Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems. (Q1421169)

Some results are obtained for the existence and subharmonic solutions to discrete Hamiltonian systems \[ \begin{aligned}\Delta x_1(n) &= -H_{x_2}(n, x_1(n+1), x_2(n)),\\ \Delta x_2(u) &= H_{x_1}(n, x_1(n+1), x_2(n))\end{aligned}\tag{1} \] by using critical point theory, where \(x_1,x_2\in \mathbb{R}^d\), \(H\in C^1(\mathbb{R}\times \mathbb{R}^d\times \mathbb{R}^d,\mathbb{R})\), \(\Delta x_i(n)= x_i(n+ 1)- x_i(n)\), \(i= 1,2\). Denote \(z= (x^T_1, x^T_2)^T\), \(H(t, z)= H(t, x_1,x_2)\) and assume the following conditions \((\text{H}_1)\) \(H(t, z)\in C^1(\mathbb{R}\times \mathbb{R}^{2d},\mathbb{R})\) and there exists a positive integer \(m\) such that \(\forall(x,z)\in \mathbb{R}\times \mathbb{R}^{2d}\), \(H(t+ m,z)= H(t,z)\), \((\text{H}_2)\) For any \((t, z)\in \mathbb{R}\times\mathbb{R}^{2d}\), \(H(t,z)\geq 0\) and \(H(t,z)= o(| z|^{2d})\) (as \(z\to 0\)); \((\text{H}_3)\) There exist some constants \(R> 0\), \(\beta> 2\) such that for any \(| z|\geq R\), \((z,H^1_z(t,z))\geq\beta H(t,z)> 0\); \((\text{H}_4)\) \(H(t,z)\) is even for the second variable \(z\), namely, \(H(t,-z) =H(t, z)\), \(\forall (t, z)\in\mathbb{R}\times\mathbb{R}^{2d}\). By assumptions \((\text{H}_1)\) and \((\text{H}_3)\) the authors see that \(H(t,z)\geq a_1| z|^\beta- a_2\), \(\forall(t,z)\in \mathbb{R}\times \mathbb{R}^{2d}\). So assumptions \((\text{H}_1)\)--\((\text{H}_3)\) imply that \(H(t, z)\) grows superquadratically both at zero and at infinity. The main results of this paper are the following two theorems: Theorem 1: Under assumption \((\text{H}_1)\)--\((\text{H}_4)\) for any given positive integer \(p\), there exist at least \(d(pm -1)\) geometrically distance nontrivial periodic solutions of the system (1) with period \(pm\). Without assumption \((\text{H}_4)\), they have Theorem (2): Under assumptions \((\text{H}_1)\)--\((\text{H}_3)\), for any given positive integer \(p\), there exist at least two nontrivial periodic solutions of (1) with period \(pm\).