Communications in Mathematical Sciences

Volume 17 (2019)

Number 7

Periodic solutions to nonlinear Euler–Bernoulli beam equations

Pages: 2005 – 2034

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n7.a10

Authors

Bochao Chen (College of Mathematics, Jilin University, Changchun, Jilin, China)

Yixian Gao (School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin, China)

Yong Li (College of Mathematics, Jilin University, Changchun, Jilin, China; and School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin, China)

Abstract

Bending vibrations of thin beams and plates can be described by nonlinear Euler–Bernoulli beam equation with $x$-dependent coefficients. In this paper we demonstrate the existence of families of time-periodic solutions to such a model by virtue of a Lyapunov–Schmidt reduction together with a Nash–Moser method. This result holds for all parameters $(\epsilon , \omega)$ in a Cantor set with asymptotically full measure as $\epsilon \to 0$.

Keywords

Euler–Bernoulli beam equations, variable coefficients, periodic solutions, Nash–Moser iteration

2010 Mathematics Subject Classification

35B10, 35L75, 58C15

Full Text (PDF format)

The research of B.C. was supported in part by NSFC Grant 11901232 and China Postdoctoral Science Foundation Funded Project 2019M651191. The research of Y.G. was supported in part by NSFC grant 11871140, 11671071, JJKH20180006KJ, JLSTDP 20190201154JC and FRFCU2412019BJ005. The research of Y.L. was supported in part by NSFC grant 11571065.

Received 29 April 2018

Accepted 1 July 2019

Published 6 January 2020