Optimal conditions to ensure the stability of periodic solutions of first order difference equations lying between lower and upper solutions

摘要

This paper is devoted to the study of the stability of the solutions of the first order implicit difference periodic equationDeltau(i) = f(i, u(i + 1)); i is an element of {0,..., N-1}, u(0) = u (N).We prove that if there exist alpha less than or equal to beta a pair of lower and upper solutions respectively of this problem together with a suitable condition on the function f and it has at most two solutions in the sector [a, beta], then one of such solutions is asymptotically stable.Moreover, it is proved the optimality of the given results, both in the conditions imposed on f and in the number of the periodic solutions. (C) 2004 Elsevier B.V All rights reserved.