Exact oscillation regions for a partial difference equation
- Chunhua Yuan^{1, 2},
- Shutang Liu^{1}Email author and
- Jian Liu^{1, 2}
https://doi.org/10.1186/s13662-015-0387-9
© Yuan et al.; licensee Springer. 2015
Received: 29 October 2014
Accepted: 26 January 2015
Published: 28 March 2015
Abstract
This paper is concerned with a partial difference equation with constant coefficients. By means of the theory of envelopes, we consider the regions of non-positive roots of its characteristic equation and obtain a necessary and sufficient condition for the oscillation of all solutions of the partial difference equation.
Keywords
MSC
1 Introduction
Partial difference equations have numerous applications in image processing, material mechanics, random walk problems, molecular orbits, finite difference schemes, as well as population dynamics [1–7]. In recent years, the oscillatory behavior of partial difference equations has been discussed in many papers (see [7–14] and the references therein).
In [15–18], Lin and Cheng gave explicit necessary and sufficient conditions for all solutions of some ordinary difference equations with constant coefficients to be oscillatory by means of the envelope theory of a family of straight lines. However, to the best of our knowledge, very little attention has been paid to the analysis of the oscillatory behavior of solutions of partial difference equations from the perspective of the envelope theory.
By a solution of (1), we mean a nontrivial double sequence \(\{A_{m,n}\}\) of real numbers which is defined for \(m \geq0\) and \(n\geq 0\) and satisfies (1) for \(m \geq0 \) and \(n\geq0\).
A solution \(\{A_{m,n}\}\) of (1) is said to be eventually positive (or negative) if \(A_{m,n} \geq0 \) (or \(A_{m,n} \leq0 \)) for large m and n. It is said to be oscillatory if it is neither eventually positive nor eventually negative.
The paper is organized as follows. Section 2 presents several useful lemmas. In Section 3, we derive the oscillation criterion for partial difference equation (1). Two examples are given to illustrate our oscillation criterion in Section 4. Finally Section 5 is devoted to concluding remarks.
2 Some lemmas
In this section, we give some preliminaries that will be needed in the next section.
From Corollary 2.9 in [14], we can easily obtain the following lemma.
Lemma 1
Lemma 2
([8])
Lemma 3
([19])
3 Main results
In this section, we establish the necessary and sufficient condition for oscillations of all solutions of (1) by the envelope theory.
Theorem 1
Proof
On the basis of Theorem 1, we can obtain the following results.
Corollary 1
Equation (6) has no positive roots if and only if \(c\geq0\) and \(a\geq0\) or \(a<0\) and \(c>a^{2}/4\).
Corollary 2
Every solution of (7) oscillates if and only if \(c\geq0\) and \(a\geq0\) or \(a<0\) and \(c>a^{2}/4\).
4 Illustrative examples
In this section, two examples are given to illustrate the applications of Theorem 1.
Example 1
Example 2
5 Conclusions
In this paper, we have introduced a novel approach to problems of oscillations of a partial difference equation. Our approach is based on the envelope theory of the family of planes. We derive effective criteria to determine oscillations of a partial difference equation by this approach. Numerical examples are given to illustrate the results presented in this paper.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of P.R. China under Grant No. 61273088. The authors thank the editor and the referees for their valuable suggestions and comments.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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