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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 $pm$.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 58E50 Applications of variational methods in infinite-dimensional spaces
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##### References:
 [1] Ahlbrandt, C. D.: Equivalence of discrete Euler equations and discrete Hamiltonian systems. J. math. Anal. appl. 180, 498-517 (1993) · Zbl 0802.39005 [2] Bohner, M.: Linear Hamiltonian difference systemsdisconjugacy and Jacobi-type conditions. J. math. Anal. appl. 199, 804-826 (1996) · Zbl 0855.39018 [3] Chang, K. C.: Critical point theory and its applications. (1980) [4] Chang, K. C.: Infinite dimensional Morse theory and multiple solution problems. (1993) · Zbl 0779.58005 [5] Chen, S.: Disconjugacy, disfocality, and oscillation of second order difference equations. J. differential equations 107, 383-394 (1994) · Zbl 0791.39001 [6] Elaydi, S. N.; Zhang, S.: Stability and periodity of difference equations with finite delay. Funkcialaj ekvac. 37, 401-413 (1994) · Zbl 0819.39006 [7] Erbe, L. H.; Yan, P.: Disconjugacy for linear Hamiltonian difference systems. J. math. Anal. appl. 167, 355-367 (1992) · Zbl 0762.39003 [8] Guo, Z.; Yu, J.: The existence of periodic and subharmonic solutions for second-order superlinear difference equations. Sci. China, ser. A 46, 506-515 (2003) · Zbl 1215.39001 [9] Hartman, P.: Difference equationsdisconjugacy, principal solutions, Green’s functions, complete monotonicity. Trans. amer. Math. soc. 246, 1-30 (1978) · Zbl 0409.39001 [10] Long, Y. M.: The index theory of Hamiltonian system and its applications. (1993) [11] Mawhin, J.; Willem, M.: Critical point theory and Hamiltonian systems. (1989) · Zbl 0676.58017 [12] Rabinowitz, P. H.: Periodic solutions of Hamiltonian systems. Comm. pure appl. Math. 31, 157-184 (1978) · Zbl 0358.70014 [13] Rabinowitz, P. H.: On subharmonics solutions of Hamiltonian systems. Comm. pure appl. Math. 33, 609-633 (1980) · Zbl 0425.34024 [14] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS. Vol. 65, Amemrican Mathematical Society, Providence, RI, 1986. · Zbl 0609.58002