#### Mohsen Timoumi, Infinitely many fast homoclinic solutions for damped vibration systems with locally defined potentials, Vol. 2018 (2018), Article ID 20, pp. 1-12

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DOI: 10.23952/cot.2018.20

Received June 1, 2018; Accepted October 5, 2018; Published November 2, 2018

Abstract. In this paper, we study the existence of infinitely many fast homoclinic solutions for a class of damped vibration system $\ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,$ $t\in\mathbb{R}$, where $q\in C(\mathbb{R},\mathbb{R})$, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is unnecessary coercive nor positive definite for all $t\in\mathbb{R}$ and $W(t,x)$ is only locally defined near the origin with respect to the second variable.