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Homoclinic orbits for a class of nonperiodic Hamiltonian systems with some twisted conditions. (English) Zbl 1295.37018

The authors consider the following first-order Hamiltonian system \[ \dot{z}=JH_{z}(t,z).\tag{1} \] In [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 24, No. 4, 589–603 (2007; Zbl 1202.37081)] C.-N. Chen and X. Hu established the Maslov index for homoclinic orbits of Hamiltonian systems. For system (1) with a kind of twist condition, Z. Liu et al. [Adv. Math. 218, No. 6, 1895–1913 (2008; Zbl 1144.37027)] obtained the existence and multiplicity of periodic solutions. Motivated by these two papers and previous works, in this paper, the authors make use of the Maslov index theory for homoclinic orbits of Hamiltonian systems to obtain the existence and multiplicity for the system (1) without periodic condition. The results in this paper are different from those in [Y. Ding and S. Li, J. Math. Anal. Appl. 189, No. 2, 585–601 (1995; Zbl 0818.34023); Y. Ding and C. Lee, J. Differ. Equations 246, No. 7, 2829–2848 (2009; Zbl 1162.70014); J. Sun et al., J. Math. Anal. Appl. 378, No. 1, 117–127 (2011; Zbl 1218.37081)] and an example is given to show this difference.

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
53D12 Lagrangian submanifolds; Maslov index
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References:

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