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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 $$q\in \mathbb R^n$$ and $$V\in C^1(\mathbb R\times\mathbb R^n,\mathbb R)$$, and $$V(t,q)=-K(t,q)+W(t,q)$$ is $$T$$-periodic in $$t$$. A map $$K$$ satisfies the “pinching” condition $$b_1|q|^2\leq K(t,q)\leq b_2|q|^2$$, $$W$$ is superlinear at infinity and $$f$$ is sufficiently small in $$L^2(\mathbb R,\mathbb R^n)$$. A homoclinic orbit is obtained as a limit of $$2kT$$-periodic solutions of a certain sequence of the second-order differential equations.

##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 70H05 Hamilton’s equations
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