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Homoclinic solutions for a class of the second order Hamiltonian systems. (English) Zbl 1080.37067
Summary: We study the existence of homoclinic orbits for the second-order Hamiltonian system $\ddot q+ V_q(t,q)= f(t)$, where $q\in \Bbb R^n$ and $V\in C^1(\Bbb R\times\Bbb R^n,\Bbb R)$, and $V(t,q)=-K(t,q)+W(t,q)$ is $T$-periodic in $t$. A map $K$ satisfies the “pinching” condition $b_1|q|^2\le K(t,q)\le b_2|q|^2$, $W$ is superlinear at infinity and $f$ is sufficiently small in $L^2(\Bbb R,\Bbb R^n)$. A homoclinic orbit is obtained as a limit of $2kT$-periodic solutions of a certain sequence of the second-order differential equations.

37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
58E05Abstract critical point theory
34C37Homoclinic and heteroclinic solutions of ODE
70H05Hamilton’s equations
Full Text: DOI
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