## 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 \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 $$pm$$.

### MSC:

 39A11 Stability of difference equations (MSC2000) 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
Full Text:

### 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), Science and Technical Press: Science and Technical Press Shanghai, China [4] Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems (1993), Birkhäuser: Birkhäuser Boston · 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), Science Press: Science Press Beijing [11] Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems (1989), Springer: Springer New York · 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.; 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.