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

Yong Li; Hongren Wang; Xue Yang

doi:10.3934/dcdsb.2018123

Article Contents

2018, Volume 23, Issue 6: 2607-2623. Doi: 10.3934/dcdsb.2018123

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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
^* Corresponding author: Yong Li

Received: June 2017

Revised: October 2017

Early access: April 2018

Published: June 2018

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).

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Abstract

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.

Keywords:

Affine-periodic solution,
Fink type conjecture,
Levinson's conjecture,
gradient or Newtonian systems with friction,
Lyapunov's methods.

Mathematics Subject Classification: Primary: 34C27; Secondary: 34C25.

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References

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