# 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 & Continuous Dynamical Systems - B, 2018, 23 (6) : 2607-2623. doi: 10.3934/dcdsb.2018123
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