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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 & 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.  Google Scholar

[2]

N. N. Bogoliubov, On some Statistical Methods in Mathematical Physics Lzv. Akad. Nauk Ukr. SSR, Kiev, 1945.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh$ä$user, Boston, 2003. Google Scholar

[6]

M. Bohner and G. Sh. Guseinov, Partial differentiation on time scales, Dynam. Syst. and Appl., 13 (2004), 351-379.   Google Scholar

[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.  Google Scholar

[8]

P. Fatou, Sur le movement d'un systáme soumis á Des forces á courte période, Bull. Soc. Math. Fance., 56 (1928), 98-139.   Google Scholar

[9]

S. Hilger, Ein Ma$β$kettenkalk$ü$ mit Anwendung auf Zentrumsmanningfaltigkeiten PhD thesis, Universit$ä$t W$ü$rzburg, 1988. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. LlibreD. 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.  Google Scholar

[13]

J. LlibreD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

N. N. Bogoliubov, On some Statistical Methods in Mathematical Physics Lzv. Akad. Nauk Ukr. SSR, Kiev, 1945.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh$ä$user, Boston, 2003. Google Scholar

[6]

M. Bohner and G. Sh. Guseinov, Partial differentiation on time scales, Dynam. Syst. and Appl., 13 (2004), 351-379.   Google Scholar

[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.  Google Scholar

[8]

P. Fatou, Sur le movement d'un systáme soumis á Des forces á courte période, Bull. Soc. Math. Fance., 56 (1928), 98-139.   Google Scholar

[9]

S. Hilger, Ein Ma$β$kettenkalk$ü$ mit Anwendung auf Zentrumsmanningfaltigkeiten PhD thesis, Universit$ä$t W$ü$rzburg, 1988. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. LlibreD. 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.  Google Scholar

[13]

J. LlibreD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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