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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]

In this paper, we study the second-order self-adjoint discrete Hamiltonian systemwhereis anreal symmetric positive definite matrix for all,is unnecessarily positive definite for allandis indefinite sign. By using fountain theorem, we establish some new criteria to guarantee that the above system has infinitely many homoclinic solutions under the assumption thatis superquadratic as. Our results generalize and improve some existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of homoclinic solutions for the second order self-adjoint discrete Hamiltonian systemwhere,andare neither autonomous nor periodic inn. By using a standard version of the Mountain Pass theorem, we give some new criteria to guarantee that the above system has at least one non-trivial homoclinic solution under the assumption thatsatisfies more general superquadratic condition. Some recent results are generalized and significantly improved. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the existence of three solutions for second-order impulsive Hamiltonian systems. By using a variational method and some critical points theorems of B. Ricceri, we obtain some new criteria for guaranteeing that the impulsive Hamiltonian systems have three solutions. Some recent results are extended and improved. Some examples are presented to demonstrate the application of our main results. [Copyright &y& Elsevier]