Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials.

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system Δ 2 u ( n − 1 ) − L ( n ) u ( n ) + ∇ W ( n , u ( n ) ) = 0, where n ∈ Z, u ∈ R N and L : ℤ → ℝ N × N are unnecessarily positive definites for all n ∈ Z. By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]