Homoclinic solutions of non-autonomous difference equations arising in hydrodynamics

Abstract

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 ∈ 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 ∞ satisfying the equation and such that { x ( n ) } n = 1 ∞ is increasing, x ( 0 ) = x ( 1 ) ∈ ( L 0 , 0 ) and lim n → ∞ x ( n ) = L . The problem is motivated by some models arising in hydrodynamics.