# American Institute of Mathematical Sciences

## 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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020310
##### References:
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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.  Google Scholar [30] D. Xu, Y. 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.  Google Scholar [31] Q. Yao, L. 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [2] L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley & Sons, New York, 1974.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] W. H. Chen, L. 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.  Google Scholar [8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge university press, Cambridge, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar [9] J. Duan, K. 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.  Google Scholar [10] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1975.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] X. Liang, L. Wang, Y. 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.  Google Scholar [18] K. Liu, Some views on recent randomized study of infinite dimensional functional differential equations (in Chinese), Sci. Sin. Math., 45 (2015), 559-566.   Google Scholar [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.  Google Scholar [20] W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar [21] X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar [22] S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics: Third Edition, American Mathematical Society, Providence, 1991. doi: 10.1090/mmono/090.  Google Scholar [23] L. Wang, Delayed Recurrent Neural Networks, Science Press, Beijing, 2008.   Google Scholar [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.   Google Scholar [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.  Google Scholar [26] Z. Wang, Y. Liu, M. Li and X. Liu, Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays, IEEE Trans. Neural Networks, 17 (2006), 814-820.   Google Scholar [27] X. Wang, K. 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.  Google Scholar [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.   Google Scholar [29] T. Wei, L. 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.  Google Scholar [30] D. Xu, Y. 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.  Google Scholar [31] Q. Yao, L. 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar
The periodic trajectory and simulation of $u_1$ and $u_2$ in Example 1
The phase graph in Example 1
The periodic trajectory and simulation of $u_1$ and $u_2$ in Example 1
The trajectory of u to Example 3.1 (left) and Example 3.2 (right) in Example 3
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