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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

-u ¨+A(t)u=λF(t,u)+μG(t,u),a.e.t[0,T]Δ(u ˙ i (t j ))=u ˙ i (t j + )-u ˙ i (t j - )=I ij (u i (t j )),i=1,2,,N,j=1,2,,l,u(0)-u(T)=u ˙(0)-u ˙(T)=0,

where A:[0,T] N×N is a continuous map from the interval [0,T] to the set of N-order symmetric matrices, λ,μ, T is a real positive number, u(t)=(u 1 (t),u 2 (t),,u N (t)), t j ,j=1,2,,l, are the instants where the impulses occur and 0=t 0 <t 1 <t 2 <<t l <t l+1 =T, I ij : (i=1,2,N, j=1,2,,l) are continuous and F,G:[0,T]× N are measurable with respect to t, for every u N , continuously differentiable in u, for almost every t[0,T] and satisfy the following standard summability condition:

sup |u|b (max|F(·,u)|,|G(·,u)|,|F(·,u)|,|G(·,u)|)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:
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