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Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects. (English) Zbl 1193.34057

The authors give sufficient conditions the existence of a solution to the following boundary value problem

u ¨(t)=F(t,u(t))a.e.t[0,T];u(0)-u(T)=u ˙(0)-u ˙(T)=0,u ˙ j (t j )=u ˙ j (t j + )-u ˙ j (t j - )=I ij (u i (t j )),i=1,2,,N;j=1,2,,p·

Here, t 0 =0<t 1 <t 2 <<t p <t p+1 =T,u(t)=(u 1 (t),u 2 (t),,u N (t)),I ij : (i=1,2,,N, j=1,2,,p) are continuous and F:[0,T]× N satisfies the following assumption:

(A) F(t,x) is measurable in t for every x N and continuously differentiable in x for a.e. t[0,T] and there exist aC( + , + ),bL 1 (0,T; + ) such that


for all x N and a.e. t[0,T].

Two illustrative examples are given.

34B37Boundary value problems for ODE with impulses
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems