Abstract
Using the least action principle, the existence of periodic solutions for some nonautonomous second order Hamiltonian systems is obtained. Using minimax methods, the multiplicity of periodic solutions is obtained. Our results extend some previous results.
Highlights
Introduction and main resultsConsider the nonautonomous second order Hamiltonian systems u(t) = ∇F(t, u(t)), a.e. t ∈ [, T], ( . )u( ) – u(T) = u ( ) – u (T) =, where the constant T >, the function F(t, x) = F (t, x) + F (t, x), and functions F, F ∈ C (R × Rn, R) with conditions that F (t + T, x) = F (t, x) and F (t + T, x) = F (t, x) hold for all t and x
Let HT = {u : [, T] → Rn | u be absolutely continuous, u( ) = u(T) and u ∈ L ([, T], Rn)} be a Hilbert space with the norm defined by u HT =
It is well known that the solution of problem ( . ) corresponds to the critical point of φ
Summary
U( ) – u(T) = u ( ) – u (T) = , where the constant T > , the function F(t, x) = F (t, x) + F (t, x), and functions F , F ∈ C (R × Rn, R) with conditions that F (t + T, x) = F (t, x) and F (t + T, x) = F (t, x) hold for all t and x. (h∗ ) there exist constants K , K > such that ≤ h(t) ≤ K t + K , ∀t ∈ [ , +∞) such that F (t, x) ≥ –s(t)h(|x|) – v(t) holds for all x ∈ Rn and a.e. t ∈ [ , T], where p(t)
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