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 q + V-q(t, q) = f (t), (HS) where V is an element of C-1(R x R-n, R), V(t, q) = -K(t, q) + W(t, q) is T-periodic in t, f is aperiodic and belongs to L-2(R, R-n). Under the assumptions that K satisfies the "pinching" condition b(1)vertical bar q vertical bar(2) <= K(t, q) <= b(2)vertical bar q vertical bar(2), W(t, q) is not globally superquadratic on q and some additionally reasonable assumptions, we give a new existence result to guarantee that (HS) has a homoclinic solution q(t) emanating from 0. The homoclinic solution q(t) is obtained as a limit of 2kT-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. (C) 2009 Elsevier Ltd. All rights reserved.