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

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We prove that if there exists αβ, 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 is stable. In several particular situations, we may claim that this solution is asymptotically stable.

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Correspondence to Alberto Cabada.

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