# American Institute of Mathematical Sciences

September  2017, 22(7): 2521-2541. doi: 10.3934/dcdsb.2017084

## 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

1 Corresponding author

Received  July 2016 Revised  September 2016 Published  March 2017

Fund Project: This work has been partially 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 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.

Citation: Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084
##### 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. Google Scholar [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. Google Scholar [3] C. Avramescu, Some remarks on a fixed point theorem of Krasnoselskii, Electron. J. Qual. Theory Differ. Equ., 5 (2003), 1-15. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] T. E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics, 77 (2005), 139-154. doi: 10.1080/10451120512331335181. Google Scholar [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. Google Scholar [17] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [28] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations World Scientific, Singapore 1995. doi: 10.1142/9789812798664. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [3] C. Avramescu, Some remarks on a fixed point theorem of Krasnoselskii, Electron. J. Qual. Theory Differ. Equ., 5 (2003), 1-15. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] T. E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics, 77 (2005), 139-154. doi: 10.1080/10451120512331335181. Google Scholar [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. Google Scholar [17] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [28] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations World Scientific, Singapore 1995. doi: 10.1142/9789812798664. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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