Minimal P-symmetric period problem of first-order autonomous Hamiltonian systems

Let P ∈ Sp(2n) satisfying P ^k = I _2n . We consider the minimal P-symmetric period problem of the autonomous nonlinear Hamiltonian system (dot xleft( t right) = JH'left( {xleft( t right)} right)). For some symplectic matrices P, we show that for any τ > 0, the above Hamiltonian system possesses a kτ periodic solution x with kτ being its period provided H satis Fies Rabinowitz's conditions on the minimal minimal P-symmetric period conjecture, together with that H is convex and H(Px) = H(x).