stability of periodic solutions of first-order difference equations lying between lower and upper solutions

We prove that if there exists α ≤ β, a pair of lower and upper solutions of the first-order discrete periodic problem Δu(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.