Báo Cáo Báo cáo hóa học STABILITY OF PERIODIC SOLUTIONS OF FIRST-ORDER DIFFERENCE EQUATIONS LYING BETWEEN LO

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 ư 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.

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