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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 $$\cases-\ddot{u}+A(t)u=\lambda \nabla F(t,u)+\mu \nabla G(t,u), \quad &\text{a.e.}\quad t\in [0,T]\\ \Delta(\dot{u}^i(t_j))=\dot{u}^i(t_j^+)-\dot{u}^i(t_j^-)=I_{ij}(u^i(t_j)), & i=1,2,\dots, N, \ j=1,2,\dots, l,\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,\endcases$$ where $A: [0,T]\to {\Bbb R}^{N\times N}$ is a continuous map from the interval $[0,T]$ to the set of $N$-order symmetric matrices, $\lambda, \mu \in {\Bbb R}$, $T$ 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 $0 = t_0 < t_1 < t_2 <\dots < t_l < t_{l+1} = T$, $I_{ij} : {\Bbb R}\to {\Bbb R}$ $(i = 1,2\dots ,N,$ $j = 1,2,\dots,l$) are continuous and $F, G:[0,T]\times {\Bbb R}^N\to {\Bbb R}$ are measurable with respect to $t,$ for every $u\in {\Bbb R}^N$, continuously differentiable in $u,$ for almost every $t\in [0, T ]$ and satisfy the following standard summability condition: $$\sup_{ |u|\le b} (\max{|F (\cdot, u)|, |G(\cdot, u)|, |\nabla F (\cdot, u)|, |\nabla G(\cdot, u)|})\in L^1 ([0, T ])$$ for all $b > 0$. A variational method and some critical points theorems are used. Examples illustrating the main results are also presented.

34B37Boundary value problems for ODE with impulses
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
58E05Abstract critical point theory
58E30Variational principles on infinite-dimensional spaces
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