Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential

Abstract

In the present paper, we deal with the existence and multiplicity of homoclinic solutions of the second-order self-adjoint discrete Hamiltonian system

where and are neither autonomous nor periodic in n. Under the assumption that is indefinite sign and subquadratic as and p(n) and L(n) are N × N real symmetric positive definite matrices for all , we establish some existence criteria to guarantee that the above system has at least one or multiple homoclinic solutions by using Clark's theorem in critical point theory.

Keywords:

• homoclinic solution
• discrete Hamiltonian system
• critical point
• Clark's theorem

2000 Mathematics Subject Classification::

• 39A11
• 58E05
• 70H05

Acknowledgements

The authors thank the referees for their valuable comments and suggestions. This work is partially supported by the NNSF (No. 10771215) of China and supported by Hunan Provincial Natural Science Foundation of China (No. 06jj5013).