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Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. (English) Zbl 0919.34046
The paper deals with the existence of homoclinic orbits for second-order time-dependent Hamiltonian systems of the type \[ \ddot{q} - L(t) q + W_q(t,q) = 0, \qquad q \in \mathbb{R} ^n. \] In nontechnical terms, \(W\) is assumed to be superquadratic at infinity and subquadratic at the origin. Also, \(L(t)\) tends to a multiple of the identity matrix as \(| t| \to \infty\). Under such conditions, the existence of a homoclinic orbit is proven. The authors apply variational methods, such as the mountain pass theorem. The result generalizes previous results, such as work of P. H. Rabinowitz and K. Tanaka [Math. Z. 206, No. 3, 473-499 (1991; Zbl 0707.58022)]. Other generalizations can be found in work of S. Wu and O. H. Miyagaki.

MSC:
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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