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2006 Planar differential systems at resonance
Alessandro Fonda, Jean Mawhin
Adv. Differential Equations 11(10): 1111-1133 (2006).

Abstract

We consider the system $$ J\dot u=\nabla { \mathcal {H}}(u)+f(u)+p(t)\,, $$ where ${ \mathcal {H}}:{{\mathbb R}} ^2\to{{\mathbb R}} $ is of class $C^1$ with locally Lipschitz continuous gradient, $f:{{\mathbb R}} ^2\to{{\mathbb R}} ^2$ is locally Lipschitz continuous and bounded, and $p:{{\mathbb R}} \to{{\mathbb R}} ^2$ is measurable, bounded and $T-$periodic. Here, $J= \begin{pmatrix} \scriptstyle 0 & \!\!\!\!\scriptstyle -1 \cr \scriptstyle 1 & \scriptstyle \!0 \end{pmatrix} $ is the standard symplectic matrix. For some classes of functions $f,$ we give new existence theorems for periodic solutions and for unbounded solutions. Applications are given to forced second-order differential equations with separated nonlinearities.

Citation

Download Citation

Alessandro Fonda. Jean Mawhin. "Planar differential systems at resonance." Adv. Differential Equations 11 (10) 1111 - 1133, 2006.

Information

Published: 2006
First available in Project Euclid: 18 December 2012

zbMATH: 1155.34020
MathSciNet: MR2279740

Subjects:
Primary: 37J45
Secondary: 34C11, 34C15, 34C25

Rights: Copyright © 2006 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.11 • No. 10 • 2006
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