We prove that if there exists α ≤ β , a pair of lower and upper solutions of the first-order discrete periodic problem Δ u ( n ) = f ( n , u ( n ) ) ; n ∈ I N ≡ { 0 , … , N − 1 } , u ( 0 ) = u ( N ) , with f a continuous N -periodic function in its first variable and such that x + f ( n , x ) is strictly increasing in x , for every n ∈ I N , then, this problem has at least one solution such that its N -periodic extension to ℕ is stable. In several particular situations, we may claim that this solution is asymptotically stable.