## 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
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### References:

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