Abstract
In this paper we define the index at infinity of an asymptotically linear autonomous Hamiltonian system. We use this index to prove the existence and bifurcation from infinity of periodic solutions of the system. We apply the degree for G-invariant strongly indefinite functionals defined by Gołȩbiewska and Rybicki in (Nonlinear Anal 74:1823–1834, 2011).
Highlights
IntroductionConsider the problem of existence of periodic solutions of the system x = J H (x),
Consider the problem of existence of periodic solutions of the system x = J H (x), (1.1)where H ∈ C2(R2N, R) is such that H is asymptotically linear at infinity, i.e. H (x) = H (∞)x + o(|x|) for |x| → ∞, where H (∞) is a symmetric matrix.One of the ideas of studying such a system is to consider an associated functional defined on an appropriate Hilbert space
Throughout this section we study the existence of periodic solutions of autonomous Hamiltonian systems of the form: x = J H (x), (3.1)
Summary
Consider the problem of existence of periodic solutions of the system x = J H (x),. One of the ideas of studying such a system is to consider an associated functional defined on an appropriate Hilbert space. Using this functional one can define an index of the stationary solution and of the infinity. Comparing these indices we can prove the existence of solutions. Such an idea has been used by many authors, see for example [1,10,15,16,21]. Supported by the National Science Center, Poland; under Grant DEC-2012/05/B/ST1/02165
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