Let T be an integer with T ≥ 3 , and let T : = { 1 , … , T } . We study the existence and uniqueness of solutions for the following two-point boundary value problems of second-order difference systems: Δ 2 u ( t − 1 ) + f ( t , u ( t ) ) = e ( t ) , t ∈ T , u ( 0 ) = u ( T + 1 ) = 0 , where e : T → ℝ n   and   f : T × ℝ n → ℝ n is a potential function satisfying f ( t , ⋅ ) ∈ C 1 ( ℝ n ) and some nonresonance conditions. The proof of the main result is based upon a mini-max theorem.