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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 $$ \dot{u}(t)=\mathcal{J}H_u(t,u), t\in \mathbb{R}, $$ where $u=(y,z)\in \mathbb{R}^{2N}, \mathcal{J}$ is the standard symplectic matrix in $\mathbb{R}^{2N}$, and $H\in C^1(\mathbb{R} \times \mathbb{R}^{2N},\mathbb{R})$ has the form $H(t,u)=\frac{1}{2}Lu \cdot u +W(t,u)$ with $L$ being a $2N\times 2N$ symmetric constant matrix, and $W\in C^1(\mathbb{R} \times \mathbb{R}^{2N},\mathbb{R})$. The main result of the paper shows, by using two recent critical point theorems for strongly indefinite functionals [{\it T. Bartsch} and {\it 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).

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