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September  2017, 22(7): 2907-2921. doi: 10.3934/dcdsb.2017156

LaSalle type stationary oscillation theorems for Affine-Periodic Systems

Hongren Wang 1,2, , Xue Yang 2,3,4,5, , Yong Li 2,3,4,, and  Xiaoyue Li 3,

1. 

College of mathematics, Jilin Normal University, Jilin 136000, China
2. 

College of Mathematics, Jilin University, Changchun 130012, China
3. 

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
4. 

Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China
5. 

Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Yong Li

Received  July 2016 Revised  March 2017 Published  May 2017

Fund Project: The second author is supported by NSFC (grant No. 11201173). The third author is supported by National Basic Research Program of China (grant No. 2013CB834100), NSFC (grant No. 11571065) and NSFC (grant No. 11171132).

The paper concerns the existence of affine-periodic solutions for affine-periodic (functional) differential systems, which is a new type of quasi-periodic solutions if they are bounded. Some more general criteria than LaSalle's one on the existence of periodic solutions are established. Some applications are also given.

Keywords: LaSalle type stationary oscillation principle, Q-affine T-periodic solutions, asymptotic stability.
Mathematics Subject Classification: Primary:34C27;Secondary:34C25.
Citation: Hongren Wang, Xue Yang, Yong Li, Xiaoyue Li. LaSalle type stationary oscillation theorems for Affine-Periodic Systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2907-2921. doi: 10.3934/dcdsb.2017156
References:
[1]

T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional-Differential Equations Mathematics in Science and Engineering, 178, Academic Press, Inc. , Orlando, FL, 1985.

[2]

O. Chadli, Q. H. Ansari and J. C. Yao, Mixed equilibrium problems and anti-periodic solutions for nonlinear evolution equations, J. Optim. Theory Appl., 168 (2016), 410-440.  doi: 10.1007/s10957-015-0707-y.

[3]

X. Chang and Y. Li, Rotating periodic solutions of second order dissipative dynamical systems, Discrete Contin. Dyn. Syst., 36 (2016), 643-652.  doi: 10.3934/dcds.2016.36.643.

[4]

Y. Chen, J. J. Nieto and D. O'Regan, Anti-periodic solutions for evolution equations associated with maximal monotone mappings, Appl. Math. Lett., 24 (2011), 302-307.  doi: 10.1016/j.aml.2010.10.010.

[5]

Y. Chen, D. O'Regan and R. P. Agarwal, Anti-periodic solutions for semilinear evolution equations in Banach spaces, J. Appl. Math. Comput., 38 (2012), 63-70.  doi: 10.1007/s12190-010-0463-y.

[6]

W. A. Coppel, Dichotomies in Stability Theory Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978.

[7]

J. R. Graef and L. Kong, Periodic solutions of first order functional differential equations, Appl. Math. Lett., 24 (2011), 1981-1985.  doi: 10.1016/j.aml.2011.05.020.

[8]

T. Haddad and T. Haddad, Existence and uniqueness of anti-periodic solutions for nonlinear third-order differential inclusions, Electron. J. Differential Equations (2013), 10 pp.

[9]

J. K. Hale, Functional Differential Equations Applied Mathematical Sciences, Vol. 3, Springer-Verlag New York, New York-Heidelberg, 1971.

[10]

S. Heidarkhani, G. A. Afrouzi, A. Hadjian and J. Henderson, Existence of infinitely many anti-periodic solutions for second-order impulsive differential inclusions, Electron. J. Differential Equations (2013), 13 pp.

[11]

J. Knežević-Miljanović, Periodic solutions of nonlinear differential equations, Adv. Dyn. Syst. Appl., 7 (2012), 89-93. 

[12]

T. Küpper, Y. Li and B. Zhang, Periodic solutions for dissipative-repulsive systems, Tohoku Math. J. (2), 52 (2000), 321-329.  doi: 10.2748/tmj/1178207816.

[13]

J. LaSalle and S. Lefschetz, Stabillity by Lyapunov's Direct Method, with Application Mathematics in Science and Engineering, Vol. 4, New York-London 1961.

[14]

Y. Li and F. Huang, Levinson's problem on affine-periodic solutions, Adv. Nonlinear Stud., 15 (2015), 241-252.  doi: 10.1515/ans-2015-0113.

[15]

Y. Li, Q. Zhou and X. Lu, Periodic solutions for functional-differential inclusions with infinite delay, Sci. China Ser. A, 37 (1994), 1289-1301. 

[16]

Y. Li, Q. Zhou and X. Lu, Periodic solutions and equilibrium states for functional-differential inclusions with nonconvex right-hand side, Quart. Appl. Math., 55 (1997), 57-68.  doi: 10.1090/qam/1433752.

[17]

J. Liang, J. Liu and T. Xiao, Periodic solutions of delay impulsive differential equations, Nonlinear Anal., 74 (2011), 6835-6842.  doi: 10.1016/j.na.2011.07.008.

[18]

A. Lipowski, B. Przeradzki and K. Szymańska-Debowska, Periodic solutions to differential equations with a generalized p-Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2593-2601.  doi: 10.3934/dcdsb.2014.19.2593.

[19]

J. Mawhin, Periodic solutions of differential and difference systems with pendulum-type nonlinearities: Variational approaches, Differential and difference equations with applications, Springer Proc. Math. Stat., Springer, New York, 47 (2013), 83-98.  doi: 10.1007/978-1-4614-7333-6_7.

[20]

X. Meng and Y. Li, Affine-periodic solutions for discrete dynamical systems, J. Appl. Anal. Comput., 5 (2015), 781-792. 

[21]

S. Padhi, John R. Graef and P. D. N. Srinivasu, Periodic Solutions of First-Order Functional Differential Equations in Population Dynamics Springer, New Delhi, 2014. doi: 10.1007/978-81-322-1895-1.

[22]

P. Sa Ngiamsunthorn, Existence of periodic solutions for differential equations with multiple delays under dichotomy condition Adv. Difference Equ. 2015 (2015), 11 pp. doi: 10.1186/s13662-015-0598-0.

[23]

C. Wang and Y. Li, Affine-periodic solutions for nonlinear dynamic equations on time scales Adv. Difference Equ. 2015 (2015), 16 pp. doi: 10.1186/s13662-015-0634-0.

[24]

C. Wang, X. Yang and Y. Li, Affine-periodic solutions for nonlinear differential equations, Rocky Mountain J. Math., 46 (2016), 1717-1737.  doi: 10.1216/RMJ-2016-46-5-1717.

[25]

J. R. Ward, Periodic solutions of first order systems, Discrete Contin. Dyn. Syst., 33 (2013), 381-389.  doi: 10.3934/dcds.2013.33.381.

[26]

R. Wu, F. Cong and Y. Li, Anti-periodic solutions for second order differential equations, Appl. Math. Lett., 24 (2011), 860-863.  doi: 10.1016/j.aml.2010.12.031.

[27]

T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions Applied Mathematical Sciences, Vol. 14. Springer-Verlag, New York-Heidelberg, 1975.

[28]

Y. Zhang, X. Yang and Y. Li, Affine-periodic solutions for dissipative systems, Abstr. Appl. Anal. (2013), Art. ID 157140, 4 pp.

show all references

References:
[1]

T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional-Differential Equations Mathematics in Science and Engineering, 178, Academic Press, Inc. , Orlando, FL, 1985.

[2]

O. Chadli, Q. H. Ansari and J. C. Yao, Mixed equilibrium problems and anti-periodic solutions for nonlinear evolution equations, J. Optim. Theory Appl., 168 (2016), 410-440.  doi: 10.1007/s10957-015-0707-y.

[3]

X. Chang and Y. Li, Rotating periodic solutions of second order dissipative dynamical systems, Discrete Contin. Dyn. Syst., 36 (2016), 643-652.  doi: 10.3934/dcds.2016.36.643.

[4]

Y. Chen, J. J. Nieto and D. O'Regan, Anti-periodic solutions for evolution equations associated with maximal monotone mappings, Appl. Math. Lett., 24 (2011), 302-307.  doi: 10.1016/j.aml.2010.10.010.

[5]

Y. Chen, D. O'Regan and R. P. Agarwal, Anti-periodic solutions for semilinear evolution equations in Banach spaces, J. Appl. Math. Comput., 38 (2012), 63-70.  doi: 10.1007/s12190-010-0463-y.

[6]

W. A. Coppel, Dichotomies in Stability Theory Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978.

[7]

J. R. Graef and L. Kong, Periodic solutions of first order functional differential equations, Appl. Math. Lett., 24 (2011), 1981-1985.  doi: 10.1016/j.aml.2011.05.020.

[8]

T. Haddad and T. Haddad, Existence and uniqueness of anti-periodic solutions for nonlinear third-order differential inclusions, Electron. J. Differential Equations (2013), 10 pp.

[9]

J. K. Hale, Functional Differential Equations Applied Mathematical Sciences, Vol. 3, Springer-Verlag New York, New York-Heidelberg, 1971.

[10]

S. Heidarkhani, G. A. Afrouzi, A. Hadjian and J. Henderson, Existence of infinitely many anti-periodic solutions for second-order impulsive differential inclusions, Electron. J. Differential Equations (2013), 13 pp.

[11]

J. Knežević-Miljanović, Periodic solutions of nonlinear differential equations, Adv. Dyn. Syst. Appl., 7 (2012), 89-93. 

[12]

T. Küpper, Y. Li and B. Zhang, Periodic solutions for dissipative-repulsive systems, Tohoku Math. J. (2), 52 (2000), 321-329.  doi: 10.2748/tmj/1178207816.

[13]

J. LaSalle and S. Lefschetz, Stabillity by Lyapunov's Direct Method, with Application Mathematics in Science and Engineering, Vol. 4, New York-London 1961.

[14]

Y. Li and F. Huang, Levinson's problem on affine-periodic solutions, Adv. Nonlinear Stud., 15 (2015), 241-252.  doi: 10.1515/ans-2015-0113.

[15]

Y. Li, Q. Zhou and X. Lu, Periodic solutions for functional-differential inclusions with infinite delay, Sci. China Ser. A, 37 (1994), 1289-1301. 

[16]

Y. Li, Q. Zhou and X. Lu, Periodic solutions and equilibrium states for functional-differential inclusions with nonconvex right-hand side, Quart. Appl. Math., 55 (1997), 57-68.  doi: 10.1090/qam/1433752.

[17]

J. Liang, J. Liu and T. Xiao, Periodic solutions of delay impulsive differential equations, Nonlinear Anal., 74 (2011), 6835-6842.  doi: 10.1016/j.na.2011.07.008.

[18]

A. Lipowski, B. Przeradzki and K. Szymańska-Debowska, Periodic solutions to differential equations with a generalized p-Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2593-2601.  doi: 10.3934/dcdsb.2014.19.2593.

[19]

J. Mawhin, Periodic solutions of differential and difference systems with pendulum-type nonlinearities: Variational approaches, Differential and difference equations with applications, Springer Proc. Math. Stat., Springer, New York, 47 (2013), 83-98.  doi: 10.1007/978-1-4614-7333-6_7.

[20]

X. Meng and Y. Li, Affine-periodic solutions for discrete dynamical systems, J. Appl. Anal. Comput., 5 (2015), 781-792. 

[21]

S. Padhi, John R. Graef and P. D. N. Srinivasu, Periodic Solutions of First-Order Functional Differential Equations in Population Dynamics Springer, New Delhi, 2014. doi: 10.1007/978-81-322-1895-1.

[22]

P. Sa Ngiamsunthorn, Existence of periodic solutions for differential equations with multiple delays under dichotomy condition Adv. Difference Equ. 2015 (2015), 11 pp. doi: 10.1186/s13662-015-0598-0.

[23]

C. Wang and Y. Li, Affine-periodic solutions for nonlinear dynamic equations on time scales Adv. Difference Equ. 2015 (2015), 16 pp. doi: 10.1186/s13662-015-0634-0.

[24]

C. Wang, X. Yang and Y. Li, Affine-periodic solutions for nonlinear differential equations, Rocky Mountain J. Math., 46 (2016), 1717-1737.  doi: 10.1216/RMJ-2016-46-5-1717.

[25]

J. R. Ward, Periodic solutions of first order systems, Discrete Contin. Dyn. Syst., 33 (2013), 381-389.  doi: 10.3934/dcds.2013.33.381.

[26]

R. Wu, F. Cong and Y. Li, Anti-periodic solutions for second order differential equations, Appl. Math. Lett., 24 (2011), 860-863.  doi: 10.1016/j.aml.2010.12.031.

[27]

T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions Applied Mathematical Sciences, Vol. 14. Springer-Verlag, New York-Heidelberg, 1975.

[28]

Y. Zhang, X. Yang and Y. Li, Affine-periodic solutions for dissipative systems, Abstr. Appl. Anal. (2013), Art. ID 157140, 4 pp.

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