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

Abstract

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∈IN≡{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∈IN, 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.