Existence and bifurcation of homoclinic solutions for Hamiltonian systems (Q2722484)

The author deals with a Hamiltonian system of the following form: NEWLINE\[NEWLINEJu'(x)+ Mu(x)-\nabla_s F\bigl(x,u(x)\bigr) =\lambda u(x),\tag{1}NEWLINE\]NEWLINE where \(J\) is a \((2N\times 2N)\) real matrix such that \(J=-J^T=-J^{-1}\) and \(H:\mathbb{R}\times \mathbb{R}\times \mathbb{R}^{2N} \to\mathbb{R}\) NEWLINE\[NEWLINEH(\lambda,x,s)= {1\over 2}(M-\lambda I)s, s-F (x,s)NEWLINE\]NEWLINE where \(M\) is a \((2N\times 2N)\) real symmetric matrix such that \(\sigma (JM)\cap i\mathbb{R}=\emptyset\) and \(F:\mathbb{R}\times \mathbb{R}^{2N} \to\mathbb{R}\) is such that \(\lim_{|s|\to 0}{F(x,s)\over|s|^2}=0\).NEWLINENEWLINENEWLINEThe author's aim is to obtain existence and bifurcation results for homoclinic solutions for a subset of such systems by imposing conditions on \(F\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00026].