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Zbl 1193.34057
Zhou, Jianwen; Li, Yongkun
Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3-4, A, 1594-1603 (2010). ISSN 0362-546X

The authors give sufficient conditions the existence of a solution to the following boundary value problem $$\cases \ddot{u}(t)=\nabla F(t,u(t))\quad &\text {a.e. }t\in [0,T];\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,\\ \triangle \dot{u}^j(t_j)=\dot{u}^j(t_j^+)-\dot{u}^j(t_j^-)=I_{ij}(u^i(t_j)), & i=1,2,\dots,N;\quad j=1,2,\dots,p. \endcases$$ 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.
[Irina V. Konopleva (Ul'yanovsk)]

MSC 2000:: *34B37 Boundary value problems with impulses
37J99 Finite-dimensional Hamiltonian etc. systems

Keywords: Hamiltonian systems; impulse; critical points

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