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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  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 & Continuous Dynamical Systems - B, 2020, 25 (7) : 2665-2697. doi: 10.3934/dcdsb.2020027
References:
[1]

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

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

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

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

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

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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