Resumen de Oscillation and Nonoscillation for Nonlinear Delay Difference Equations by Phase Plane Analysis

Pati Iwaasa, Hideaki Matsunaga

• This study is devoted to the investigation on the oscillation problem of nonlinear delay difference equations of the form x(n + 1) − ax(n) + f (n, x(n − k)) = 0, n = 0, 1, 2,... . Here, a is a positive number, k is a positive integer, and f (n, x) is a continuous function with respect to the second variable that satisfies suitable conditions. A pair of oscillation and nonoscillation theorems for the equation is established.

  This study also improves the lower bounded condition in an oscillation theorem and the upper bounded condition in a nonoscillation theorem for the equation when the nonlinear function f (n, x) is separable. The obtained theorems are proved by transforming the equation into a certain difference system and using a phase plane analysis.