## 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
$\begin{cases} \ddot{u}(t)=\nabla F(t,u(t))\quad &\text{a.e. }t\in [0,T];\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,\\ \triangle \dot{u}^j(t_j)=\dot{u}^j(t_j^+)-\dot{u}^j(t_j^-)=I_{ij}(u^i(t_j)), & i=1,2,\dots,N;\quad j=1,2,\dots,p. \end{cases}$
Here, $$t_0=0<t_1<t_2<\cdots<t_p<t_{p+1}=T, u(t)=(u^1(t),u^2(t),\dots,u^N(t)), I_{ij}:\mathbb{R}\to \mathbb{R}$$ $$(i=1,2,\dots,N$$, $$j=1,2,\dots,p)$$ are continuous and $$F:[0,T]\times \mathbb{R}^N \to \mathbb{R}$$ satisfies the following assumption:
(A) $$F(t,x)$$ is measurable in $$t$$ for every $$x\in \mathbb{R}^N$$ and continuously differentiable in $$x$$ for a.e. $$t \in [0,T]$$ and there exist $$a\in C(\mathbb{R}^+,\mathbb{R}^+), b\in L^1(0,T;\mathbb{R}^+)$$ such that
$|F(t,x)|\leq a(|x|)b(x),\quad |\nabla F(t,x)|\leq a(|x|)b(x)$
for all $$x\in \mathbb{R}^N$$ and a.e. $$t\in [0,T]$$.
Two illustrative examples are given.

### MSC:

 34B37 Boundary value problems with impulses for ordinary differential equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

### Keywords:

Hamiltonian systems; impulse; critical points
Full Text:

### References:

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