Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 128, pp. 1-9.

Title: Homoclinic solutions for a class of  second order 
       non-autonomous systems

Authors: Rong Yuan    (Beijing Normal Univ., Beijing, China)
         Ziheng Zhang (Beijing Normal Univ., Beijing, China)

Abstract:
 This article concerns the existence of homoclinic solutions
 for the second order non-autonomous system
 $$
 \ddot q+A \dot q-L(t)q+W_{q}(t,q)=0,
 $$
 where $A$ is a skew-symmetric constant matrix, $L(t)$ is a symmetric
 positive definite matrix depending continuously on $t\in \mathbb{R}$,
 $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$.
 We assume that $W(t,q)$ satisfies the global Ambrosetti-Rabinowitz
 condition, that the norm of $A$ is sufficiently small and
 that $L$ and $W$ satisfy additional hypotheses. We prove the
 existence of at least one nontrivial homoclinic solution, and
 the existence of infinitely  many homoclinic solutions if $W(t,q)$
 is even in $q$. Recent results in the literature are generalized
 and improved.

Submitted October 14, 2008. Published October 07, 2009.
Math Subject Classifications: 34C37, 35A15, 37J45.
Key Words: Homoclinic solutions; critical point; variational methods;
           mountain pass theorem.