×

Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. (English) Zbl 1370.34137

Stochastic differential equations have many applications in science and engineering, and have been receiving much attention over the last decades. In this paper, the authors establish sufficient conditions ensuring existence and continuous dependence of mild solutions to first order stochastic impulsive differential equation with delays in a real separable Hilbert space. The approach is based on Banach’s fixed point theorem and Krasnoselski-Schaefer type fixed point theorem.

MSC:

34K50 Stochastic functional-differential equations
34K45 Functional-differential equations with impulses
34K30 Functional-differential equations in abstract spaces
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] H. M. Ahmed, Semilinear neutral fractional stochastic integro-differential equations with nonlocal conditions,, J. Theoret. Probab., 28, 667 (2015) · Zbl 1326.34115 · doi:10.1007/s10959-013-0520-1
[2] E. Alos, Stochastic calculus with respect to Gaussian processes,, Ann. Probab., 29, 766 (2001) · Zbl 1015.60047 · doi:10.1214/aop/1008956692
[3] C. Avramescu, Some remarks on a fixed point theorem of Krasnoselskii,, Electron. J. Qual. Theory Differ. Equ., 5, 1 (2003) · Zbl 1028.47039
[4] J. Bao, Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients,, Comput. Math. Appl., 59, 207 (2010) · Zbl 1189.60122 · 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, 81 (1956) · Zbl 0070.08201 · doi:10.1007/BF02022967
[6] A. Boudaoui, Existence of mild solutions to stochastic delay evolution equations with a fractional Brownian motion and impulses,, Stoch. Anal. Appl., 33, 244 (2015) · Zbl 1327.35460 · doi:10.1080/07362994.2014.981641
[7] A. Boudaoui, Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay., Math. Meth. Appl. Sci., 39, 1435 (2016) · Zbl 1338.34154 · doi:10.1002/mma.3580
[8] A. Boudaoui, Impulsive neutral functional differential equations driven by a fractional Brownian motion with unbounded delay,, Appl. Anal., 95, 2039 (2016) · Zbl 1356.34079 · doi:10.1080/00036811.2015.1086756
[9] B. Boufoussi, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space,, Statist. Probab. Lett., 82, 1549 (2012) · Zbl 1248.60069 · doi:10.1016/j.spl.2012.04.013
[10] G. Cao, Successive approximations of infinite dimensional SDES with jump,, Stoch. Dyn., 5, 609 (2005) · Zbl 1082.60048 · doi:10.1142/S0219493705001584
[11] T. Caraballo, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,, Nonlinear Anal., 74, 3671 (2011) · Zbl 1218.60053 · 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, 745 (2013) · Zbl 1279.60078 · doi:10.1007/s11464-013-0300-3
[13] M. M. El-Borai, On some fractional stochastic delay differential equations,, Comput. Math. Appl., 59, 1165 (2010) · Zbl 1189.60117 · doi:10.1016/j.camwa.2009.05.004
[14] G. R. Gautam, Existence result of fractional functional integrodifferential equation with not instantaneous impulse,, Int. J. Adv. Appl. Math. Mech, 1, 11 (2014) · Zbl 1359.34087
[15] T. E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations,, Stochastics, 77, 139 (2005) · Zbl 1115.60064 · doi:10.1080/10451120512331335181
[16] J. R. Graef, <em>Impulsive Differential Inclusions. A Fixed Point Approach</em>,, De Gruyter Series in Nonlinear Analysis and Applications (2013) · Zbl 1285.34002 · doi:10.1515/9783110295313
[17] J. K. Hale, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21, 11 (1978) · Zbl 0383.34055
[18] E. Hernández, On a new class of abstract impulsive differential equations,, Proc. Amer. Math. Soc., 141, 1641 (2013) · Zbl 1266.34101 · doi:10.1090/S0002-9939-2012-11613-2
[19] F. Jiang, 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, 1590 (2011) · Zbl 1217.60054 · doi:10.1016/j.camwa.2011.01.027
[20] V. Lakshmikantham, <em>Theory of Impulsive Differential Equations</em>,, Series in Modern Applied Mathematics (1989) · Zbl 0718.34011 · doi:10.1142/0906
[21] X. Li, An impulsive delay differential inequality and applications,, Comput. Math. Appl., 64, 1875 (2012) · Zbl 1268.34159 · doi:10.1016/j.camwa.2012.03.013
[22] X. Li, On the global exponential stability of impulsive functional differential equations with infinite delays or finite delays,, Commun. Nonlinear Sci. Numer. Simul., 19, 442 (2014) · Zbl 1470.34198 · doi:10.1016/j.cnsns.2013.07.011
[23] Y. Mishura, <em>Stochastic Calculus for Fractional Brownian Motion and Related Topics</em>,, Lecture Notes in Mathematics, 1929 (2008) · Zbl 1138.60006 · doi:10.1007/978-3-540-75873-0
[24] D. Nualart, <em>The Malliavin Calculus and Related Topics,</em>, 2nd ed. Probability and its Applications (New York). Springer-Verlag (2006) · Zbl 1099.60003
[25] A. Pazy, <em>Semigroups of Linear Operators and Applications to Partial Differential Equations</em>., Applied Mathematical Sciences, 44 (1983) · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1
[26] M. Pierri, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses,, Appl. Math. Comp., 219, 6743 (2013) · Zbl 1293.34019 · doi:10.1016/j.amc.2012.12.084
[27] R. Sakthivel, Asymptotic stability of impulsive stochastic partial differential equations with infinite delays,, J. Math. Anal. Appl., 356, 1 (2009) · Zbl 1166.60037 · doi:10.1016/j.jmaa.2009.02.002
[28] A. M. Samoilenko, <em>Impulsive Differential Equations</em>,, World Scientific (1995) · Zbl 0837.34003 · doi:10.1142/9789812798664
[29] G. Shen, Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space,, J. Korean Statist. Soc., 44, 123 (2015) · Zbl 1311.60073 · doi:10.1016/j.jkss.2014.06.002
[30] T. Taniguchi, Successive approximations to solutions of stochastic differential equations,, J. Differential Equations, 96, 152 (1992) · Zbl 0744.34052 · doi:10.1016/0022-0396(92)90148-G
[31] S. Tindel, Stochastic evolution equations with fractional Brownian motion,, Probab. Theory Related Fields, 127, 186 (2003) · Zbl 1036.60056 · doi:10.1007/s00440-003-0282-2
[32] J. R. Wang, On a new class of impulsive fractional differential equations,, Appl. Math. Comput., 242, 649 (2014) · Zbl 1334.34022 · doi:10.1016/j.amc.2014.06.002
[33] Z. Yan, Existence of solutions for impulsive partial stochastic neutral integro-differential equations with state-dependent delay,, Collect. Math., 64, 235 (2013) · Zbl 1272.34107 · 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, 126 (2014) · Zbl 1309.60065 · doi:10.1016/j.jmaa.2014.02.016
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.