On four types of solutions of a second-order difference equation

Abstract

The second-order non-autonomous difference equationwhere h . 0 is a parameter and f is continuous and has three real zeros L 0 , 0 , L is investigated. The equation is a discretization of differential equations arising in hydrodynamics or in the nonlinear field theory. This paper provides conditions for f which guarantee that the equation can have four types of solutions -escape, homoclinic, damped and non-monotonous.