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

Zhou, Jianwen; Li, Yongkun

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3-4, 1594-1603 (2010).

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.

Reviewer: Irina V. Konopleva (Ul’yanovsk)

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

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