Periodic solutions of asymptotically linear systems without symmetry
The author studies the existence of t-periodic solutions of the nonautonomous Hamiltonian system \(-J\dot z=H_ z(t,z)\) where \(H\in C^ 1({\mathbb{R}}^{2n+1},{\mathbb{R}})\) is T-periodic in t and asymptotically quadratic, i.e. (*) \(H_ z(t,z)=b_{\infty}(t)z+g(t,z)\) where \(b_{\infty}(t)\) is a symmetric matrix 2n\(\times 2n\) and g(t,z)/\(| z| \to 0\) as \(| z| \to \infty\). Let \(L_{\infty}\) be the ''linearized operator at infinity'', i.e. \(L_{\infty}z=-J\dot z- b_{\infty}(t)z\), and \(\sigma (L_{\infty})\) denote the spectrum of \(L_{\infty}\). The author considers the nonresonant case \((0\not\in \sigma (L_{\infty}))\) and the resonant case \((0\in \sigma (L_{\infty}))\). In the second case strong resonance is assumed, i.e. G(t,z)\(\to 0\) as \(| z| \to \infty\) where \(G(t,z)=H(t,z)- (b_{\infty}(t)z| z)_{{\mathbb{R}}^{2n}}\). The existence of one T- periodic solution is proved by using min-max arguments. Moreover, if 0 is an equilibrium point of the Hamiltonian vector field, the existence of a non trivial T-periodic solution of (*) is proved under additional assumptions. An analogous result in the nonresonant case has been proved \textit{H.Amann} and \textit{E. Zehnder} [Manuscr. Math. 32, 149-189 (1980; Zbl 0443.70019)].
scientific article; zbMATH DE number 3951033
scientific article