Advances in Difference Equations
Volume 2005 (2005), Issue 3, Pages 333-343

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

Alberto Cabada, Victoria Otero-Espinar, and Dolores Rodríguez-Vivero

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.


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));nIN{0,,N1},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 nIN, 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.