Copyright © 2010 Lukáš Rachůnek and Irena Rachůnková. 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.

Abstract

The paper provides conditions sufficient for the existence of strictly increasing solutions of the second-order nonautonomous difference equation x(n+1)=x(n)+(n/(n+1))2(x(n)-x(n-1)+h2f(x(n))), n∈N, where h>0 is a parameter and f is Lipschitz continuous and has three real zeros L0<0<L. In particular we prove that for each sufficiently small h>0 there exists a solution {x(n)}n=0∞ such that {x(n)}n=1∞ is increasing, x(0)=x(1)∈(L0,0), and lim⁡n→∞x(n)>L. The problem is motivated by some models arising in hydrodynamics.