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

• Pati Iwaasa ^[2] ; Hideaki Matsunaga ^[1]
  1. [1] Osaka Prefecture University

     Osaka Prefecture University

     Sakai Ku, Japón

     
  2. [2] Nippon Telegraph and Telephone West Corporation
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
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• Resumen

  • 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.