Advances in Difference Equations
Volume 2005 (2005), Issue 3, Pages 333-343
doi:10.1155/ADE.2005.333
Stability of periodic solutions of first-order difference
equations lying between lower and upper solutions
Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Galicia, Santiago de Compostela 15782, Spain
Received 8 January 2004; Revised 2 September 2004
Copyright © 2005 Alberto Cabada et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.