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

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Abstract

This study is devoted to the investigation on the oscillation problem of nonlinear delay difference equations of the form \(x(n+1)-ax(n)+f(n,x(n-k))=0\), \(n=0,1,2,\dots \). 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.

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Acknowledgements

The authors would like to thank the referees for carefully reading the manuscript and making useful suggestions. The authors would also like to thank Editage (www.editage.com) for English language editing.

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Authors and Affiliations

  1. Nippon Telegraph and Telephone West Corporation, Osaka, 540-8511, Japan

    Pati Iwaasa

  2. Department of Mathematical Sciences, Osaka Prefecture University, Sakai, 599-8531, Japan

    Hideaki Matsunaga

Authors
  1. Pati Iwaasa
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  2. Hideaki Matsunaga
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Correspondence to Hideaki Matsunaga.

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Funding

The second author’s work was supported by JSPS KAKENHI Grants-in-Aid for Scientific Research (C) (Grant Number JP19K03524).

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All data analyzed in this study are included in this paper.

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The authors declare that they have no conflicts of interest in this work.

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Dedicated to Professor Jitsuro Sugie on the occasion of his 65th birthday.

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Iwaasa, P., Matsunaga, H. Oscillation and Nonoscillation for Nonlinear Delay Difference Equations by Phase Plane Analysis. Qual. Theory Dyn. Syst. 21, 120 (2022). https://doi.org/10.1007/s12346-022-00652-4

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