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New existence and multiplicity of homoclinic solutions for second order non-autonomous systems

In this paper, we study the second order non-autonomous system ¨ u(t) + A ˙ u(t) L(t)u(t) +rW(t, u(t)) = 0, 8t2 R, where A is an antisymmetric N N constant matrix, L 2 C(R,R N N ) may not be uniformly positive definite for all t 2 R, and W(t, u) is allowed to be sign-changing and local superquadratic. Under some simple assumptions on A, L and W, we establish some existence criteria to guarantee that the above system has at least one homoclinic solution or infinitely many homoclinic solutions by using the mountain pass theorem or the fountain theorem, respectively. Recent results in the literature are generalized