Homoclinic solutions of non-autonomous difference equations arising in hydrodynamics

The paper deals with the second-order non-autonomous difference equation x(n+1)=x(n)+(nn+1)2(x(n)−x(n−1)+h2f(x(n))),n∈N, where h>0h>0 is a parameter and ff is Lipschitz continuous and has three real zeros L0<00h>0 there exists a homoclinic solution of the above equation. The homoclinic solution is a sequence {x(n)}n=0∞ satisfying the equation and such that {x(n)}n=1∞ is increasing, x(0)=x(1)∈(L0,0)x(0)=x(1)∈(L0,0) and limn→∞x(n)=Llimn→∞x(n)=L. The problem is motivated by some models arising in hydrodynamics.