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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.

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
Full Text: DOI
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