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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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