In this paper, the Conley conjecture, which were recently proved by Franks and Handel [FrHa] (for surfaces of positive genus), Hingston [Hi] (for tori) and Ginzburg [Gi] (for closed symplectically aspherical manifolds), is proved for C1-Hamiltonian systems on the cotangent bundle of a C3-smooth compact manifoldM without boundary, of a time 1-periodic C2-smooth Hamiltonian H : R × T ∗M → R which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If H also satisfies H(−t, q,−p) = H(t, q, p) for any (t, q, p) ∈ R × T ∗M , it is shown that the time-one map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of T ∗M . If M is C5-smooth and dimM > 1, H is of C4 class and independent of time t, then for any τ > 0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of τ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of H, L : R×TM → R, which is proved to be strongly convex and to have quadratic growth in the velocities yet. ∗Partially supported by the NNSF 10671017 of China and the Program for New Century Excellent Talents of the Education Ministry of China.
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