Existence of Homoclinic Solutions for a Class of Damped Vibration Problems

Abstract

In this paper, we consider the existence of homoclinic solution for a class of damped vibration problem

$$\begin{aligned} \ddot{x}(t)+(q(t)I_{N\times N}+B){\dot{x}}(t)+\left( \frac{1}{2} q(t)B-A(t)\right) x(t)+H_{x}(t,x(t))=f(t). \end{aligned}$$

For every $k\in {\mathbb {N}}$, we obtain the 2kT-periodic solution $x_{k}$ by a standard minimizing argument. By taking the limit of $\{x_{k}\}$, we get a solution $x_{0}$ of this problem. We prove that $x_{0}$ satisfies $x_{0}\rightarrow 0$ and ${\dot{x}}_{0}\rightarrow 0$ as $t\rightarrow \pm \infty$, and therefore $x_{0}$ is a homoclinic solution of the problem.