Periodic and subharmonic solutions for duffing equation with a singularity
Zhibo Cheng Jingli Ren
Discrete & Continuous Dynamical Systems - A 2012, 32(5): 1557-1574 doi: 10.3934/dcds.2012.32.1557
This paper is devoted to the existence and multiplicity of periodic and subharmonic solutions for a superlinear Duffing equation with a singularity. In this manner, various preceding theorems are improved and sharpened. Our proof is based on a generalized version of the Poincaré-Birkhoff twist theorem.
keywords: superlinear; singularity harmonic and subharmonic solution Poincaré-Birkhoff twist theorem. Duffing equation
Positive periodic solution for Brillouin electron beam focusing system
Jingli Ren Zhibo Cheng Stefan Siegmund
Discrete & Continuous Dynamical Systems - B 2011, 16(1): 385-392 doi: 10.3934/dcdsb.2011.16.385
An experimental conjecture on the existence of positive periodic solutions for the Brillouin electron beam focusing system $x''+a(1+\cos2t)x=\frac{1}{x}$ for $0 < a < 1$ is proved, using a topological degree theorem by Mawhin.
keywords: Brillouin electron beam focusing system singularity. Liénard equation
Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations
Pedro J. Torres Zhibo Cheng Jingli Ren
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 2155-2168 doi: 10.3934/dcds.2013.33.2155
We analyze the non-degeneracy of the linear $2n$-order differential equation $u^{(2n)}+\sum\limits_{m=1}^{2n-1}a_{m}u^{(m)}=q(t)u$ with potential $q(t)\in L^p(\mathbb{R}/T\mathbb{Z})$, by means of new forms of the optimal Sobolev and Wirtinger inequalities. The results is applied to obtain existence and uniqueness of periodic solution for the prescribed nonlinear problem in the semilinear and superlinear case.
keywords: superlinear uniqueness $2n$-order differential equation. Non-degeneracy semilinear

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