# American Institute of Mathematical Sciences

August  2018, 23(6): 2607-2623. doi: 10.3934/dcdsb.2018123

## Fink type conjecture on affine-periodic solutions and Levinson's conjecture to Newtonian systems

 1 School of Mathematics, Jilin University, Changchun 130012, China 2 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China 3 Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China 4 Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012, China 5 College of mathematics, Jilin Normal University, Jilin 136000, China

* Corresponding author: Yong Li

Received  June 2017 Revised  October 2017 Published  August 2018 Early access  April 2018

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

This paper concerns the existence of affine-periodic solutions for differential systems (including functional differential equations) and Newtonian systems with friction. This is a kind of pattern solutions in time-space, which may be periodic, anti-periodic, subharmonic or quasi periodic corresponding to rotation motions. Fink type conjecture is verified and Lyapunov's methods are given. These results are applied to study gradient systems and Newtonian (including Rayleigh or Lienard) systems. Levinson's conjecture to Newtonian systems is proved.

Citation: Yong Li, Hongren Wang, Xue Yang. Fink type conjecture on affine-periodic solutions and Levinson's conjecture to Newtonian systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2607-2623. doi: 10.3934/dcdsb.2018123
##### References:
 [1] V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk, 18 (1963), 13-40. [2] J.-M. Belley and É Bondo, Anti-periodic solutions of Liénard equations with state dependent impulses, J. Differential Equations, 261 (2016), 4164-4187.  doi: 10.1016/j.jde.2016.06.020. [3] 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. [4] T. A. Burton and B. Zhang, Uniform ultimate boundedness and periodicity in functional-differential equations, Tohoku Math. J.(2), 42 (1990), 93-100.  doi: 10.2748/tmj/1178227696. [5] T. A. Burton and S. Zhang, Unified boundedness, periodicity, and stability in ordinary and functional-differential equations, Ann. Mat. Pura Appl.(4), 145 (1986), 129-158.  doi: 10.1007/BF01790540. [6] 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. [7] J. Chu and F. Wang, Prevalence of stable periodic solutions for Duffing equations, J. Differential Equations, 260 (2016), 7800-7820.  doi: 10.1016/j.jde.2016.02.003. [8] F. Cong, T. Küpper, Y. Li and J. You, KAM-type theorem on resonant surfaces for nearly integrable Hamiltonian systems, J. Nonlinear Sci., 10 (2000), 49-68.  doi: 10.1007/s003329910003. [9] A. M. Fink, Convergence and almost periodicity of solutions of forced lienard equation, SIAM. J. Appl. Math., 26 (1974), 26-34.  doi: 10.1137/0126004. [10] A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530. [11] 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. [12] N. Levinson, Transformation theory of non-linear differential equations of the second order, Ann. of Math.(2), 45 (1944), 723-737.  doi: 10.2307/1969299. [13] 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. [14] Y. Li and Y. Yi, A quasi-periodic Poincaré's theorem, Math. Ann., 326 (2003), 649-690.  doi: 10.1007/s00208-002-0399-0. [15] F. Lin, Uniformly asymptotic stability and existence of periodic solutions and almost periodic solutions, Sci. China Ser. A, 37 (1994), 933-945. [16] X. Meng and Y. Li, Affine-periodic solutions for discrete dynamical systems, J. Appl. Anal. Comput., 5 (2015), 781-792. [17] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. Ⅱ, 1962 (1962), 1-20. [18] 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. [19] 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. [20] H. Wang, X. Yang, Y. Li and X. Li, LaSalle type stationary oscillation theorems for affine-periodic systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2907-2921.  doi: 10.3934/dcdsb.2017156. [21] J. Xing, X. Yang and Y. Li, Affine-periodic solutions by averaging methods, Sci. China Math., 61 (2018), 439-452.  doi: 10.1007/s11425-016-0455-1. [22] 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. [23] P. Zabreiko and M. Krasnosel'skii, Iteration of operators and fixed points, Dokl. Akad. Nauk SSSR, 196 (1971), 1006-1009. [24] 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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##### References:
 [1] V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk, 18 (1963), 13-40. [2] J.-M. Belley and É Bondo, Anti-periodic solutions of Liénard equations with state dependent impulses, J. Differential Equations, 261 (2016), 4164-4187.  doi: 10.1016/j.jde.2016.06.020. [3] 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. [4] T. A. Burton and B. Zhang, Uniform ultimate boundedness and periodicity in functional-differential equations, Tohoku Math. J.(2), 42 (1990), 93-100.  doi: 10.2748/tmj/1178227696. [5] T. A. Burton and S. Zhang, Unified boundedness, periodicity, and stability in ordinary and functional-differential equations, Ann. Mat. Pura Appl.(4), 145 (1986), 129-158.  doi: 10.1007/BF01790540. [6] 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. [7] J. Chu and F. Wang, Prevalence of stable periodic solutions for Duffing equations, J. Differential Equations, 260 (2016), 7800-7820.  doi: 10.1016/j.jde.2016.02.003. [8] F. Cong, T. Küpper, Y. Li and J. You, KAM-type theorem on resonant surfaces for nearly integrable Hamiltonian systems, J. Nonlinear Sci., 10 (2000), 49-68.  doi: 10.1007/s003329910003. [9] A. M. Fink, Convergence and almost periodicity of solutions of forced lienard equation, SIAM. J. Appl. Math., 26 (1974), 26-34.  doi: 10.1137/0126004. [10] A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530. [11] 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. [12] N. Levinson, Transformation theory of non-linear differential equations of the second order, Ann. of Math.(2), 45 (1944), 723-737.  doi: 10.2307/1969299. [13] 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. [14] Y. Li and Y. Yi, A quasi-periodic Poincaré's theorem, Math. Ann., 326 (2003), 649-690.  doi: 10.1007/s00208-002-0399-0. [15] F. Lin, Uniformly asymptotic stability and existence of periodic solutions and almost periodic solutions, Sci. China Ser. A, 37 (1994), 933-945. [16] X. Meng and Y. Li, Affine-periodic solutions for discrete dynamical systems, J. Appl. Anal. Comput., 5 (2015), 781-792. [17] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. Ⅱ, 1962 (1962), 1-20. [18] 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. [19] 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. [20] H. Wang, X. Yang, Y. Li and X. Li, LaSalle type stationary oscillation theorems for affine-periodic systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2907-2921.  doi: 10.3934/dcdsb.2017156. [21] J. Xing, X. Yang and Y. Li, Affine-periodic solutions by averaging methods, Sci. China Math., 61 (2018), 439-452.  doi: 10.1007/s11425-016-0455-1. [22] 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. [23] P. Zabreiko and M. Krasnosel'skii, Iteration of operators and fixed points, Dokl. Akad. Nauk SSSR, 196 (1971), 1006-1009. [24] 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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