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October  2017, 10(5): 959-971. doi: 10.3934/dcdss.2017050

## Existence of periodic solutions of dynamic equations on time scales by averaging

 a. College of Mathematics, Jilin University, Changchun, 130012, China b. School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China c. State Key Laboratory of Automotive Simulation and control, Jilin University, Changchun, 130012, China

Received  December 2016 Revised  January 2017 Published  June 2017

Fund Project: The first author was supposed by NSFC (grant No. 11301541). The second author was supposed by National Basic Research Program of China (grant No. 2013CB834100), NSFC (grant No. 11571065), NSFC (grant No. 11171132). The fourth author was supposed by NSFC (grant No. 11201173).

In this paper, we study the existence of periodic solutions for perturbed dynamic equations on time scales. Our approach is based on the averaging method. Further, we extend some averaging theorem to periodic solutions of dynamic equations on time scales to $k-$th order in $\varepsilon$. More precisely, results of higher order averaging for finding periodic solutions are given via the topological degree theory.

Citation: Ruichao Guo, Yong Li, Jiamin Xing, Xue Yang. Existence of periodic solutions of dynamic equations on time scales by averaging. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 959-971. doi: 10.3934/dcdss.2017050
##### References:
 [1] M. Adivar and Y. N. Raffoul, Existence results for periodic solutions of intego-dynamic equations on time scales, Ann. Mat. Pura. Appl., 188 (2009), 543-559.  doi: 10.1007/s10231-008-0088-z. [2] N. N. Bogoliubov, On some Statistical Methods in Mathematical Physics Lzv. Akad. Nauk Ukr. SSR, Kiev, 1945. [3] N. N. Bogoliubov and N. Krylov, The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Acad. Sci. , Kiev, 1934. [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications Birkh$ä$user, Boston, 2001. doi: 10.1007/978-1-4612-0201-1. [5] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh$ä$user, Boston, 2003. [6] M. Bohner and G. Sh. Guseinov, Partial differentiation on time scales, Dynam. Syst. and Appl., 13 (2004), 351-379. [7] A. Buica and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22.  doi: 10.1016/j.bulsci.2003.09.002. [8] P. Fatou, Sur le movement d'un systáme soumis á Des forces á courte période, Bull. Soc. Math. Fance., 56 (1928), 98-139. [9] S. Hilger, Ein Ma$β$kettenkalk$ü$ mit Anwendung auf Zentrumsmanningfaltigkeiten PhD thesis, Universit$ä$t W$ü$rzburg, 1988. [10] Y. Li and C. Wang, Almost periodic functions on time scales and applications Discrete Dyn. Nat. Soc., 2011 (2011), Art. ID 727068, 20 pp. doi: 10.1155/2011/727068. [11] C. Lizama and J. G. Mesquita, Almost automorphic solutions of dynamic equations on time scales, J. Funct. Anal., 265 (2013), 2267-2311.  doi: 10.1016/j.jfa.2013.06.013. [12] J. Llibre, D. D Novaes and M. A Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearly, 27 (2014), 563-583.  doi: 10.1088/0951-7715/27/3/563. [13] J. Llibre, D. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. Math., 139 (2015), 229-244.  doi: 10.1016/j.bulsci.2014.08.011. [14] A. Slavík, Averaging dynamic equations on time scales, J. Math. Anal. Appl., 388 (2012), 996-1012.  doi: 10.1016/j.jmaa.2011.10.043. [15] C. Wang and Y. Li, Affine-periodic solutions for nonlinear differential equations on time scales, Adv. Differ. Equ., 2015 (2015), 286-302.  doi: 10.1186/s13662-015-0634-0.

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##### References:
 [1] M. Adivar and Y. N. Raffoul, Existence results for periodic solutions of intego-dynamic equations on time scales, Ann. Mat. Pura. Appl., 188 (2009), 543-559.  doi: 10.1007/s10231-008-0088-z. [2] N. N. Bogoliubov, On some Statistical Methods in Mathematical Physics Lzv. Akad. Nauk Ukr. SSR, Kiev, 1945. [3] N. N. Bogoliubov and N. Krylov, The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Acad. Sci. , Kiev, 1934. [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications Birkh$ä$user, Boston, 2001. doi: 10.1007/978-1-4612-0201-1. [5] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh$ä$user, Boston, 2003. [6] M. Bohner and G. Sh. Guseinov, Partial differentiation on time scales, Dynam. Syst. and Appl., 13 (2004), 351-379. [7] A. Buica and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22.  doi: 10.1016/j.bulsci.2003.09.002. [8] P. Fatou, Sur le movement d'un systáme soumis á Des forces á courte période, Bull. Soc. Math. Fance., 56 (1928), 98-139. [9] S. Hilger, Ein Ma$β$kettenkalk$ü$ mit Anwendung auf Zentrumsmanningfaltigkeiten PhD thesis, Universit$ä$t W$ü$rzburg, 1988. [10] Y. Li and C. Wang, Almost periodic functions on time scales and applications Discrete Dyn. Nat. Soc., 2011 (2011), Art. ID 727068, 20 pp. doi: 10.1155/2011/727068. [11] C. Lizama and J. G. Mesquita, Almost automorphic solutions of dynamic equations on time scales, J. Funct. Anal., 265 (2013), 2267-2311.  doi: 10.1016/j.jfa.2013.06.013. [12] J. Llibre, D. D Novaes and M. A Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearly, 27 (2014), 563-583.  doi: 10.1088/0951-7715/27/3/563. [13] J. Llibre, D. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. Math., 139 (2015), 229-244.  doi: 10.1016/j.bulsci.2014.08.011. [14] A. Slavík, Averaging dynamic equations on time scales, J. Math. Anal. Appl., 388 (2012), 996-1012.  doi: 10.1016/j.jmaa.2011.10.043. [15] C. Wang and Y. Li, Affine-periodic solutions for nonlinear differential equations on time scales, Adv. Differ. Equ., 2015 (2015), 286-302.  doi: 10.1186/s13662-015-0634-0.
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