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December  2020, 25(12): 4737-4754. doi: 10.3934/dcdsb.2020122

## Periodic solutions for SDEs through upper and lower solutions

 1 School of Mathematics and Statistics & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China 2 School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China 3 School of Mathematics, Jilin University, Changchun 130012, China 4 State Key Laboratory of Automotive Simulation and Control, Jilin University, 130025, China

* Corresponding author: Yong Li

Received  November 2019 Revised  January 2020 Published  March 2020

Fund Project: The first author is supported NSFC grant 11601043, China Postdoctoral Science Foundation (Grant No. 2016M590243 and 2019T120226). The second author was supported by NSFC grant 11201173. The third author was supported by National Basic Research Program of China (Grant No. 2013CB834100) and NSFC grants 11171132 and 11571065

We study a kind of better recurrence than Kolmogorov's one: periodicity recurrence, which corresponds periodic solutions in distribution for stochastic differential equations. On the basis of technique of upper and lower solutions and comparison principle, we obtain the existence of periodic solutions in distribution for stochastic differential equations (SDEs). Hence this provides an effective method how to study the periodicity of stochastic systems by analyzing deterministic ones. We also illustrate our results.

Citation: Chunyan Ji, Yang Xue, Yong Li. Periodic solutions for SDEs through upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4737-4754. doi: 10.3934/dcdsb.2020122
##### References:
 [1] X. Bai and J. Jiang, Comparison theorem for stochastic functional differential equations and applications, J. Dynam. Differential Equations, 29 (2017), 1-24.  doi: 10.1007/s10884-014-9406-x.  Google Scholar [2] S. R. Bernfeld and V. Lakshmtkantham, An Introduction to Nonlinear Boundary Value Problems. Mathematics in Science and Engineering, Vol. 109, Academic Press, Inc., New YorkLondon, 1974.  Google Scholar [3] R. Buckdahn and S. Peng, Ergodic backward stochastic differential equations and associated partial differential equations, in Seminar on Stochastic Analysis: Random Fields and Applications, Vol. 45, Birkhäuser, Basel, 1999, 73–85. doi: 10.1007/978-3-0348-8681-9_6.  Google Scholar [4] M. R. Cândido, J. Llibre and D. D. Novaes, Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction, Nonlinearity, 30 (2017), 3560-3586.  doi: 10.1088/1361-6544/aa7e95.  Google Scholar [5] F. Chen, Y. Han, Y. Li and X. Yang, Periodic solutions of Fokker-Planck equations, J. Differential Equations, 263 (2017), 285-298.  doi: 10.1016/j.jde.2017.02.032.  Google Scholar [6] C. Feng, H. Zhao and B. Zhou, Pathwise random periodic solutions of stochastic differential equations, J. Differential Equations, 251 (2011), 119-149.  doi: 10.1016/j.jde.2011.03.019.  Google Scholar [7] C. Feng, Y. Wu and H. Zhao, Anticipating random periodic solutions-I. SDEs with multiplicative linear noise, J. Funct. Anal., 271 (2016), 365-417.  doi: 10.1016/j.jfa.2016.04.027.  Google Scholar [8] J.-M. Fokam, Multiplicity and regularity of large periodic solutions with rational frequency for a class of semilinear monotone wave equations, Proc. Amer. Math. Soc., 145 (2017), 4283-4297.  doi: 10.1090/proc/12760.  Google Scholar [9] P. Gao, Some periodic type solutions for stochastic reaction-diffusion equation with cubic nonlinearities, Comput. Math. Appl., 74 (2017), 2281-2297.  doi: 10.1016/j.camwa.2017.07.005.  Google Scholar [10] N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633.   Google Scholar [11] M. Ji, W. Qi, Z. Shen and Y. Yi, Existence of periodic probability solutions to Fokker-Planck equations with applications, J Funct. Anal., 277 (2019), 108281, 41 pp. doi: 10.1016/j.jfa.2019.108281.  Google Scholar [12] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2$^nd$ edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar [13] S. K. Kaul and A. S. Vatsala, Monotone method for integro-differential equation with periodic boundary conditions, Appl. Anal., 21 (1986), 297-305.  doi: 10.1080/00036818608839598.  Google Scholar [14] R.Z. Khasminskii, Stochastic Stability of Differential Equations, 2$^nd$ edition, Stochastic Modelling and Applied Probability, Vol. 66, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar [15] M. A. Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$, The Operator of Translations Along Trajectories of Differential Equations. Translations of Mathematical Monographs, Vol. 19, American Mathematical Society, Providence, R.I., 1968. doi: 10.1090/mmono/019.  Google Scholar [16] V. Lakshmikantham and S. Leela, Remarks on first and second periodic boundary value problems, Nonlinear Anal., 8 (1984), 281-287.  doi: 10.1016/0362-546X(84)90050-6.  Google Scholar [17] Y. Li, H. Z. Wang, X. R. Lü and X. G. Lu, Periodic solutions for functional-differential equations with infinite lead and delay, Appl. Math. Comput., 70 (1995), 1-28.  doi: 10.1016/0096-3003(94)00131-M.  Google Scholar [18] Y. Li, F. Cong, Z. Lin and W. Liu, Periodic solutions for evolution equations, Nonlinear Anal., 36 (1999) 275–293. doi: 10.1016/S0362-546X(97)00626-3.  Google Scholar [19] Z. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 266 (2014), 1115-1149.  doi: 10.1016/j.jfa.2013.11.011.  Google Scholar [20] Z. Liu and W. Wang, Favard separation method for almost periodic stochastic differential equations, J. Differential Equations, 260 (2016), 8109-8136.  doi: 10.1016/j.jde.2016.02.019.  Google Scholar [21] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chinchester, 1997.  Google Scholar [22] O. Mellah and P. R. De Fitte, Counterexamples to mean square almost periodicity of the solutions of some SDEs with almost periodic coefficients, Electron. J. Differential Equations, 91 (2013), 7 pp.  Google Scholar [23] S. E. A. Mohammed, Stochastic Functional Differential Equations. Research Notes in Mathematics, Vol. 99, Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar [24] S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380.  doi: 10.1016/j.spa.2005.08.004.  Google Scholar [25] G. Da Prato and C. Tudor, Periodic and almost periodic solutions for semilinear stochastic equations, Stochastic Anal. Appl., 13 (1995), 13-33.  doi: 10.1080/07362999508809380.  Google Scholar [26] V. Šeda, J. J. Nieto and M. Gera, Periodic boundary value problems for nonlinear higher order ordinary differential equations, Appl. Math. Comput., 48 (1992), 71-82.  doi: 10.1016/0096-3003(92)90019-W.  Google Scholar [27] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995.  Google Scholar [28] M. Tarallo and Z. Zhou, Limit periodic upper and lower solutions in a generic sense, Discrete Contin. Dyn. Syst., 38 (2018), 293-309.  doi: 10.3934/dcds.2018014.  Google Scholar [29] C. Tudor, Almost periodic solutions of affine stochastic evolution equations, Stochastics Stochastics Rep., 38 (1992), 251-266.  doi: 10.1080/17442509208833758.  Google Scholar [30] H. Zhao and Z.-H. Zheng, Random periodic solutions of random dynamical systems, J. Differential Equations, 246 (2009), 2020-2038.  doi: 10.1016/j.jde.2008.10.011.  Google Scholar

show all references

##### References:
 [1] X. Bai and J. Jiang, Comparison theorem for stochastic functional differential equations and applications, J. Dynam. Differential Equations, 29 (2017), 1-24.  doi: 10.1007/s10884-014-9406-x.  Google Scholar [2] S. R. Bernfeld and V. Lakshmtkantham, An Introduction to Nonlinear Boundary Value Problems. Mathematics in Science and Engineering, Vol. 109, Academic Press, Inc., New YorkLondon, 1974.  Google Scholar [3] R. Buckdahn and S. Peng, Ergodic backward stochastic differential equations and associated partial differential equations, in Seminar on Stochastic Analysis: Random Fields and Applications, Vol. 45, Birkhäuser, Basel, 1999, 73–85. doi: 10.1007/978-3-0348-8681-9_6.  Google Scholar [4] M. R. Cândido, J. Llibre and D. D. Novaes, Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction, Nonlinearity, 30 (2017), 3560-3586.  doi: 10.1088/1361-6544/aa7e95.  Google Scholar [5] F. Chen, Y. Han, Y. Li and X. Yang, Periodic solutions of Fokker-Planck equations, J. Differential Equations, 263 (2017), 285-298.  doi: 10.1016/j.jde.2017.02.032.  Google Scholar [6] C. Feng, H. Zhao and B. Zhou, Pathwise random periodic solutions of stochastic differential equations, J. Differential Equations, 251 (2011), 119-149.  doi: 10.1016/j.jde.2011.03.019.  Google Scholar [7] C. Feng, Y. Wu and H. Zhao, Anticipating random periodic solutions-I. SDEs with multiplicative linear noise, J. Funct. Anal., 271 (2016), 365-417.  doi: 10.1016/j.jfa.2016.04.027.  Google Scholar [8] J.-M. Fokam, Multiplicity and regularity of large periodic solutions with rational frequency for a class of semilinear monotone wave equations, Proc. Amer. Math. Soc., 145 (2017), 4283-4297.  doi: 10.1090/proc/12760.  Google Scholar [9] P. Gao, Some periodic type solutions for stochastic reaction-diffusion equation with cubic nonlinearities, Comput. Math. Appl., 74 (2017), 2281-2297.  doi: 10.1016/j.camwa.2017.07.005.  Google Scholar [10] N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633.   Google Scholar [11] M. Ji, W. Qi, Z. Shen and Y. Yi, Existence of periodic probability solutions to Fokker-Planck equations with applications, J Funct. Anal., 277 (2019), 108281, 41 pp. doi: 10.1016/j.jfa.2019.108281.  Google Scholar [12] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2$^nd$ edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar [13] S. K. Kaul and A. S. Vatsala, Monotone method for integro-differential equation with periodic boundary conditions, Appl. Anal., 21 (1986), 297-305.  doi: 10.1080/00036818608839598.  Google Scholar [14] R.Z. Khasminskii, Stochastic Stability of Differential Equations, 2$^nd$ edition, Stochastic Modelling and Applied Probability, Vol. 66, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar [15] M. A. Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$, The Operator of Translations Along Trajectories of Differential Equations. Translations of Mathematical Monographs, Vol. 19, American Mathematical Society, Providence, R.I., 1968. doi: 10.1090/mmono/019.  Google Scholar [16] V. Lakshmikantham and S. Leela, Remarks on first and second periodic boundary value problems, Nonlinear Anal., 8 (1984), 281-287.  doi: 10.1016/0362-546X(84)90050-6.  Google Scholar [17] Y. Li, H. Z. Wang, X. R. Lü and X. G. Lu, Periodic solutions for functional-differential equations with infinite lead and delay, Appl. Math. Comput., 70 (1995), 1-28.  doi: 10.1016/0096-3003(94)00131-M.  Google Scholar [18] Y. Li, F. Cong, Z. Lin and W. Liu, Periodic solutions for evolution equations, Nonlinear Anal., 36 (1999) 275–293. doi: 10.1016/S0362-546X(97)00626-3.  Google Scholar [19] Z. Liu and K. Sun, Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 266 (2014), 1115-1149.  doi: 10.1016/j.jfa.2013.11.011.  Google Scholar [20] Z. Liu and W. Wang, Favard separation method for almost periodic stochastic differential equations, J. Differential Equations, 260 (2016), 8109-8136.  doi: 10.1016/j.jde.2016.02.019.  Google Scholar [21] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chinchester, 1997.  Google Scholar [22] O. Mellah and P. R. De Fitte, Counterexamples to mean square almost periodicity of the solutions of some SDEs with almost periodic coefficients, Electron. J. Differential Equations, 91 (2013), 7 pp.  Google Scholar [23] S. E. A. Mohammed, Stochastic Functional Differential Equations. Research Notes in Mathematics, Vol. 99, Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar [24] S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380.  doi: 10.1016/j.spa.2005.08.004.  Google Scholar [25] G. Da Prato and C. Tudor, Periodic and almost periodic solutions for semilinear stochastic equations, Stochastic Anal. Appl., 13 (1995), 13-33.  doi: 10.1080/07362999508809380.  Google Scholar [26] V. Šeda, J. J. Nieto and M. Gera, Periodic boundary value problems for nonlinear higher order ordinary differential equations, Appl. Math. Comput., 48 (1992), 71-82.  doi: 10.1016/0096-3003(92)90019-W.  Google Scholar [27] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995.  Google Scholar [28] M. Tarallo and Z. Zhou, Limit periodic upper and lower solutions in a generic sense, Discrete Contin. Dyn. Syst., 38 (2018), 293-309.  doi: 10.3934/dcds.2018014.  Google Scholar [29] C. Tudor, Almost periodic solutions of affine stochastic evolution equations, Stochastics Stochastics Rep., 38 (1992), 251-266.  doi: 10.1080/17442509208833758.  Google Scholar [30] H. Zhao and Z.-H. Zheng, Random periodic solutions of random dynamical systems, J. Differential Equations, 246 (2009), 2020-2038.  doi: 10.1016/j.jde.2008.10.011.  Google Scholar
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