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) + (n/n + 1)(2) (x(n) - x(n - 1) + h(2)f (x(n))), n is an element of N, where h > 0 is a parameter and f is Lipschitz continuous and has three real zeros L-0 < 0 < L. We provide conditions for f under which for each sufficiently small h > 0 there exists a homoclinic solution of the above equation. The homoclinic solution is a sequence {x(n)}(n=0)(infinity) satisfying the equation and such that {x(n)}(n=1)(infinity) is increasing. x(0) = x(1) is an element of (L-0, 0) and lim(n ->infinity)x(n) = L. The problem is motivated by some models arising in hydrodynamics. (C) 2010 Elsevier Ltd. All rights reserved.