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 ) + 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 . 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 ) ∈ ( L 0 , 0 ), and lim ⁡ n → ∞ x ( n ) > L . The problem is motivated by some models arising in hydrodynamics.