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July  2020, 25(7): 2665-2697. doi: 10.3934/dcdsb.2020027

The existence and exponential behavior of solutions to time fractional stochastic delay evolution inclusions with nonlinear multiplicative noise and fractional noise

1. 

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

2. 

College of Science, Northwest A & F University Yangling, Shaanxi 712100, China

* Corresponding author: Yejuan Wang

Received  March 2019 Published  July 2020 Early access  April 2020

Fund Project: This work was supported by NSF of China (Grant Nos. 41875084, 11801452, 11571153), the Fundamental Research Funds for the Central Universities under Grants No. lzujbky-2018-ot03 and lzujbky-2018-it58, the Natural Science Basic Research Plan in Shaanxi Province of China under Grant No. 2019JQ-196, and the Doctor Fund of Northwest A & F University under Grant No. 2452018017

This article is devoted to study time fractional stochastic evolution inclusions with infinite delays driven by a nonlinear multiplicative noise and a fractional Brownian motion with Hurst parameter $ H\in(\frac{1}{2},1) $. First of all, we investigate the local and global existence of mild solutions to such evolution inclusions by using the fractional resolvent operator theory and some new results on the measure of noncompactness for the stochastic integral term. Further, we prove that every mild solution decays exponentially to zero in the sense of mean-square topology.

Citation: Yajing Li, Yejuan Wang. The existence and exponential behavior of solutions to time fractional stochastic delay evolution inclusions with nonlinear multiplicative noise and fractional noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2665-2697. doi: 10.3934/dcdsb.2020027
References:
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N. U. Ahmed, Nonlinear stochastic differential inclusions on Banach space, Stoch. Anal. Appl., 12 (1994), 1-10.  doi: 10.1080/07362999408809334.

[2]

D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal., 69 (2008), 3692-3705.  doi: 10.1016/j.na.2007.10.004.

[3]

E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces (Ph.D. thesis), University Press Facilities, Eindhoven University of Technology, 2001.

[4]

P. Balasubramaniam and P. Tamilalagan, Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi's function, Appl. Math. Comput., 256 (2015), 232-246.  doi: 10.1016/j.amc.2015.01.035.

[5]

M. BenchohraS. Litimein and G. N'Guérékata, On fractional integro-differential inclusions with state-dependent delay in Banach spaces, Appl. Anal., 92 (2013), 335-350.  doi: 10.1080/00036811.2011.616496.

[6]

A. BoudaouiT. Caraballo and A. Ouahab, Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay, Math. Methods Appl. Sci., 39 (2016), 1435-1451.  doi: 10.1002/mma.3580.

[7]

B. Boufoussi and S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statist. Probab. Lett., 82 (2012), 1549-1558.  doi: 10.1016/j.spl.2012.04.013.

[8]

Á. Cartea and D. del-Castillo-Negrete, Fluid limit of the continuous-time random walk with general Lévy jump distribution functions, Phys. Rev. E, 76 (2007), 041105. doi: 10.1103/PhysRevE.76.041105.

[9]

P. M. Carvalho-Neto, Fractional Differential Equations: A Novel Study of Local and Global Solutions in Banach Spaces, PhD thesis, Universidade de São Paulo, São Carlos, 2013.

[10]

P. M. Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980.  doi: 10.1016/j.jde.2015.04.008.

[11]

A. Chadha and D. N. Pandey, Existence of the mild solution for impulsive neutral stochastic fractional integro-differential inclusions with nonlocal conditions, Mediterr. J. Math., 13 (2016), 1005-1031.  doi: 10.1007/s00009-015-0558-7.

[12]

P. Y. Chen and Y. X. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728.  doi: 10.1007/s00033-013-0351-z.

[13]

R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430.  doi: 10.1016/0022-0396(71)90004-0.

[14]

K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992.

[15]

R. Friedrich, F. Jenko, A. Baule and S. Eule, Anomalous diffusion of inertial, weakly damped particles, Phys. Rev. Lett., 96 (2006), 230601. doi: 10.1103/PhysRevLett.96.230601.

[16]

T. Guendouzi, Existence and controllability results for fractional stochastic semilinear differential inclusions, Differ. Equ. Dyn. Syst., 23 (2015), 225-240.  doi: 10.1007/s12591-014-0217-7.

[17]

T. Guendouzi and L. Bousmaha, Approximate controllability of fractional neutral stochastic functional integro-differential inclusions with infinite delay, Qual. Theory Dyn. Syst., 13 (2014), 89-119.  doi: 10.1007/s12346-014-0107-y.

[18]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: de Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter, Berlin, New York, 2001.

[19]

T. D. Ke and D. Lan, Global attractor for a class of functional differential inclusions with Hille-Yosida operators, Nonlinear Anal., 103 (2014), 72-86.  doi: 10.1016/j.na.2014.03.006.

[20]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V., Amsterdam, 2006.

[21]

M. Kisielewicz, Stochastic Differential Inclusions and Applications, Springer, New York, 2013. doi: 10.1007/978-1-4614-6756-4.

[22]

K. X. Li, Stochastic delay fractional evolution equations driven by fractional Brownian motion, Math. Methods Appl. Sci., 38 (2015), 1582-1591.  doi: 10.1002/mma.3169.

[23]

Y. Li and B. Liu, Existence of solution of nonlinear neutral stochastic differential inclusions with infinite delay, Stoch. Anal. Appl., 25 (2007), 397-415.  doi: 10.1080/07362990601139610.

[24] N. G. Lloyd, Degree Theory, Cambridge University Press, 1978. 
[25]

F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, Ser. Adv. Math. Appl. Sci., 23 (1994), 246-251. 

[26] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300.
[27]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093.

[28]

M. Michta, On connections between stochastic differential inclusions and set-valued stochastic differential equations driven by semimartingales, J. Differential Equations, 262 (2017), 2106-2134.  doi: 10.1016/j.jde.2016.10.039.

[29]

Y. S. Mishura, Stocastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0.

[30]

D. Nguyen, Asymptotic behavior of linear fractional stochastic differential equations with time-varying delays, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 1-7.  doi: 10.1016/j.cnsns.2013.06.004.

[31]

T. A. NguyenD. K. Tran and N. Q. Nguyen, Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3637-3654.  doi: 10.3934/dcdsb.2016114.

[32]

M. Niu and B. Xie, Regularity of a fractional partial differential equation driven by space-time white noise, Proc. Amer. Math. Soc., 138 (2010), 1479-1489.  doi: 10.1090/S0002-9939-09-10197-1.

[33]

I. Podlubny, Fractional Difierential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, California, USA, 1999.

[34]

Y. RenL. Y. Hu and R. Sakthivel, Controllability of impulsive neutral stochastic functional differential inclusions with infinite delay, J. Comput. Appl. Math., 235 (2011), 2603-2614.  doi: 10.1016/j.cam.2010.10.051.

[35]

R. SakthivelY. RenA. Debbouche and N. I. Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal., 95 (2016), 2361-2382.  doi: 10.1080/00036811.2015.1090562.

[36]

C. Sandra, Stationary Hamilton-Jacobi equations in Hilbert spaces and applications to a stochastic optimal control problem, SIAM J. Control Optim., 40 (2001), 824-852.  doi: 10.1137/S0363012999359949.

[37]

X. B. Shu and F. Xu, The existence of solutions for impulsive fractional partial neutral differential equations, J. Math., 2013 (2013), Art. ID 147193, 9 pp. doi: 10.1155/2013/147193.

[38]

I. M. Sokolov and R. Metzler, Towards deterministic equations for Lévy walks: The fractional material derivative, Phys. Rev. E, 67 (2003), 010101. doi: 10.1103/PhysRevE.67.010101.

[39]

P. Tamilalagan and P. Balasubramaniam, Moment stability via resolvent operators of fractional stochastic differential inclusions driven by fractional Brownian motion, Appl. Math. Comput., 305 (2017), 299-307.  doi: 10.1016/j.amc.2017.02.013.

[40]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.

[41]

Z. M. Yan, Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces, IMA J. Math. Control Inform., 30 (2013), 443-462.  doi: 10.1093/imamci/dns033.

show all references

References:
[1]

N. U. Ahmed, Nonlinear stochastic differential inclusions on Banach space, Stoch. Anal. Appl., 12 (1994), 1-10.  doi: 10.1080/07362999408809334.

[2]

D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal., 69 (2008), 3692-3705.  doi: 10.1016/j.na.2007.10.004.

[3]

E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces (Ph.D. thesis), University Press Facilities, Eindhoven University of Technology, 2001.

[4]

P. Balasubramaniam and P. Tamilalagan, Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi's function, Appl. Math. Comput., 256 (2015), 232-246.  doi: 10.1016/j.amc.2015.01.035.

[5]

M. BenchohraS. Litimein and G. N'Guérékata, On fractional integro-differential inclusions with state-dependent delay in Banach spaces, Appl. Anal., 92 (2013), 335-350.  doi: 10.1080/00036811.2011.616496.

[6]

A. BoudaouiT. Caraballo and A. Ouahab, Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay, Math. Methods Appl. Sci., 39 (2016), 1435-1451.  doi: 10.1002/mma.3580.

[7]

B. Boufoussi and S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statist. Probab. Lett., 82 (2012), 1549-1558.  doi: 10.1016/j.spl.2012.04.013.

[8]

Á. Cartea and D. del-Castillo-Negrete, Fluid limit of the continuous-time random walk with general Lévy jump distribution functions, Phys. Rev. E, 76 (2007), 041105. doi: 10.1103/PhysRevE.76.041105.

[9]

P. M. Carvalho-Neto, Fractional Differential Equations: A Novel Study of Local and Global Solutions in Banach Spaces, PhD thesis, Universidade de São Paulo, São Carlos, 2013.

[10]

P. M. Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980.  doi: 10.1016/j.jde.2015.04.008.

[11]

A. Chadha and D. N. Pandey, Existence of the mild solution for impulsive neutral stochastic fractional integro-differential inclusions with nonlocal conditions, Mediterr. J. Math., 13 (2016), 1005-1031.  doi: 10.1007/s00009-015-0558-7.

[12]

P. Y. Chen and Y. X. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728.  doi: 10.1007/s00033-013-0351-z.

[13]

R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430.  doi: 10.1016/0022-0396(71)90004-0.

[14]

K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992.

[15]

R. Friedrich, F. Jenko, A. Baule and S. Eule, Anomalous diffusion of inertial, weakly damped particles, Phys. Rev. Lett., 96 (2006), 230601. doi: 10.1103/PhysRevLett.96.230601.

[16]

T. Guendouzi, Existence and controllability results for fractional stochastic semilinear differential inclusions, Differ. Equ. Dyn. Syst., 23 (2015), 225-240.  doi: 10.1007/s12591-014-0217-7.

[17]

T. Guendouzi and L. Bousmaha, Approximate controllability of fractional neutral stochastic functional integro-differential inclusions with infinite delay, Qual. Theory Dyn. Syst., 13 (2014), 89-119.  doi: 10.1007/s12346-014-0107-y.

[18]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: de Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter, Berlin, New York, 2001.

[19]

T. D. Ke and D. Lan, Global attractor for a class of functional differential inclusions with Hille-Yosida operators, Nonlinear Anal., 103 (2014), 72-86.  doi: 10.1016/j.na.2014.03.006.

[20]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V., Amsterdam, 2006.

[21]

M. Kisielewicz, Stochastic Differential Inclusions and Applications, Springer, New York, 2013. doi: 10.1007/978-1-4614-6756-4.

[22]

K. X. Li, Stochastic delay fractional evolution equations driven by fractional Brownian motion, Math. Methods Appl. Sci., 38 (2015), 1582-1591.  doi: 10.1002/mma.3169.

[23]

Y. Li and B. Liu, Existence of solution of nonlinear neutral stochastic differential inclusions with infinite delay, Stoch. Anal. Appl., 25 (2007), 397-415.  doi: 10.1080/07362990601139610.

[24] N. G. Lloyd, Degree Theory, Cambridge University Press, 1978. 
[25]

F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, Ser. Adv. Math. Appl. Sci., 23 (1994), 246-251. 

[26] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300.
[27]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.  doi: 10.1137/1010093.

[28]

M. Michta, On connections between stochastic differential inclusions and set-valued stochastic differential equations driven by semimartingales, J. Differential Equations, 262 (2017), 2106-2134.  doi: 10.1016/j.jde.2016.10.039.

[29]

Y. S. Mishura, Stocastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0.

[30]

D. Nguyen, Asymptotic behavior of linear fractional stochastic differential equations with time-varying delays, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 1-7.  doi: 10.1016/j.cnsns.2013.06.004.

[31]

T. A. NguyenD. K. Tran and N. Q. Nguyen, Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3637-3654.  doi: 10.3934/dcdsb.2016114.

[32]

M. Niu and B. Xie, Regularity of a fractional partial differential equation driven by space-time white noise, Proc. Amer. Math. Soc., 138 (2010), 1479-1489.  doi: 10.1090/S0002-9939-09-10197-1.

[33]

I. Podlubny, Fractional Difierential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, California, USA, 1999.

[34]

Y. RenL. Y. Hu and R. Sakthivel, Controllability of impulsive neutral stochastic functional differential inclusions with infinite delay, J. Comput. Appl. Math., 235 (2011), 2603-2614.  doi: 10.1016/j.cam.2010.10.051.

[35]

R. SakthivelY. RenA. Debbouche and N. I. Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal., 95 (2016), 2361-2382.  doi: 10.1080/00036811.2015.1090562.

[36]

C. Sandra, Stationary Hamilton-Jacobi equations in Hilbert spaces and applications to a stochastic optimal control problem, SIAM J. Control Optim., 40 (2001), 824-852.  doi: 10.1137/S0363012999359949.

[37]

X. B. Shu and F. Xu, The existence of solutions for impulsive fractional partial neutral differential equations, J. Math., 2013 (2013), Art. ID 147193, 9 pp. doi: 10.1155/2013/147193.

[38]

I. M. Sokolov and R. Metzler, Towards deterministic equations for Lévy walks: The fractional material derivative, Phys. Rev. E, 67 (2003), 010101. doi: 10.1103/PhysRevE.67.010101.

[39]

P. Tamilalagan and P. Balasubramaniam, Moment stability via resolvent operators of fractional stochastic differential inclusions driven by fractional Brownian motion, Appl. Math. Comput., 305 (2017), 299-307.  doi: 10.1016/j.amc.2017.02.013.

[40]

R. N. WangD. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.

[41]

Z. M. Yan, Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces, IMA J. Math. Control Inform., 30 (2013), 443-462.  doi: 10.1093/imamci/dns033.

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