Advances in Differential Equations

Planar differential systems at resonance

Alessandro Fonda and Jean Mawhin

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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.

Article information

Adv. Differential Equations Volume 11, Number 10 (2006), 1111-1133.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
Secondary: 34C11: Growth, boundedness 34C15: Nonlinear oscillations, coupled oscillators 34C25: Periodic solutions


Fonda, Alessandro; Mawhin, Jean. Planar differential systems at resonance. Adv. Differential Equations 11 (2006), no. 10, 1111--1133.

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