By using the critical point theory, we establish some existence criteria to guarantee that the nonlinear difference equation Δ[p(n)(Δx(n-1))δ]-q(n)(x(n))δ=f(n,x(n)) has at least one homoclinic solution, where n∈ℤ, x(n)∈ℝ, and f:ℤ×ℝ→ℝ is non periodic in n. Our conditions on the nonlinear term f(n,x(n)) are rather relaxed, and we generalize some existing results in the literature.