| Summary: | Abstract By applying the mountain pass theorem in critical point theory, the existence of fast homoclinic solutions is obtained for the following second-order damped vibration system: u¨(t)+q(t)u˙(t)−L(t)u(t)−a(t)|u(t)|p−2u(t)+∇W(t,u(t))=0, $$\ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)-a(t) \bigl\vert u(t) \bigr\vert ^{p-2}u(t)+\nabla W\bigl(t,u(t)\bigr)=0, $$ where p∈(2,+∞) $p\in(2,+\infty)$, t∈R $t\in{ \mathbb {R}}$, u∈RN $u\in{ \mathbb {R}}^{N}$, L(t) $L(t)$ is a positive definite symmetric matrix-valued function for all t∈R $t\in{ \mathbb {R}}$, W∈C1(R×RN,R) $W\in C^{1}({\mathbb {R}}\times{ \mathbb {R}}^{N},{\mathbb {R}})$ is not periodic in t, a(t) $a(t)$ is a continuous, positive function on R ${\mathbb {R}}$ and q:R→R $q:{\mathbb {R}}\rightarrow{ \mathbb {R}}$ is a continuous function and Q(t)=∫0tq(s)ds $Q(t)=\int_{0}^{t}q(s)\,ds$ with lim|t|→+∞Q(t)=+∞ $\lim_{ \vert t \vert \rightarrow+\infty}Q(t)=+\infty$.
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