By Antonio Ambrosetti (auth.), J. P. Aubin C.E.R.E.M.A.D.E., A. Bensoussan C.E.R.E.M.A.D.E., I. Ekeland C.E.R.E.M.A.D.E. (eds.)

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**Extra resources for Advances in Hamiltonian Systems**

**Example text**

This happens if every h E «(:4. B) } H has the form identity + compact. In any case the construction of the homeomorphism n as in Def. 3). are delicate points (cf. 6) and 1. [B2 1. [BCFI 1 and [FHR j). 1. the following result has been In a recent work. [BCFl obtained. 7 Theorem. sentation Tw Let Hbe a real Hilbert space. on which a unitary repre- of the group Coo acts. ) is the linear product in E • L is bounded selfadjoint. operator and ~ E CI(E,R) • ~(O) = 0 • is a functional whose Frechet derivative is compact.

The norm ~nd A Variational Approach. the euclidean scalar product, respectively, in~2N. In [9 I Rabinowitz 'has proved the following. (Rabinowitz [9 I). 1 Suppose H E CI (R2N,R) satisfies (I. 2. As pointed out by Rabinowitz (see for example [16 J), T , (H) has a T-periodic solution z satisfying 1Iz11 >~. 1. 1), an upper bound, say To ' for the minimal period of ~ solution of (H) could exist. This occurs, for example, if H(~) = ~(1~12) , ~ E COO , ~'(z) V z >0 and ~'(O) = I , ~'(+oo) = 00 • >0 , The energy surfaces are spheres and the solutions of (H) (at the energy level h :: H(z(t») are ~(hl/2) (~ cos y(h)t + n sin y(h)t , ~ sin y(h)t + - n cos y (h) t) where corresponding periods, Inl2 = I , ~ = ~-I , y(h) = 2/~'(h) • The TI ~'(h) , are bounded from above.

BCFI ] V. BENCI - A. CAPOZZI - D. FORTUNADO, Periodic solutions of Hamiltonian systems with a prescribed period, Preprint. [ BCF2 ] V. BENCI - A. CAPOZZI - D. FORTUNADO, Periodic solutions for a class of Hamiltonian systems, to appear. [ BFI] V. BENCI - D. FORTUNADO, Un teorema di molteplicita per un'equazione ellittica non lineare su varieta simmetriche, Proceedings of the Symposium "Metodi 42 asintotici e topologici in problem diff. non lineari", L 'Aquila (1981). [ BF2 1 V. BENCI - D. FORTUNADO, The dual method in critical point theory.