The concept of resonance with the first eigenvalue (λ= 0) of the scalar T-periodic problem x″ +λx = 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 S 1. By means of degree methods, an existence result at double resonance for a planar system of the kind Ju′ = F(t; u); J = 1 00 -1 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  and including some classical results for the scalar case.

Garrione, M. (2012). Resonance at the first eigenvalue for first-order systems in the plane: Vanishing Hamiltonians and the Landesman-lazer condition. DIFFERENTIAL AND INTEGRAL EQUATIONS, 25(5-6), 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 (λ= 0) of the scalar T-periodic problem x″ +λx = 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 S 1. By means of degree methods, an existence result at double resonance for a planar system of the kind Ju′ = F(t; u); J = 1 00 -1 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  and including some classical results for the scalar case.
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Resonance; positively homogeneous Hamiltonians; Landesman-Lazer condition
English
2012
505
526
22
Garrione, M. (2012). Resonance at the first eigenvalue for first-order systems in the plane: Vanishing Hamiltonians and the Landesman-lazer condition. DIFFERENTIAL AND INTEGRAL EQUATIONS, 25(5-6), 505-526.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10281/62277`
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