb.2018321 |
[20] |
Tomás Caraballo, María J. Garrido–Atienza, Björn Schmalfuss, José Valero. Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 439-455. doi: 10.3934/dcdsb.2010.14.439 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]