Geodesic
Asymptotic Behavior of Neutral Stochastic Partial Functional Integro--Differential Equations Driven by a Fractional Brownian Motion
Caraballo, Tomás1 ; Diop, Mamadou Abdoul2 ; Ndiaye, Abdoul Aziz3
1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. Apdo. de Correos, 1160, 41080-Sevilla, Spain
2 Département de Mathématiques, Université Gaston Berger de Saint-Louis, , UFR SAT, 234, Saint-Louis, Sénégal
3 Département de Mathématiques, Université Gaston Berger de Saint-Louis, UFR SAT 234, Saint-Louis, Sénégal
Journal of nonlinear sciences and its applications, Tome 7 (2014) no. 6, p. 407-421.

Voir la notice de l'article provenant de la source International Scientific Research Publications

This paper deals with the existence, uniqueness and asymptotic behavior of mild solutions to neutral stochastic delay functional integro-differential equations perturbed by a fractional Brownian motion BH, with Hurst parameter $H \in ( \frac{1}{2} , 1)$. The main tools for the existence of solution is a fixed point theorem and the theory of resolvent operators developed in Grimmer [R. Grimmer, Trans. Amer. Math. Soc., 273 (1982), 333-349.], while a Gronwall-type lemma plays the key role for the asymptotic behavior. An example is provided to illustrate the results of this work.
DOI : 10.22436/jnsa.007.06.04
Classification : 60H15, 60G15, 60J65
Keywords: Resolvent operators, \(C_0\)-semigroup, Wiener process, Mild solutions, Fractional Brownian motion, Exponential decay of solutions.

Caraballo, Tomás 1 ; Diop, Mamadou Abdoul 2 ; Ndiaye, Abdoul Aziz 3

1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. Apdo. de Correos, 1160, 41080-Sevilla, Spain
2 Département de Mathématiques, Université Gaston Berger de Saint-Louis, , UFR SAT, 234, Saint-Louis, Sénégal
3 Département de Mathématiques, Université Gaston Berger de Saint-Louis, UFR SAT 234, Saint-Louis, Sénégal 
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Caraballo, Tomás; Diop, Mamadou Abdoul; Ndiaye, Abdoul Aziz. Asymptotic Behavior of Neutral Stochastic Partial Functional Integro--Differential Equations Driven by a Fractional Brownian Motion. Journal of nonlinear sciences and its applications, Tome 7 (2014) no. 6, p. 407-421. doi : 10.22436/jnsa.007.06.04. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.007.06.04/

[1] Alos, E.; Mazet, O.; Nualart, D. Stochastic calculus with respect to Gaussian processes, Ann. Probab., Volume 29 (1999), pp. 766-801

[2] Bao, J.; Hou, Z. Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Computers and Mathematics with Applications, Volume 59 (2010), pp. 207-214

[3] Boufoussi, B.; Hajji, S. Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statistics and Probability Letters, Volume 82 (2012), pp. 1549-1558

[4] Cao, G.; He, K.; Zhang, X. Successive approximations of infinite dimensional SDES with jump, Stoch. Syst., Volume 5 (2005), pp. 609-619

[5] Caraballo, T.; Diop, M. A. Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion , Front. Math. China, Volume 8 (2013), pp. 745-760

[6] 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., Volume 74 (2011), pp. 3671-3684

[7] Caraballo, T.; Márquez-Durán, A. M. Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay, Dyn. Partial Differ. Equ., Volume 10 (2013), pp. 267-281

[8] Caraballo, T.; Real, J.; T. Taniguchi The exponential stability of neutral stochastic delay partial differential equations, Discrete Contin. Dyn. Syst., Volume 18 (2007), pp. 295-313

[9] Chen, H. Integral inequality and exponential stability for neutral stochastic partial differential equations, Journal of Inequalities and Applications, Volume 2009 (2009), pp. 1-297478

[10] Prato, G. Da; Zabczyk, J. Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 1992

[11] Govindan, T. E. Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics , Volume 77 (2005), pp. 139-154

[12] Grimmer, R. Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., Volume 273 (1982), pp. 333-349

[13] Jiang, F.; Y. Shen A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients, Computers and Mathematics with Applications , Volume 61 (2011), pp. 1590-1594

[14] Mishura, Y. Stochastic Calculus for Fractional Brownian Motion and Related Topics, in: Lecture Notes in Mathematics no. 1929, , 2008

[15] Nualart, D. The Malliavin Calculus and Related Topics, second edition, Springer-Verlag, Berlin, 2006

[16] T. Taniguchi Successive approximations of solutions of stochastic differential equations, J. Differential Equations, Volume 96 (1992), pp. 152-169

[17] Tindel, S.; Tudor, C.; Viens, F. Stochastic evolution equations with fractional brownian motion, Probab. Theory Related Fields, Volume 127 (2) (2003), pp. 186-204

[18] Turo, J. Successive approximations of solutions to stochastic functional differential equations, Period. Math. Hungar, Volume 30 (1995), pp. 87-96

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