Advances in Difference Equations
Volume 2010 (2010), Article ID 714891, 11 pages
doi:10.1155/2010/714891

Strictly increasing solutions of nonautonomous difference equations arising in hydrodynamics

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.