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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  December 2020 Early access  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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [21] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chinchester, 1997. [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. [23] S. E. A. Mohammed, Stochastic Functional Differential Equations. Research Notes in Mathematics, Vol. 99, Pitman (Advanced Publishing Program), Boston, MA, 1984. [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. [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. [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. [27] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. [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. [29] C. Tudor, Almost periodic solutions of affine stochastic evolution equations, Stochastics Stochastics Rep., 38 (1992), 251-266.  doi: 10.1080/17442509208833758. [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.

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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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [21] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chinchester, 1997. [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. [23] S. E. A. Mohammed, Stochastic Functional Differential Equations. Research Notes in Mathematics, Vol. 99, Pitman (Advanced Publishing Program), Boston, MA, 1984. [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. [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. [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. [27] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. [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. [29] C. Tudor, Almost periodic solutions of affine stochastic evolution equations, Stochastics Stochastics Rep., 38 (1992), 251-266.  doi: 10.1080/17442509208833758. [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.
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