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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

Δx 1 (n)=-H x 2 (n,x 1 (n+1),x 2 (n)),Δx 2 (u)=H x 1 (n,x 1 (n+1),x 2 (n))(1)

by using critical point theory, where x 1 ,x 2 d , HC 1 (× d × d ,), Δx i (n)=x i (n+1)-x i (n), i=1,2.

Denote z=(x 1 T ,x 2 T ) T , H(t,z)=H(t,x 1 ,x 2 ) and assume the following conditions

(H 1 ) H(t,z)C 1 (× 2d ,) and there exists a positive integer m such that (x,z)× 2d , H(t+m,z)=H(t,z),

(H 2 ) For any (t,z)× 2d , H(t,z)0 and H(t,z)=o(|z| 2d ) (as z0);

(H 3 ) There exist some constants R>0, β>2 such that for any |z|R, (z,H z 1 (t,z))βH(t,z)>0;

(H 4 ) H(t,z) is even for the second variable z, namely, H(t,-z)=H(t,z), (t,z)× 2d . By assumptions (H 1 ) and (H 3 ) the authors see that H(t,z)a 1 |z| β -a 2 , (t,z)× 2d . So assumptions (H 1 )(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 (H 1 )(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 (H 4 ), they have

Theorem (2): Under assumptions (H 1 )(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