Fast homoclinic solutions for some second order non-autonomous systems

This paper is concerned with the existence of homoclinic solutions for the following second order non-autonomous systemequation(FHS)q¨+cq˙+Uq(t,q)=0, where q∈Rnq∈Rn, c⩾0c⩾0 is a constant and U(t,q)∈C1(R×Rn,R)U(t,q)∈C1(R×Rn,R). Under some reasonable assumptions, for the first time we introduce the concept of fast homoclinic solutions of (FHS) and establish some criteria to guarantee the existence of fast homoclinic solutions of (FHS). Besides that, in some particular case, the uniqueness of homoclinic solutions of (FHS) is also obtained. Recent results in the literature are generalized and significantly improved.