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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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