Nonoscillatory solutions of a second-order difference equation of Poincaré type

Poincaré’s classical theorem about the convergence of ratios of successive values of solutions applies if the characteristic roots of the associated limiting equation are simple and have different moduli. In this work, it is shown that for the nonoscillatory solutions the conclusion of Poincaré’s theorem is also true in the case where the limiting equation has a double positive characteristic root.