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 [7] 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
GARRIONE, MAURIZIO
2012
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 [7] and including some classical results for the scalar case.
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