Abstract
In the present paper, we deal with the existence of infinitely many homoclinic solutions for the second-order self-adjoint discrete Hamiltonian system
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 operator, p, L : Z → RN ×N and W : Z × RN → R, W (n, x) is continuously differentiable in x for every n ∈ Z
We say that a solution u(n) of system ( . ) is homoclinic if u(n) → as n → ±∞
Under assumption (L) above, we will use the symmetric mountain pass theorem to study the existence of infinitely many homoclinic solutions for ( . ) in the case, where W satisfies the following weaker assumptions than (W ) as x → and (AR) as |x| → ∞
Summary
The existence and the multiplicity of homoclinic solutions of system We are interested in the case when L(n) is not global positive definite and satisfies the following assumption. Under assumption (L) above, we will use the symmetric mountain pass theorem to study the existence of infinitely many homoclinic solutions for
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