Strictly Increasing Solutions of Nonautonomous Difference Equations Arising in Hydrodynamics

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 limn→∞x n > L. The problem is motivated by some models arising in hydrodynamics.