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Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential. (English) Zbl 1118.37032
Summary: We are concerned with the existence of homoclinic solutions for the second-order Hamiltonian q ¨-V q (t,q)=f(t), where t and q n . A potential VC 1 (× n ,) is T-periodic in t, coercive in q and the integral of V(·,0) over [0,T] is equal to 0. A function f: n is continuous, bounded, square integrable and f0. We will show that there exists a solution q 0 such that q 0 (t)0 and q ˙ 0 (t)0, as t±. Although q0 is not a solution of our system, we are to call q 0 a homoclinic solution. It is obtained as a limit of 2kT-periodic orbits of a sequence of the second-order differential equations.

MSC:
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
34C37Homoclinic and heteroclinic solutions of ODE
58E05Abstract critical point theory