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The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. (English) Zbl 1198.34036
This paper is concerned with the study of multiplicity of solutions for perturbed impulsive Hamiltonian boundary value problems of the form
\[ \begin{cases}-\ddot{u}+A(t)u=\lambda \nabla F(t,u)+\mu \nabla G(t,u), \quad &\text{a.e.}\quad t\in [0,T]\\ \Delta(\dot{u}^i(t_j))=\dot{u}^i(t_j^+)-\dot{u}^i(t_j^-)=I_{ij}(u^i(t_j)), & i=1,2,\dots, N, \;j=1,2,\dots, l,\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,\end{cases} \] where \(A: [0,T]\to {\mathbb R}^{N\times N}\) is a continuous map from the interval \([0,T]\) to the set of \(N\)-order symmetric matrices, \(\lambda, \mu \in {\mathbb R}\), \(T\) is a real positive number, \(u(t) = (u^1(t), u^2(t),\dots , u^N (t))\), \(t_j, j = 1, 2, \dots , l\), are the instants where the impulses occur and \(0 = t_0 < t_1 < t_2 <\dots < t_l < t_{l+1} = T\), \(I_{ij} : {\mathbb R}\to {\mathbb R}\) \((i = 1,2\dots ,N,\) \(j = 1,2,\dots,l\)) are continuous and \(F, G:[0,T]\times {\mathbb R}^N\to {\mathbb R}\) are measurable with respect to \(t,\) for every \(u\in {\mathbb R}^N\), continuously differentiable in \(u,\) for almost every \(t\in [0, T ]\) and satisfy the following standard summability condition:
\[ \sup_{ |u|\leq b} (\max{|F (\cdot, u)|, |G(\cdot, u)|, |\nabla F (\cdot, u)|, |\nabla G(\cdot, u)|})\in L^1 ([0, T ]) \] for all \(b > 0\). A variational method and some critical points theorems are used. Examples illustrating the main results are also presented.

MSC:
34B37 Boundary value problems with impulses for ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E30 Variational principles in infinite-dimensional spaces
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