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.


34C25 Periodic solutions to ordinary differential equations
Full Text: DOI


[1] Ekeland, I., (Convexity Methods in Hamiltonian Mechanics (1990), Springer: Springer New York) · Zbl 0707.70003
[2] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, (CBMS Regional Conference Series in Mathematics, Vol. 65 (1986), American Mathematical Society: American Mathematical Society Berlin) · Zbl 0152.10003
[3] Amann, H.; Zehnder, E., Periodic solutions of asymptotically linear Hamiltonian systems, Manuscripta Math., 32, 149-189 (1980) · Zbl 0443.70019
[4] Chang, K. C., Solutions of asymptotically linear operator equations via Morse theory, Communs pure appl. Math., 34, 693-712 (1981) · Zbl 0444.58008
[5] Conley, C.; Zehnder, E., Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Communs pure appl. math., 37, 207-253 (1984) · Zbl 0559.58019
[6] Shujie, Li; Liu, J. Q., Morse theory and asymptotic linear Hamiltonian system, J. diff. Eqns, 78, 53-73 (1989) · Zbl 0672.34037
[7] Long, Y. M., Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems, Sci. China Ser. A, 33, 1409-1419 (1990) · Zbl 0736.58022
[8] Long, Y. M.; Zehnder, E., Morse-theory for forced oscillations of asymptotically linear Hamiltonian systems, (Stochastic Processes, Physics and Geometry (1990), World Sci. Publishing: World Sci. Publishing Providence, RI), 528-563
[9] Benci, V., On critical point theory for indefinite functionals in the presence of symmetries, Trans. Am. math. Soc., 274, 533-572 (1982) · Zbl 0504.58014
[10] Benci, V.; Capozzi, A.; Fortunato, D., On asymptotically quadratic Hamiltonian systems, (Equadiff · Zbl 0525.70021
[11] Costa, D.; Willem, M., Lusternik-Schnirelman theory and asymptotically linear Hamiltonian systems, (Differential Equations: Qualitative Theory (1987), North-Holland: North-Holland Berlin), 179-191 · Zbl 0622.34044
[12] Benci, V., A geometrical index for the group \(S^1\) and some applications to the study of periodic solutions of ordinary differential equations, Communs pure appl. Math., 34, 393-432 (1981) · Zbl 0447.34040
[13] Benci, V.; Capozzi, A.; Fortunato, D., Periodic solutions of Hamiltonian systems with superquadratic potential, Annali mat. pura appl., 143, 1-46 (1986) · Zbl 0632.34036
[14] Fadell, E.; Husseini, S., Relative cohomological index theories, Adv. math., 64, 1-31 (1987) · Zbl 0619.58012
[15] Degiovanni, M.; Olian Fannio, L., Multiple periodic solutions of autonomous semilinear wave equations, Topol. Meth. Nonlinear Analysis, 4, 427-428 (1994) · Zbl 0836.35094
[16] Fadell, E.; Rabinowitz, P. H., Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45, 139-174 (1978) · Zbl 0403.57001
[17] Fadell, E., Lectures in cohomological index theories of \(G\)-spaces with applications to critical point theory, Rc. Semin. Dip. mat. Univ. Calabria, 6 (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.