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Morse theory and asymptotic linear Hamiltonian system. (English) Zbl 0672.34037
We try to quote from the Introduction in order to give an idea on the paper avoiding technical details. “We consider the existence of $2\pi$- periodic solutions of Hamiltonian systems $-J\dot z=H'(z,t)$ $(*)\quad \vert H'(s,t)-h\sb{\infty}s\vert /\vert s\vert \to 0,$ as $\vert s\vert \to \infty$ $\vert H'(s,t)-h\sb 0s\vert /\vert s\vert \to 0,$ as $\vert s\vert \to 0$ $(h\sb{\infty},h\sb 0$ are constant symmetric matrices and $H'$ denotes the gradient of H with respect to the first 2n variables). Given a constant 2n$\times 2n$ matrix h we define an index $i\sp-(h)$. measuring the difference between the “sizes” of the negative spaces of the operators -Jd/dt-h and -Jd/dt. We define $i\sp 0(h)$ which is just the dimension of the null space of -Jd/dt-h. Let $H: R\sp{2n}\times R\to R$ be $C\sp 1,2\pi$-periodic in t and satisfying (*). If $i\sp 0(h\sb 0)=i\sp 0(h\sb{\infty})=0$ and $i\sp-(h\sb 0)\ne i\sp-(h\sb{\infty})$ then there exists at least one nontrivial periodic solution”. Morse theory and Galerkin method are combined.
Reviewer: A.Halanay

34C25Periodic solutions of ODE
70H05Hamilton’s equations
Full Text: DOI
[1] Amann, H.; Zehender, E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. scuola norm. Sup. Pisa cl. Sci 8, 539-603 (1980) · Zbl 0452.47077
[2] Amann, H.; Zehender, E.: Periodic solutions of an asymptotical linear Hamiltonian system. Manuscripta math. 32, 149-189 (1980)
[3] Benci, V.: The direct method in the study of periodic solutions of Hamiltonian systems with prescribed period. Advances in Hamiltonian systems (1983) · Zbl 0542.70019
[4] Benci, V.; Rabinowitz, P. H.: Critical point theorems for indefinite functional. Invent. math. 52, 336-352 (1979) · Zbl 0465.49006
[5] Castro, A.; Lazer, A. C.: Critical point theory and the number of solutions of a nonlinear Dirichlet problem. Ann. mat. Pura appl. 120, 113-137 (1979) · Zbl 0426.35038
[6] Chang, K. C.: Solutions of asymptotically linear operator equations via Morse theory. Comm. pure appl. Math. 34, 693-712 (1981) · Zbl 0444.58008
[7] A. C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, to appear. · Zbl 0619.58011
[8] Shujie Li and J. Q. Liu, ”Nontrivial critical Points for Asymptotically Quadratic Function,” IC/86/390. · Zbl 0767.35025
[9] Shujie Li and J. Q. Liu, ”Infinite Dimensional Local Linking and Some Applications to the Hamiltonian Systems,” ISAS-International School for Advanced Studies, 39/87/M. · Zbl 0697.58020
[10] Rabinowitz, P. H.: Periodic solutions of Hamiltonian systems. Comm. pure appl. Math. 31, 157-184 (1978) · Zbl 0358.70014