Homoclinic solutions of some second order non-autonomous systems

In this paper we investigate the existence of homoclinic solutions for the following second order non-autonomous system q̈+Aq̇−L(t)q+Wq(t,q)=0,(DS) where AA is an antisymmetric constant matrix, L∈C(R,Rn2)L∈C(R,Rn2) is a symmetric and positive definite matrix for all t∈Rt∈R, W(t,q)=a(t)V(q)W(t,q)=a(t)V(q) such that a:R→Ra:R→R is a continuous function and V∈C1(Rn,R)V∈C1(Rn,R). Assuming that V(q)V(q) is subquadratic as ∣q∣→+∞∣q∣→+∞ and some technical assumptions on AA and LL, we establish two existence criteria to guarantee that (DS) has at least one nontrivial homoclinic solution by using a standard minimizing argument. Besides that, in some particular case, for the first time the uniqueness of homoclinic solutions of (DS) is also obtained. Recent results in the literature are generalized and significantly improved.