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Multiple periodic solutions of asymptotically linear Hamiltonian systems. (English) Zbl 0851.34047

The authors consider the autonomous Hamiltonian system \((*)\) \(z'= JH_z(z)\), where \(H: \mathbb{R}^{2n}\to \mathbb{R}\) is a function of class \(C^1\), \(J= (\begin{smallmatrix} O\\ \text{Id}\end{smallmatrix} \begin{smallmatrix} -\text{Id}\\ O\end{smallmatrix})\) and \(H_z\) denotes the gradient of \(H\).
The authors find a multiplicity result, namely, with appropriate conditions on \(H\), \(H_z\), \(H_{zz}(0)\) and \(H_{zz}(\infty)\) there exist at least \[ \frac12 v\left((T/2\pi) H_{zz}(\infty), (T/2\pi) H_{zz}(0)\right) \] nonconstant \(T\)-periodic solutions of \((*)\) (Theorem 1.2) and respectively there exist at least \({1\over 2} v((T/2\pi) H_{zz}(0), (T/2\pi) H_{zz}(\infty))\) nonconstant \(T\)-periodic solutions of \((*)\) (Theorem 1.8), where \(H_{zz}(0)\) and \(H_{zz}(\infty)\) denote two symmetric linear operators and for \(B, C: \mathbb{R}^{2n}\to \mathbb{R}^{2n}\), \(v(B,C)\) is defined by \[ v(B, C):= \sum^{+ \infty}_{k= -\infty} (\underline m(ikJ+ B)- \overline m(ikJ+ C)), \] \(\underline m(A)\) being the number of strictly negative eigenvalues of an operator \(A\) and \(\overline m(A)\) the number of nonpositive eigenvalues of \(A\).
Also an abstract critical point theorem (Theorem 1.14) is shown for which the authors give an application to semilinear hyperbolic equations in a previous paper.

MSC:

34C25 Periodic solutions to ordinary differential equations
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