## Differential and Integral Equations

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

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

#### 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