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

Yajing Li; Yejuan Wang

doi:10.3934/dcdsb.2020027

Previous Article
Probabilistic continuity of a pullback random attractor in time-sample
DCDS-B Home
This Issue
Next Article
Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy

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

Yajing Li ^1,2, and Yejuan Wang ^1,,

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.

Keywords: Fractional stochastic evolution inclusion, infinite delay, measure of noncompactness, exponential decay, Caputo fractional derivative, nonlinear multiplicative noise, fractional Brownian motion.

Mathematics Subject Classification: Primary: 34A08, 34K09, 34K50, 35K58.

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. Benchohra, S. 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. Boudaoui, T. 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. Nguyen, D. 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. Ren, L. 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. Sakthivel, Y. Ren, A. 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. Wang, D. 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. Benchohra, S. 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. Boudaoui, T. 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. Nguyen, D. 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. Ren, L. 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. Sakthivel, Y. Ren, A. 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. Wang, D. 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

[1]	Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021001
[2]	Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318
[3]	Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020376
[4]	Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020104
[5]	Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282
[6]	Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020383
[7]	S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020435
[8]	Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020 doi: 10.3934/naco.2020054
[9]	Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020432
[10]	Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020321
[11]	Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088
[12]	Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021 doi: 10.3934/nhm.2021003
[13]	Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020379
[14]	Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020466
[15]	Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020388
[16]	Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021014
[17]	Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020456
[18]	Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265
[19]	Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268
[20]	Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors