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On the existence of homoclinic solutions for almost periodic second order systems. (English) Zbl 0873.58032
The existence of at least one homoclinic solution for a second order Lagrangian system is proved. The potential is an almost periodic function of time. This result generalizes some known existence theorems when the potential is periodic of time. One uses variational methods to establish this result. Solutions are found as critical points of a suitable functional. The absence of a group of symmetries for which the functional is invariant (as in the case of periodic potentials) is replaced by the study of problems “at infinity” and a suitable use of a property introduced by V. Coti Zelati, I. Ekeland and E. Séré [Math. Ann. 288, No. 1, 133-160 (1990; Zbl 0731.34050)] and E. Séré [Math. Z. 209, No. 1, 27-42 (1992; Zbl 0739.58023)].

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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References:
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