Homoclinic solutions for some second order non-autonomous Hamiltonian systems without the globally superquadratic condition

This paper deals with the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system equation(HS)q̈+Vq(t,q)=f(t), where V∈C1(R×Rn,R)V∈C1(R×Rn,R), V(t,q)=−K(t,q)+W(t,q)V(t,q)=−K(t,q)+W(t,q) is TT-periodic in tt, ff is aperiodic and belongs to L2(R,Rn)L2(R,Rn). Under the assumptions that KK satisfies the “pinching” condition b1|q|2≤K(t,q)≤b2|q|2b1|q|2≤K(t,q)≤b2|q|2, W(t,q)W(t,q) is not globally superquadratic on qq and some additionally reasonable assumptions, we give a new existence result to guarantee that (HS) has a homoclinic solution q(t)q(t) emanating from 00. The homoclinic solution q(t)q(t) is obtained as a limit of 2kT2kT-periodic solutions of a sequence of the second order differential equations and these periodic solutions are obtained by the use of a standard version of the Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.