Differential and Integral Equations

Resonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition

Maurizio Garrione

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Abstract

The concept of resonance with the first eigenvalue ($\lambda = 0$) of the scalar $T$-periodic problem $$ x''+\lambda x =0, \ \ \ x(0)=x(T), \; x'(0)=x'(T) $$ is considered for first-order planar systems, by dealing with positively homogeneous Hamiltonians which can vanish at some points on $\mathbb{S}^1$. By means of degree methods, an existence result at double resonance for a planar system of the kind $$ Ju'=F(t, u), \quad J= \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} \right), $$ is then proved, under the assumption that $F(t, u)$ is controlled from below by the gradient of one of such Hamiltonians described above, complementing the main theorem in [7] and including some classical results for the scalar case.

Article information

Source
Differential Integral Equations, Volume 25, Number 5/6 (2012), 505-526.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356012676

Mathematical Reviews number (MathSciNet)
MR2951738

Zentralblatt MATH identifier
1265.34058

Subjects
Primary: 34C25: Periodic solutions 37C27: Periodic orbits of vector fields and flows 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods

Citation

Garrione, Maurizio. Resonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition. Differential Integral Equations 25 (2012), no. 5/6, 505--526. https://projecteuclid.org/euclid.die/1356012676


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