STABILITY OF PERIODIC SOLUTIONS OF FIRST-ORDER DIFFERENCE EQUATIONS LYING BETWEEN LOWER AND UPPER SOLUTIONS ALBERTO CABADA, VICTORIA OTERO-ESPINAR, ´ AND DOLORES RODRIGUEZ-VIVERO Received 8 January 2004 and in revised form 2 September 2004 We prove that if there exists α ≤ β, a pair of lower and upper solutions of the first-order discrete periodic problem ∆u(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. | STABILITY OF PERIODIC SOLUTIONS OF FIRST-ORDER DIFFERENCE EQUATIONS LYING BETWEEN LOWER AND UpPER SOLUTIONS ALBERTO CABADA VICTORIA OTERO-ESPINAR AND DOLORES RODRIGUEZ-VIVERO Received 8 January 2004 and in revised form 2 September 2004 We prove that if there exists a f a pair of lower and upper solutions of the first-order discrete periodic problem Au n f n u n n e 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 e 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. 1. Introduction It is well known that one of the most important concepts in the qualitative theory of differential and difference equations is the stability of the solutions of the treated problems. Classical tools as approximation by linear equations or Lyapunov functions have been developed for both type of equations see 7 for ordinary differential equations and 8 for difference ones. More recently some authors as among others de Coster and Habets 6 Nieto 9 or Ortega 10 have proved the stability of solutions of adequate ordinary differential equations that lie between a pair of lower and upper solutions. In this case fixed points theorems and degree and index theory are the fundamental arguments to deduce the mentioned stability results. Stability for order-preserving operators defined on Banach spaces have been obtained by Dancer in 4 and Dancer and Hess in 5 . On these papers the authors describe the assymptotic behavior of the iterates that lie between a lower and an upper solution of suitable operators. Our purpose is to ensure the stability of at least one periodic solution of a first-order difference equation. We will prove such result by using a monotone nondecreasing operator. In this case the defined operator does not verify the conditions imposed in 5 . The .
