Some results are obtained for the existence and subharmonic solutions to discrete Hamiltonian systems
by using critical point theory, where , , , .
Denote , and assume the following conditions
and there exists a positive integer such that , ,
For any , and (as );
There exist some constants , such that for any , ;
is even for the second variable , namely, , . By assumptions and the authors see that , . So assumptions – imply that grows superquadratically both at zero and at infinity. The main results of this paper are the following two theorems:
Theorem 1: Under assumption – for any given positive integer , there exist at least geometrically distance nontrivial periodic solutions of the system (1) with period .
Without assumption , they have
Theorem (2): Under assumptions –, for any given positive integer , there exist at least two nontrivial periodic solutions of (1) with period .