DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 1, Number 2, April 1995 pp. 277–283 PERIODIC SOLUTIONS TO UNBOUNDED HAMILTONIAN SYSTEM V. Barbu* University of Ia¸si, 6600 Ia¸si, Romania Abstract. The existence of periodic solutions for unbounded Hamiltonian systems in Hilbert spaces is studied. 1. Introduction. This work concerns the Hamiltonian system ‰ y 0 + Ay ∈ ∂ p H(t, y, p), t ∈ (0,T ) p 0 - A * p ∈-∂ y H(t, y, p), t ∈ (0,T ) (1.1) with periodic conditions y(0) = y(T ), p(0) = p(T ) (1.2) in a real Hilbert space X with the scalar product h·, ·i and the norm denoted |·|. Here H : [0,T ] × X × X -→ R is assumed to satisfy the following hypothesis: (H 1 ) H = H(t, x, y) is continuous and convex in (x, y) ∈ X × X, measurable in t and H(t, x, y) ≤ ω(|x| 2 + |y| 2 )+ C, ∀ x, y ∈ X, a.e. t ∈ (),T ) (1.3) for some ω> 0 and C ∈ R. Moreover, there are α, β ∈ X and γ ∈ L ∞ (0,T ) such that H(t, x, y) ≥hα, xi + hβ,yi + γ (t), ∀x, y ∈ X, a.e. t ∈ [0,T ] (1.4) In (1.1), ∂H(t, ·, ·)=(∂ y H(t, ·, ·),∂ p H(t, ·, ·)) stands for the subdifferential of func- tion H(t, ·, ·) (see e.g. [1],[2]). As regards the linear operator A : D(A) ⊂ X -→ X we shall assume that -A is the infinitesimal generator of a C 0 –semigroup on X and A * is the adjoint of A. As a matter of fact, the system (1.1),(1.2) will be studied here in two special cases (parabolic and hyperbolic) to be treated separately in sections 2 and 3 below. Some related results have been obtained by a different method in [3]. 2. The parabolic nonresonant case. We shall assume here that A satisfies the following assumption: (H 2 ) A is self–adjoint, hAy, yi≥-μ|y| 2 , ∀ y ∈ D(A) (2.1) for some μ ≥ 0 and (λI + A) -1 is compact for λ>μ. *This work was done while visiting Ohio University. 277