Abstract
Under the assumptions that W(n,x) is indefinite sign and subquadratic as |x|→+∞ and L(n) satisfies lim inf | n | → + ∞ [ | n | ν − 2 inf | x | = 1 ( L ( n ) x , x ) ] >0 for some constant ν<2, we establish a theorem on the existence of infinitely many homoclinic solutions for the second-order self-adjoint discrete Hamiltonian system △ [ p ( n ) △ u ( n − 1 ) ] −L(n)u(n)+∇W ( n , u ( n ) ) =0, where p(n) and L(n) are N×N real symmetric matrices for all n∈Z, and p(n) is always positive definite.MSC:39A11, 58E05, 70H05.
Highlights
Consider the second-order self-adjoint discrete Hamiltonian system p(n) u(n – ) – L(n)u(n) + ∇W n, u(n) =, ( . )where n ∈ Z, u ∈ RN, u(n) = u(n + ) – u(n) is the forward difference, p, L : Z → RN ×N and W : Z × RN → R, W (n, x) is continuously differentiable in x for every n ∈ Z.As usual, we say that a solution u(n) of system ( . ) is homoclinic if u(n) → as n → ±∞
|x|=1 for some constant ν < 2, we establish a theorem on the existence of infinitely many homoclinic solutions for the second-order self-adjoint discrete Hamiltonian system p(n) u(n – 1) – L(n)u(n) + ∇W(n, u(n)) = 0, where p(n) and L(n) are N × N real symmetric matrices for all n ∈ Z, and p(n) is always positive definite
We say that a solution u(n) of system ( . ) is homoclinic if u(n) → as n → ±∞
Summary
|x|=1 for some constant ν < 2, we establish a theorem on the existence of infinitely many homoclinic solutions for the second-order self-adjoint discrete Hamiltonian system p(n) u(n – 1) – L(n)u(n) + ∇W(n, u(n)) = 0, where p(n) and L(n) are N × N real symmetric matrices for all n ∈ Z, and p(n) is always positive definite. The existence and multiplicity of nontrivial homoclinic solutions for problem
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