Advances in Difference Equations
Volume 2010 (2010), Article ID 714891, 11 pages
Research Article

Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics

Department of Mathematics, Faculty of Science, Palacký University, tř. 17. listopadu 12, 77146 Olomouc, Czech Republic

Received 19 December 2009; Revised 14 February 2010; Accepted 10 March 2010

Academic Editor: Ağacik Zafer

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.


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))), nN, 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 limnx(n)>L. The problem is motivated by some models arising in hydrodynamics.