### Existence and uniquenes results for systems of impulsive functional stochastic differential equations driven by fractional Brownian motion with multiple delay

DOI: http://dx.doi.org/10.12775/TMNA.2018.009

#### Abstract

#### Keywords

#### References

E. Alos and D. Nualart, Stochastic calculus with respect to the fractional Brownian motion, Ann. Probab. 29 (2001), 766–801.

M. Benchohra, J. Henderson and S.K. Ntouyas, Impulsive Differential Equations and Inclusions, Vol. 2, Hindawi Publishing Corporation, New York, 2006.

A.T. Bharucha-Reid, Random Integral Equations, Academic Press, New York, 1972.

F. Biagini, Y. Hu, B. Øksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, London, UK, 2008.

I. Bihari, A generalisation of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar. 7 (1956), 81–94.

T. Blouhi, T. 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), no. 5, 792–834.

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.

J. Cao, R. Maheswari and A. Chandrasekar, Exponential H∞ filtering analysis for discrete-time switched neural networks with random delays using sojourn probabilities, Sci. China. Tech. Sci. 59 (2016), no. 3, 387–402.

J. Cao, Q. Yang, Z. Huang and Q. Liu, Asymptotically almost periodic solutions of stochastic functional differential equations, Appl. Math. Comput. 218 (2011), 1499–1511.

P. Carmona, L. Coutin and G. Montseny, Stochastic integration with respect to fractional Brownian motion, Ann. Inst. H. Poincaré B Probabilités et Statistiques 39 (2003), 27–68.

R. Cristescu, Order Structures in Normed Vector Spaces, Editura Ştiinţifică şi Enciclopedică, Bucureşti, 1983 (in Romanian).

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.

B. Davis, On the integrability of the martingale square function, Israel J. Math. 8 (1970), 187–190.

T.E. Duncan, Y. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion, J. Cont. Optim. 38 (2000), no. 2, 582–612.

T.C. Gard, Introduction to Stochastic Differential Equations, Marcel Dekker, New York, 1988.

I.I. Gikhman and A. Skorokhod, Stochastic Differential Equations, Springer–Verlag, 1972.

J.R. Graef, J. Henderson and A. Ouahab, Impulsive differential inclusions. A fixed point approach, De Gruyter Ser. in Nonlinear Anal. Appl., Vol. 20, de Gruyter, Berlin, 2013.

C. Guilan and H. Kai, On a type of stochastic differential equations driven by countably many Brownian motions, J. Funct. Anal. 203 (2003), 262–285.

A. Halanay and D. Wexler, Teoria Calitativa a Sistemelor Impulsuri, Editura Academiei Republicii Socialiste România, Bucharest, 1968 (in Romanian).

V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

C. Li, J. Shi and J. Sun, Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks, Nonlinear Anal. 74 (2011), 3099–3111.

B. Liu, X. Liu, K. Teo and K.Q. Wang, Razumikhin-type theorems on exponential stability of impulsive delay systems, IMA J. Appl. Math. 71 (2006), 47–61.

J. Liu, X. Liu and W.-C. Xie, Existence and uniqueness results for impulsive hybrid stochastic delay systems, Commun. Appl. Nonl. Anal. 17 (2010), 37–54.

M. Liu and K. Wang, On a stochastic logistic equation with impulsive perturbations, Comput. Math. Appl. 63 (2012), 871–886.

X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997.

P. Millar, Warwick martingale integrals, Trans. Amer. Math. Soc. 133 (1968), 145–166.

V.D. Milman and A.A. Myshkis, On the stability of motion in the presence of impulses, Sib. Math. J. 1 (1960), 233–237. (in Russian)

Y. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Springer Science Business Media, 2008.

A.A. Novikov, The moment inequalities for stochastic integrals, Teor. Verojatnost. Primenen. 16 (1971), 548–551. (in Russian)

A. Ouahab, Local and global existence and uniqueness results for impulsive functional differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006), 456–472.

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 4th ed., Springer, Berlin, 1995.

L. Pan and J. Cao, Exponential stability of impulsive stochastic functional differential equations, J. Math. Anal. Appl. 382 (2011), 672–685.

L. Pan and J. Cao, Exponential stability of stochastic functional differential equations with Markovian switching and delayed impulses via Razumikhin method, Adv. Differential Equations 61 (2012).

A.I. Perov, On the Cauchy problem for a system of ordinary differential equations, Priblizhen. Met. Reshen. Differ. Uravn. 2 (1964), 115–134. (in Russian).

R. Precup, Methods Nonlinear Integral Equations, Kluwer, Dordrecht, 2000.

I.A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevances, Fixed Point Theory 9 (2008), 541–559.

F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Related Fields 97 (1993), no. 3, 403–421.

R. Sakthivel and J. Luo, Asymptotic stability of nonlinear impulsive stochastic differential equations, Statist. Probab. Lett. 79 (2009), 1219–1223.

A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.

H. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers, London, 1991.

C.P. Tsokos and W.J. Padgett, Random Integral Equations with Applications to Life Sciences and Engineering, Academic Press, New York, 1974.

S.J. Wu, X.L. Guo and S.Q. Lin, Existence and uniqueness of solutions to random impulsive differential systems, Acta. Math. Appl. Sinica 22 (2006), 595–600.

Y. Xu, R. Guo, D. Liu, H. Zhang and J. Duan, Stochastic averaging principle for dynamical systems with fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), 1197–1212.

F. Yao, J. Cao, P. Cheng and Li. Qiu, Generalized average dwell time approach to stability and input-to-state stability of hybrid impulsive stochastic differential systems, Nonlinear Anal. Hybrid Syst. 22 (2016), 147–160.

F. Yao, J. Cao, L. Qiu and P.Cheng, Exponential stability analysis for stochastic delayed differential systems with impulsive effects: Average impulsive interval approach, Asian J. Control 19 (2017), 74–86.

Q. Zhu and B. Song, Exponential stability for impulsive nonlinear stochastic differential equations with mixed delays, Nonlinear Anal. 12 (2011), 2851–2860.

### Refbacks

- There are currently no refbacks.