Non-densely defined impulsive neutral stochastic functional differential equations driven by fBm in Hilbert space with infinite delay

Abstract

We study a class of non-densely defined impulsive neutral stochastic functional differential equations driven by an independent cylindrical fractional Brownian motion (fBm) with Hurst parameter H∈ (1/2, 1) in the Hilbert space. We prove the existence and uniqueness of the integral solution for this kind of equations with the coefficients satisfying some non-Lipschitz conditions. The results are obtained by using the method of successive approximation.

Key words： Stochastic functional differential equation; non-densely defined operator; cylindrical fractional Brownian motion (fBm); impulsive effect

References

1 Abada N, Benchohra M, Hadda H. Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J Differential Equations, 2009, 246: 3834-3863

2 Adimy M, Ezzinbi K, Ouhinou A. Variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay. J Math Anal Appl, 2006, 317: 668-689

3 Bihari I. A generalization of a lemma of Bellman and its application to uniqueness problem of differential equations. Acta Math Acad Sci, 1956, 7: 71-94

4 Benchohra M, Gatsori E, Henderson J, Ntouyas S K. Nondensely defined evolution impulsive differential inclusions with nonlocal conditions. J Math Anal Appl, 2003, 285: 307-325

5 Benchohra M, Gorniewicz L. Existence results for nondensely defined impulsive semilinear functional differential inclusions with infinite delay. J Fixed Point Theory Appl, 2007, 2: 11-51

6 Benchohra M, Ntouyas S K, Ouahab A. On nondensely defined semilinear stochastic functional differential equations with nonlocal conditions. J Appl Math Stoch Anal, 2006, Art ID 69584

7 Boufoussi B, Hajji S. Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space. Statist Probab Lett, 2012, 82: 1549-1558

8 Caraballo T, Garrido-Atienza M J, Taniguchi T. The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal, 2011, 74: 3671-3684

9 Da Prato G, Sinestrari E. Differential operators with non-dense domains. Ann Sc Norm Super Pisa Cl Sci, 1987, 14: 285-344

10 Dai D, Heyde C C. Ito’s formula with respect to fractional Brownian motion and its application. J Appl Math Stoch Anal, 1996, 9: 439-448

11 Duncan T E, Pasik-Duncan B. Fractional Brownian motion and stochastic equations in Hilbert spaces. Stoch Dyn, 2002, 2: 225-250

12 Feyel D, de la Pradelle A. On fractional Brownian processes. Potential Anal, 1999, 10: 273-288

13 Hernandez E, Keck D N, McKibben M A. On a class of measure-dependent stochastic evolution equations driven by fBm. J Appl Math Stoch Anal, 2007, Art ID 69747 (26pp)

14 Hale J K, Kato J. Phase spaces for retarded equations with infinite delay. Funkcial Ekvac, 1978, 21: 11-41

15 Hale J K, Lunel S M V. Introduction to Function Differential Equations. Berlin: Springer-Verlag, 1991

16 Kellerman H, Hieber M. Integrated semigroups. J Funct Anal, 1989, 84: 160-180

17 Maslowski B, Nualart D. Evolution equations driven by a fractional Brownian motion. J Funct Anal, 2003, 202: 277-305

18 Nieto J J, Rodriguez-Lopez R. New comparison results for impulsive integro-differential equations and applications. J Math Anal Appl, 2007, 328: 1343-1368

19 Nualart D. The Malliavin Calculus and Related Topics. 2nd ed. Berlin: Springer-Verlag, 2006

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

21 Wu J. Theory and Applications of Partial Functional Differential Equations. New York: Springer-Verlag, 1996

22 Yosida K. Functional Analysis. 6th ed. Berlin: Springer-Verlag, 1980