# American Institute of Mathematical Sciences

September  2018, 38(9): 4657-4674. doi: 10.3934/dcds.2018204

## Lyapunov stability for regular equations and applications to the Liebau phenomenon

 1 School of Mathematics and Physics, Changzhou University, Changzhou 213164, China 2 Department of Mathematics, Nanjing University, Nanjing 210093, China 3 Departamento de Matemáticas, Universidade de Vigo, 32004, Pabellón 3, Campus de Ourense, Spain 4 Department of Functional Analysis, Faculty of Mathematics and Natural Sciences, University of Rzeszów, Pigonia 1, 35-959 Rzeszów, Poland

* Corresponding author: J. A. Cid

Received  December 2017 Revised  March 2018 Published  June 2018

Fund Project: F. Wang was sponsored by Qing Lan Project of Jiangsu Province, and was supported by the National Natural Science Foundation of China (Grant No. 11501055 and No. 11401166), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 15KJB110001), China Postdoctoral Science Foundation funded project (Grant No. 2017M610315), Hainan Natural Science Foundation (Grant No.117005), Jiangsu Planned Projects for Postdoctoral Research Funds. J. A. Cid was partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Project MTM2017-85054-C2-1-P. M. Zima was partially supported by the Centre for Innovation and Transfer of Natural Science and Engineering Knowledge of University of Rzeszów.

We study the existence and stability of periodic solutions of two kinds of regular equations by means of classical topological techniques like the Kolmogorov-Arnold-Moser (KAM) theory, the Moser twist theorem, the averaging method and the method of upper and lower solutions in the reversed order. As an application, we present some results on the existence and stability of $T$-periodic solutions of a Liebau-type equation.

Citation: Feng Wang, José Ángel Cid, Mirosława Zima. Lyapunov stability for regular equations and applications to the Liebau phenomenon. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4657-4674. doi: 10.3934/dcds.2018204
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##### References:
$2\pi$-periodic solution of equation (39) with $b = 1.55$ and $c = 0.4$
$2\pi$-periodic positive solution of equation (40) with $b = 3/2$ and $c = 0.133333$
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