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 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 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 so-obtained results are in the same direction as the ones proved by the authors in [3] for the first-order implicit difference equation ∆u(i) = f (i,u(i + 1)) coupled with periodic boundary conditions. In that situation, we give some optimal conditions on function f and on the number of the possible periodic solutions of the considered problem, that warrant the existence of at least one stable solution. The arguments there are different Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:3 (2005) 333–343 DOI: 10.1155/ADE.2005.333
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Stability of periodic solutions
from the ones used in this paper because in that situation, due to the lack of uniqueness of solutions of the initial problem, the discrete operator considered here cannot be defined. The paper is organized as follows. In Section 2, we present some fundamental properties of the set of solutions of initial and periodic problems. In Section 3, we prove the existence of at least one N-periodic stable solution. Finally, Section 4 is devoted to give some examples that point out the, in some sense, optimality of the obtained results. 2. Preliminaries This paper is devoted to study the stability, by using the method of lower and upper solutions, of the
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