Infinitely Many Fast Homoclinic Solutions for Damped Vibration Systems with Combined Nonlinearities

This article  concerns the existence of fast homoclinic solutions for the following damped vibration system\begin{equation*}\frac{d}{dt}(P(t)\dot{u}(t))+q(t)P(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\end{equation*}where $P,L\in C\left(\mathbb{R},\mathbb{R}^{N^{2}}\right)$ are symmetric and positive definite matrices, $q\in C\left(\mathbb{R},\mathbb{R}\right)$ and $W\in C^{1}\left(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R}\right)$. Applying the Fountain Theorem and Dual Fountain Theorem, we prove the above system possesses two different sequences of fast homoclinic solutions when $L$ satisfies a new coercive condition and the potential $W(t,x)$ is combined nonlinearity.