How to find initial data generating bounded solutions of discrete equations

This contribution concerns the asymptotic behavior of solutions of a first-order difference equation. We are looking for a solution whose graph stays in a given domain. It is supposed that all the boundary points of this domain are the so called points of strict egress. Under this supposition, it has been already proved that the existence of at least one solution the graph of which stays in the given domain is guaranteed. The main aim of this article is to find the concrete value of the initial condition which generates such a solution. The method we introduce resembles the well-known bisection method for finding roots of equations.