September  2021, 26(9): 4727-4743. doi: 10.3934/dcdsb.2020310

Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications

1. 

School of Mathematical Sciences, Ocean University of China, Qingdao, Shandong 266100, China

2. 

Department of Mathematics, University of Dundee, Dundee DD1 4HN, UK

3. 

Ministry of Education Key Laboratory of Marine Genetics and Breeding, College of Marine Life Science, Ocean University of China, Qingdao, Shandong 266100, China

* Corresponding authors: Linshan Wang and Yangfan Wang

Received  January 2019 Revised  June 2020 Published  September 2021 Early access  October 2020

Fund Project: The authors are supported by the China Scholarship Council under Grant 201906330009, the Fundamental Research Funds for the Central Universities under Grant 201861005, the National Key Research and Development Program of China under Grant 2018YFD0901601, the National Natural Science Foundation of China under Grant 11771014 and Grant 31772844, and the Major Basic Research Projects of Shandong Natural Science Foundation under Grant 2018A07

In this paper, we focus on the mild periodic solutions to a class of delayed stochastic reaction-diffusion differential equations. First, the key issues of Markov property in Banach space $ C $, $ p $-uniformly boundedness, and $ p $-point dissipativity of mild solutions $ \boldsymbol{u}_t $ to the equations are discussed. Then, the theorems of existence-uniqueness and exponential stability in the mean-square sense of the mild periodic solutions are established by using the dissipative theory and the operator semigroup technique, and the relevant results about the existence of mild periodic solutions in the quoted literature are generalized. Next, the given theoretical results are successfully applied to the delayed stochastic reaction-diffusion Hopfield neural networks, and some easy-to-test criteria of exponential stability for the mild periodic solution to the networks are obtained. Finally, some examples are presented to demonstrate the feasibility of our results.

Citation: Qi Yao, Linshan Wang, Yangfan Wang. Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4727-4743. doi: 10.3934/dcdsb.2020310
References:
[1]

G. Adomian and R. Rach, Nonlinear stochastic differential delay equations, J. Math. Anal. Appl., 91 (1983), 94-101.  doi: 10.1016/0022-247X(83)90094-X.

[2]

L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley & Sons, New York, 1974.

[3]

H. Bao and J. Cao, Delay-distribution-dependent state estimation for discrete-time stochastic neural networks with random delay, Neural Networks, 24 (2011), 19-28. 

[4]

E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, J. Comput. Appl. Math., 125 (2000), 297-307.  doi: 10.1016/S0377-0427(00)00475-1.

[5]

J. Cao, New results concerning exponential stability and periodic solutions of delayed cellular neural networks, Phys. Lett. A, 307 (2003), 136-147.  doi: 10.1016/S0375-9601(02)01720-6.

[6]

T. Caraballo and K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stochastic Anal. Appl., 17 (1999), 743-763.  doi: 10.1080/07362999908809633.

[7]

W. H. ChenL. Liu and X. Lu, Intermittent synchronization of reaction-diffusion neural networks with mixed delays via Razumikhin technique, Nonlinear Dynam., 87 (2017), 535-551.  doi: 10.1007/s11071-016-3059-8.

[8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[9]

J. DuanK. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z.

[10] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1975. 
[11]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, 74. Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.

[12]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[13]

K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75.  doi: 10.1215/kjm/1250524705.

[14]

R. Jahanipur, Stochastic functional evolution equations with monotone nonlinearity: Existence and stability of the mild solutions, J. Differential Equations, 248 (2010), 1230-1255.  doi: 10.1016/j.jde.2009.12.012.

[15]

J. Lei and M. C. Mackey, Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system, SIAM J. Appl. Math., 67 (2006/07), 387-407.  doi: 10.1137/060650234.

[16]

X. Li, Existence and global exponential stability of periodic solution for impulsive Cohen-Grossberg-type BAM neural networks with continuously distributed delays, Appl. Math. Comput., 215 (2009), 292-307.  doi: 10.1016/j.amc.2009.05.005.

[17]

X. LiangL. WangY. Wang and R. Wang, Dynamical behavior of delayed reaction-diffusion Hopfield neural networks driven by infinite dimensional Wiener processes, IEEE Trans. Neural Netw. Learn. Syst., 27 (2016), 1816-1826.  doi: 10.1109/TNNLS.2015.2460117.

[18]

K. Liu, Some views on recent randomized study of infinite dimensional functional differential equations (in Chinese), Sci. Sin. Math., 45 (2015), 559-566. 

[19]

Z. Liu and L. Liao, Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays, J. Math. Anal. Appl., 290 (2004), 247-262.  doi: 10.1016/j.jmaa.2003.09.052.

[20]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.

[21]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[22]

S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics: Third Edition, American Mathematical Society, Providence, 1991. doi: 10.1090/mmono/090.

[23] L. Wang, Delayed Recurrent Neural Networks, Science Press, Beijing, 2008. 
[24]

L. Wang, Global well-posedness and stability of the mild solutions for a class of stochastic partial functional differential equations (in Chinese), Sci. Sin. Math., 47 (2017), 371-382. 

[25]

L. Wang and Y. Gao, Global exponential robust stability of reaction-diffusion interval neural networks with time-varying delays, Phys. Lett. A, 350 (2006), 342-348.  doi: 10.1016/j.physleta.2005.10.031.

[26]

Z. WangY. LiuM. Li and X. Liu, Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays, IEEE Trans. Neural Networks, 17 (2006), 814-820. 

[27]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.

[28]

L. Wang and D. Xu, Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays, Sci. China Ser. F, 46 (2003), 466-474. 

[29]

T. WeiL. Wang and Y. Wang, Existence, uniqueness and stability of mild solutions to stochastic reaction-diffusion Cohen-Grossberg neural networks with delays and Wiener processes, Neurocomputing, 239 (2017), 19-27.  doi: 10.1016/j.neucom.2017.01.069.

[30]

D. XuY. Huang and Z. Yang, Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Contin. Dyn. Syst., 24 (2009), 1005-1023.  doi: 10.3934/dcds.2009.24.1005.

[31]

Q. YaoL. Wang and Y. Wang, Existence-uniqueness and stability of reaction-diffusion stochastic Hopfield neural networks with S-type distributed time delays, Neurocomputing, 275 (2018), 470-477. 

[32]

B. Zhang and K. Gopalsamy, On the periodic solution of $n$-dimensional stochastic population models, Stoch. Anal. Appl., 18 (2000), 323-331.  doi: 10.1080/07362990008809671.

[33]

Q. Zhu and B. Song, Exponential stability of impulsive nonlinear stochastic differential equations with mixed delays, Nonlinear Anal. Real World Appl., 12 (2011), 2851-2860.  doi: 10.1016/j.nonrwa.2011.04.011.

show all references

References:
[1]

G. Adomian and R. Rach, Nonlinear stochastic differential delay equations, J. Math. Anal. Appl., 91 (1983), 94-101.  doi: 10.1016/0022-247X(83)90094-X.

[2]

L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley & Sons, New York, 1974.

[3]

H. Bao and J. Cao, Delay-distribution-dependent state estimation for discrete-time stochastic neural networks with random delay, Neural Networks, 24 (2011), 19-28. 

[4]

E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, J. Comput. Appl. Math., 125 (2000), 297-307.  doi: 10.1016/S0377-0427(00)00475-1.

[5]

J. Cao, New results concerning exponential stability and periodic solutions of delayed cellular neural networks, Phys. Lett. A, 307 (2003), 136-147.  doi: 10.1016/S0375-9601(02)01720-6.

[6]

T. Caraballo and K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stochastic Anal. Appl., 17 (1999), 743-763.  doi: 10.1080/07362999908809633.

[7]

W. H. ChenL. Liu and X. Lu, Intermittent synchronization of reaction-diffusion neural networks with mixed delays via Razumikhin technique, Nonlinear Dynam., 87 (2017), 535-551.  doi: 10.1007/s11071-016-3059-8.

[8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.
[9]

J. DuanK. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.  doi: 10.1007/s10884-004-7830-z.

[10] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1975. 
[11]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, 74. Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.

[12]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[13]

K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75.  doi: 10.1215/kjm/1250524705.

[14]

R. Jahanipur, Stochastic functional evolution equations with monotone nonlinearity: Existence and stability of the mild solutions, J. Differential Equations, 248 (2010), 1230-1255.  doi: 10.1016/j.jde.2009.12.012.

[15]

J. Lei and M. C. Mackey, Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system, SIAM J. Appl. Math., 67 (2006/07), 387-407.  doi: 10.1137/060650234.

[16]

X. Li, Existence and global exponential stability of periodic solution for impulsive Cohen-Grossberg-type BAM neural networks with continuously distributed delays, Appl. Math. Comput., 215 (2009), 292-307.  doi: 10.1016/j.amc.2009.05.005.

[17]

X. LiangL. WangY. Wang and R. Wang, Dynamical behavior of delayed reaction-diffusion Hopfield neural networks driven by infinite dimensional Wiener processes, IEEE Trans. Neural Netw. Learn. Syst., 27 (2016), 1816-1826.  doi: 10.1109/TNNLS.2015.2460117.

[18]

K. Liu, Some views on recent randomized study of infinite dimensional functional differential equations (in Chinese), Sci. Sin. Math., 45 (2015), 559-566. 

[19]

Z. Liu and L. Liao, Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays, J. Math. Anal. Appl., 290 (2004), 247-262.  doi: 10.1016/j.jmaa.2003.09.052.

[20]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.

[21]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[22]

S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics: Third Edition, American Mathematical Society, Providence, 1991. doi: 10.1090/mmono/090.

[23] L. Wang, Delayed Recurrent Neural Networks, Science Press, Beijing, 2008. 
[24]

L. Wang, Global well-posedness and stability of the mild solutions for a class of stochastic partial functional differential equations (in Chinese), Sci. Sin. Math., 47 (2017), 371-382. 

[25]

L. Wang and Y. Gao, Global exponential robust stability of reaction-diffusion interval neural networks with time-varying delays, Phys. Lett. A, 350 (2006), 342-348.  doi: 10.1016/j.physleta.2005.10.031.

[26]

Z. WangY. LiuM. Li and X. Liu, Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays, IEEE Trans. Neural Networks, 17 (2006), 814-820. 

[27]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.

[28]

L. Wang and D. Xu, Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays, Sci. China Ser. F, 46 (2003), 466-474. 

[29]

T. WeiL. Wang and Y. Wang, Existence, uniqueness and stability of mild solutions to stochastic reaction-diffusion Cohen-Grossberg neural networks with delays and Wiener processes, Neurocomputing, 239 (2017), 19-27.  doi: 10.1016/j.neucom.2017.01.069.

[30]

D. XuY. Huang and Z. Yang, Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Contin. Dyn. Syst., 24 (2009), 1005-1023.  doi: 10.3934/dcds.2009.24.1005.

[31]

Q. YaoL. Wang and Y. Wang, Existence-uniqueness and stability of reaction-diffusion stochastic Hopfield neural networks with S-type distributed time delays, Neurocomputing, 275 (2018), 470-477. 

[32]

B. Zhang and K. Gopalsamy, On the periodic solution of $n$-dimensional stochastic population models, Stoch. Anal. Appl., 18 (2000), 323-331.  doi: 10.1080/07362990008809671.

[33]

Q. Zhu and B. Song, Exponential stability of impulsive nonlinear stochastic differential equations with mixed delays, Nonlinear Anal. Real World Appl., 12 (2011), 2851-2860.  doi: 10.1016/j.nonrwa.2011.04.011.

Figure 1.  The periodic trajectory and simulation of $ u_1 $ and $ u_2 $ in Example 1
Figure 2.  The phase graph in Example 1
Figure 3.  The periodic trajectory and simulation of $ u_1 $ and $ u_2 $ in Example 1
Figure 4.  The trajectory of u to Example 3.1 (left) and Example 3.2 (right) in Example 3
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