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Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process
1. | Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012-Sevilla, Spain |
2. | Université d'Abomey-Calavi(UAC), Institut de Mathématiques et de Sciences Physiques(IMSP), 01 B.P. 613, Porto-Novo, République du Bénin |
3. | Université Gaston Berger de Saint-Louis, UFR SAT Département de Mathématiques, B.P234, Saint-Louis, Sénégal |
The existence and uniqueness of mild solution of an impulsive stochastic system driven by a Rosenblatt process is analyzed in this work by using the Banach fixed point theorem and the theory of resolvent operator developed by R. Grimmer in [
References:
[1] |
E. Alos, O. Maze and D. Nualart,
Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29 (2001), 766-801.
doi: 10.1214/aop/1008956692. |
[2] |
G. Arthi, J. H. Park and H. Y. Jung,
Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion, Communications in Nonlinear Science and Numerical Simulations, 32 (2016), 145-157.
doi: 10.1016/j.cnsns.2015.08.014. |
[3] |
P. Balasubramaniam, M. Syed Ali and J. H. Kim,
Faedo-Galerkin approximate solutions for stochastic semilinear integrodifferential equations, Computers and Mathematics with Applications, 58 (2009), 48-57.
doi: 10.1016/j.camwa.2009.03.084. |
[4] |
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 Analysis, 74 (2011), 3671-3684.
doi: 10.1016/j.na.2011.02.047. |
[5] |
H. B. Chen, Integral inequality and exponential stability for neutral stochastic partial differential equations with delays, Journal of Inequalities and Applications, 2009 (2009), Art. ID 297478, 15 pp.
doi: 10.1155/2009/297478. |
[6] |
H. B. Chen,
Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Statistics and Probability Letters, 80 (2010), 50-56.
doi: 10.1016/j.spl.2009.09.011. |
[7] |
H. B. Chen, The asymptotic behavior for second-order neutral stochastic partial differential equations with infinite delay, Discrete Dynamics in Nature and Society, 2011 (2011), Art. ID 584510, 15 pp.
doi: 10.1155/2011/584510. |
[8] |
J. Cui, L. T. Yan and X. C. Sun,
Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps, Statistics and Probability Letters, 81 (2011), 1970-1977.
doi: 10.1016/j.spl.2011.08.010. |
[9] |
M. Dieye, M. A. Diop and K. Ezzinbi,
On exponential stability of mild solutions for some stochastic partial integrodifferential equations, Statistics Probability Letters, 123 (2017), 61-76.
doi: 10.1016/j.spl.2016.10.031. |
[10] |
R. L. Dobrushin and P. Major,
Non-central limit theorems for nonlinear functionals of Gaussian fields, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 50 (1979), 27-52.
doi: 10.1007/BF00535673. |
[11] |
N. T. Dung,
Stochastic Volterra integro-differential equations driven by fractional Brownian motion in a Hilbert space, Stochastics, 87 (2015), 142-159.
doi: 10.1080/17442508.2014.924938. |
[12] |
R. C. Grimmer,
Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349.
doi: 10.1090/S0002-9947-1982-0664046-4. |
[13] |
F. Jiang and Y. Shen,
Stability of impulsive stochastic neutral partial differential equations with infinite delays, Asian Journal of Control, 14 (2012), 1706-1709.
doi: 10.1002/asjc.491. |
[14] |
A. N. Kolmogorov,
The Wiener spiral and some other interesting curves in Hilbert space, Dokl. Akad. Nauk SSSR, 26 (1940), 115-118.
|
[15] |
I. Kruk, F. Russo and C. A. Tudor,
Wiener integrals, Malliavin calculus and covariance measure structure, Journal of Functional Analysis, 249 (2007), 92-142.
doi: 10.1016/j.jfa.2007.03.031. |
[16] |
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
doi: 10.1142/0906. |
[17] |
N. N. Leonenko and V. V. Ahn,
Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence, Journal of Applied Mathematics and Stochastic Analysis, 14 (2001), 27-46.
doi: 10.1155/S1048953301000041. |
[18] |
J. Liang, J. H. Liu and T.-J. Xiao,
Nonlocal problems for integrodifferential equations, Dynamics of Continuous, Discrete and Impulsive Systems, Series A, Mathematical Analysis, 15 (2008), 815-824.
|
[19] |
M. Maejima and C. A. Tudor,
Wiener integrals with respect to the Hermite process and a non central limit theorem, Stochastic Analysis and Applications, 25 (2007), 1043-1056.
doi: 10.1080/07362990701540519. |
[20] |
M. Maejima and C. A. Tudor,
Selfsimilar processes with stationary increments in the second Wiener chaos, Probability and Mathematical Statistics, 32 (2012), 167-186.
|
[21] |
M. Maejima and C. A. Tudor,
On the distribution of the Rosenblatt process, Statistics and Probability Letters, 83 (2013), 1490-1495.
doi: 10.1016/j.spl.2013.02.019. |
[22] |
B. B. Mandelbrot and J. W. Van Ness,
Fractional Brownian motions, fractional noises and applications, SIAM Review, 10 (1968), 422-437.
doi: 10.1137/1010093. |
[23] |
V. Pipiras and M. S. Taqqu,
Regularization and integral representations of Hermite processes, Statistics and Probability Letters, 80 (2010), 2014-2023.
doi: 10.1016/j.spl.2010.09.008. |
[24] |
J. Prüss, Evolutionary Integral Equations and Applications, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[25] |
Y. Ren, W. S. Yin and R. Sakthivel,
Stabilization of stochastic differential equations driven by $G$-Brownian motion with feedback control based on discrete-time state observation, Automatica J. IFAC, 95 (2018), 146-151.
doi: 10.1016/j.automatica.2018.05.039. |
[26] |
R. Sakthivel, P. Revathi, Y. Ren and G. J. Shen,
Retarded stochastic differential equations with infinite delay driven by Rosenblatt process, Stochastic Analysis and Applications, 36 (2018), 304-323.
doi: 10.1080/07362994.2017.1399801. |
[27] |
A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 14. World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
doi: 10.1142/9789812798664. |
[28] |
M. Ali Syed, Robust stability of stochastic fuzzy impulsive recurrent neural networks with time-varying delays, Iranian Journal of Fuzzy Systems, 11 (2014), 1–13, 97. |
[29] |
M. S. Taqqu,
Weak convergence to the fractional Brownian motion and to the Rosenblatt process, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 31 (1974/75), 287-302.
doi: 10.1007/BF00532868. |
[30] |
S. Tindel, C. A. Tudor and F. Viens,
Stochastic evolution equations with fractional Brownian motion, Probability Theory and Related Fields, 127 (2003), 186-204.
doi: 10.1007/s00440-003-0282-2. |
[31] |
C. A. Tudor,
Analysis of the Rosenblatt process, ESAIM Probab. Stat., 12 (2008), 230-257.
doi: 10.1051/ps:2007037. |
[32] |
L. T. Yan and G. J. Shen,
On the collision local time of sub-fractional Brownian motions, Stat. Probab. Lett., 80 (2010), 296-308.
doi: 10.1016/j.spl.2009.11.003. |
[33] |
H. Yang and F. Jiang, Exponential stability of mild solutions to impulsive stochastic neutral partial differential equations with memory, Advances in Difference Equations, 2013 (2013), Art. ID 148, 9 pp.
doi: 10.1186/1687-1847-2013-148. |
show all references
Dedicated to Prof. Dr. Juan J. Nieto on the occasion of his 60th birthday
References:
[1] |
E. Alos, O. Maze and D. Nualart,
Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29 (2001), 766-801.
doi: 10.1214/aop/1008956692. |
[2] |
G. Arthi, J. H. Park and H. Y. Jung,
Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion, Communications in Nonlinear Science and Numerical Simulations, 32 (2016), 145-157.
doi: 10.1016/j.cnsns.2015.08.014. |
[3] |
P. Balasubramaniam, M. Syed Ali and J. H. Kim,
Faedo-Galerkin approximate solutions for stochastic semilinear integrodifferential equations, Computers and Mathematics with Applications, 58 (2009), 48-57.
doi: 10.1016/j.camwa.2009.03.084. |
[4] |
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 Analysis, 74 (2011), 3671-3684.
doi: 10.1016/j.na.2011.02.047. |
[5] |
H. B. Chen, Integral inequality and exponential stability for neutral stochastic partial differential equations with delays, Journal of Inequalities and Applications, 2009 (2009), Art. ID 297478, 15 pp.
doi: 10.1155/2009/297478. |
[6] |
H. B. Chen,
Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Statistics and Probability Letters, 80 (2010), 50-56.
doi: 10.1016/j.spl.2009.09.011. |
[7] |
H. B. Chen, The asymptotic behavior for second-order neutral stochastic partial differential equations with infinite delay, Discrete Dynamics in Nature and Society, 2011 (2011), Art. ID 584510, 15 pp.
doi: 10.1155/2011/584510. |
[8] |
J. Cui, L. T. Yan and X. C. Sun,
Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps, Statistics and Probability Letters, 81 (2011), 1970-1977.
doi: 10.1016/j.spl.2011.08.010. |
[9] |
M. Dieye, M. A. Diop and K. Ezzinbi,
On exponential stability of mild solutions for some stochastic partial integrodifferential equations, Statistics Probability Letters, 123 (2017), 61-76.
doi: 10.1016/j.spl.2016.10.031. |
[10] |
R. L. Dobrushin and P. Major,
Non-central limit theorems for nonlinear functionals of Gaussian fields, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 50 (1979), 27-52.
doi: 10.1007/BF00535673. |
[11] |
N. T. Dung,
Stochastic Volterra integro-differential equations driven by fractional Brownian motion in a Hilbert space, Stochastics, 87 (2015), 142-159.
doi: 10.1080/17442508.2014.924938. |
[12] |
R. C. Grimmer,
Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349.
doi: 10.1090/S0002-9947-1982-0664046-4. |
[13] |
F. Jiang and Y. Shen,
Stability of impulsive stochastic neutral partial differential equations with infinite delays, Asian Journal of Control, 14 (2012), 1706-1709.
doi: 10.1002/asjc.491. |
[14] |
A. N. Kolmogorov,
The Wiener spiral and some other interesting curves in Hilbert space, Dokl. Akad. Nauk SSSR, 26 (1940), 115-118.
|
[15] |
I. Kruk, F. Russo and C. A. Tudor,
Wiener integrals, Malliavin calculus and covariance measure structure, Journal of Functional Analysis, 249 (2007), 92-142.
doi: 10.1016/j.jfa.2007.03.031. |
[16] |
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
doi: 10.1142/0906. |
[17] |
N. N. Leonenko and V. V. Ahn,
Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence, Journal of Applied Mathematics and Stochastic Analysis, 14 (2001), 27-46.
doi: 10.1155/S1048953301000041. |
[18] |
J. Liang, J. H. Liu and T.-J. Xiao,
Nonlocal problems for integrodifferential equations, Dynamics of Continuous, Discrete and Impulsive Systems, Series A, Mathematical Analysis, 15 (2008), 815-824.
|
[19] |
M. Maejima and C. A. Tudor,
Wiener integrals with respect to the Hermite process and a non central limit theorem, Stochastic Analysis and Applications, 25 (2007), 1043-1056.
doi: 10.1080/07362990701540519. |
[20] |
M. Maejima and C. A. Tudor,
Selfsimilar processes with stationary increments in the second Wiener chaos, Probability and Mathematical Statistics, 32 (2012), 167-186.
|
[21] |
M. Maejima and C. A. Tudor,
On the distribution of the Rosenblatt process, Statistics and Probability Letters, 83 (2013), 1490-1495.
doi: 10.1016/j.spl.2013.02.019. |
[22] |
B. B. Mandelbrot and J. W. Van Ness,
Fractional Brownian motions, fractional noises and applications, SIAM Review, 10 (1968), 422-437.
doi: 10.1137/1010093. |
[23] |
V. Pipiras and M. S. Taqqu,
Regularization and integral representations of Hermite processes, Statistics and Probability Letters, 80 (2010), 2014-2023.
doi: 10.1016/j.spl.2010.09.008. |
[24] |
J. Prüss, Evolutionary Integral Equations and Applications, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[25] |
Y. Ren, W. S. Yin and R. Sakthivel,
Stabilization of stochastic differential equations driven by $G$-Brownian motion with feedback control based on discrete-time state observation, Automatica J. IFAC, 95 (2018), 146-151.
doi: 10.1016/j.automatica.2018.05.039. |
[26] |
R. Sakthivel, P. Revathi, Y. Ren and G. J. Shen,
Retarded stochastic differential equations with infinite delay driven by Rosenblatt process, Stochastic Analysis and Applications, 36 (2018), 304-323.
doi: 10.1080/07362994.2017.1399801. |
[27] |
A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 14. World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
doi: 10.1142/9789812798664. |
[28] |
M. Ali Syed, Robust stability of stochastic fuzzy impulsive recurrent neural networks with time-varying delays, Iranian Journal of Fuzzy Systems, 11 (2014), 1–13, 97. |
[29] |
M. S. Taqqu,
Weak convergence to the fractional Brownian motion and to the Rosenblatt process, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 31 (1974/75), 287-302.
doi: 10.1007/BF00532868. |
[30] |
S. Tindel, C. A. Tudor and F. Viens,
Stochastic evolution equations with fractional Brownian motion, Probability Theory and Related Fields, 127 (2003), 186-204.
doi: 10.1007/s00440-003-0282-2. |
[31] |
C. A. Tudor,
Analysis of the Rosenblatt process, ESAIM Probab. Stat., 12 (2008), 230-257.
doi: 10.1051/ps:2007037. |
[32] |
L. T. Yan and G. J. Shen,
On the collision local time of sub-fractional Brownian motions, Stat. Probab. Lett., 80 (2010), 296-308.
doi: 10.1016/j.spl.2009.11.003. |
[33] |
H. Yang and F. Jiang, Exponential stability of mild solutions to impulsive stochastic neutral partial differential equations with memory, Advances in Difference Equations, 2013 (2013), Art. ID 148, 9 pp.
doi: 10.1186/1687-1847-2013-148. |
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