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Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero. (English) Zbl 1218.37081

The authors consider the following first-order Hamiltonian system

u ˙(t)=𝒥H u (t,u),t,

where u=(y,z) 2N ,𝒥 is the standard symplectic matrix in 2N , and HC 1 (× 2N ,) has the form H(t,u)=1 2Lu·u+W(t,u) with L being a 2N×2N symmetric constant matrix, and WC 1 (× 2N ,). The main result of the paper shows, by using two recent critical point theorems for strongly indefinite functionals [T. Bartsch and Y. Ding, Math. Nachr. 279, No. 12, 1267–1288 (2006; Zbl 1117.58007)], that if the technical working assumptions (L 1 ), (H 1 )-(H 5 ) hold, then the considered Hamiltonian system has at least one homoclinic orbit (Theorem 1.1).

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