P-symmetric Subharmonic Solutions for Nonlinear Hamiltonian Systems

In this paper, we prove that for each positive k ≡ 1 mod m there exists a P-symmetric kmτ-periodic solution x_k for asymptotically linear mτ-periodic Hamiltonian systems, which are nonautonomous and endowed with a P-symmetry. If the P-symmetric Hamiltonian function is semi-positive, one can prove, under a new iteration inequality of the Maslov-type P-index, that \({x_{{k_1}}}\) and \({x_{{k_2}}}\) are geometrically distinct for k₁/k₂ ≥ (2n + 1)m + 1; and \({x_{{k_1}}}\), \({x_{{k_2}}}\) are geometrically distinct for k₁/k₂ ≥ m + 1 provided \({x_{{k_1}}}\) is non-degenerate.