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Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems. (English) Zbl 1053.39011
Some results are obtained for the existence and subharmonic solutions to discrete Hamiltonian systems $$\aligned\Delta x_1(n) &= -H_{x_2}(n, x_1(n+1), x_2(n)),\\ \Delta x_2(u) &= H_{x_1}(n, x_1(n+1), x_2(n))\endaligned\tag1$$ by using critical point theory, where $x_1,x_2\in \bbfR^d$, $H\in C^1(\bbfR\times \bbfR^d\times \bbfR^d,\bbfR)$, $\Delta x_i(n)= x_i(n+ 1)- x_i(n)$, $i= 1,2$. Denote $z= (x^T_1, x^T_2)^T$, $H(t, z)= H(t, x_1,x_2)$ and assume the following conditions $(\text{H}_1)$ $H(t, z)\in C^1(\bbfR\times \bbfR^{2d},\bbfR)$ and there exists a positive integer $m$ such that $\forall(x,z)\in \bbfR\times \bbfR^{2d}$, $H(t+ m,z)= H(t,z)$, $(\text{H}_2)$ For any $(t, z)\in \bbfR\times\bbfR^{2d}$, $H(t,z)\ge 0$ and $H(t,z)= o(\vert z\vert^{2d})$ (as $z\to 0$); $(\text{H}_3)$ There exist some constants $R> 0$, $\beta> 2$ such that for any $\vert z\vert\ge R$, $(z,H^1_z(t,z))\ge\beta H(t,z)> 0$; $(\text{H}_4)$ $H(t,z)$ is even for the second variable $z$, namely, $H(t,-z) =H(t, z)$, $\forall (t, z)\in\bbfR\times\bbfR^{2d}$. By assumptions $(\text{H}_1)$ and $(\text{H}_3)$ the authors see that $H(t,z)\ge a_1\vert z\vert^\beta- a_2$, $\forall(t,z)\in \bbfR\times \bbfR^{2d}$. 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 $pm$.

39A11Stability of difference equations (MSC2000)
58E50Applications of variational methods in infinite-dimensional spaces
Full Text: DOI
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