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Homoclinic and heteroclinic orbits for a class of Hamiltonian systems. (English) Zbl 0791.34042
The goal of this paper is to establish the existence of a rich structure of homoclinic and heteroclinic solutions for the Hamiltonian system \(\ddot q+V_ q(t,q)=0\), where the potential \(V(t,q)\) satisfies the conditions \((\text{V}_ 1)\;V \in C^ 2(\mathbb{R}^ 1 \times \mathbb{R}^ n, \mathbb{R}^ 1)\) and is \(T\)-periodic in \(t\); \((\text{V}_ 2)\) There is a point \(\xi \in \mathbb{R}^ n \backslash \{0\}\) such that \(\forall t \in \mathbb{R}^ 1\), (i) \(V(t,0)=0=V(t,\xi)\), (ii) \(V_ q (t,0)= 0= V_ q(t,\xi)\), (iii) \(V_{qq} (t,0)\) and \(V_{qq} (t,\xi)\) are negatively definite; \((\text{V}_ 3)\;V(t,q)<0\) for \(q \neq 0\), \(\xi\); \((\text{V}_ 4)\) There is a constant \(V_ 0<0\) such that \(\varlimsup_{| q | \to \infty} V(t,q) \leq V_ 0\). The technique of the proof is based on the calculus of variations and is useful to finding actual solutions of an equation near an approximate solution.

MSC:
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E99 Variational problems in infinite-dimensional spaces
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