- Research
- Open Access
On the difference equation \(x_{n+1}=ax_{n-l}+bx_{n-k}+f ( x_{n-l},x_{n-k} )\)
- Mahmoud A. E. Abdelrahman^{1},
- George E. Chatzarakis^{2},
- Tongxing Li^{3, 4}Email author and
- Osama Moaaz^{1}
https://doi.org/10.1186/s13662-018-1880-8
© The Author(s) 2018
- Received: 17 February 2018
- Accepted: 8 November 2018
- Published: 21 November 2018
Abstract
Keywords
- Difference equation
- Equilibrium point
- Local stability
- Periodic solution
MSC
- 39A10
- 39A23
- 39A30
1 Introduction
Difference equations describe the observed evolution of a phenomenon at discrete time steps. Thus, difference equations are used as discrete models of the workings of physical or artificial systems. The asymptotic behavior of the solutions of linear difference equations is a qualitative property having important applications in many areas, including control theory, mathematical biology, neural networks, and so forth. We cannot use numerical methods to study the asymptotic behavior of all solutions of a given equation due to the global nature of that behavior. Therefore, the analytical study of those qualitative properties has been attracting considerable interest from mathematicians and engineers, as the only method to gain insight into those properties.
The results in this paper make three main contributions to the study of linear difference equations. First, we formulate a general class of difference equations as a means of establishing general theorems for the asymptotic behavior of its solutions and the solutions of equations that are special cases of the studied equation. Second, we study the asymptotic behavior of the solutions of this more general class of difference equations using an efficient method introduced in [12] and modified in [19]. Theorem 3.2 establishes how this method can be applied to equation (1.1). In particular, this method is also valid and can be applied to several classes of difference equations for which the classical method fails to give results. Moreover, we consider difference equations with real coefficients and initial values which extend and slightly improve previous results. Third, we can use our analysis to check and verify the results obtained by other researchers.
For the basic definitions and auxiliary lemmas we use for establishing our results, namely equilibrium points, local stability, and periodicity of the solutions, we refer the reader to [1, 9, 17, 18]. For the convenience of the reader, we present below some related results.
Lemma 1.1
(see [17, Theorem 1.3.7])
Lemma 1.2
(see [9, Corollary 4])
Let \(f:\mathbb{R} _{+}^{n}\rightarrow \mathbb{R} \) be continuous and differentiable on \(\mathbb{R} _{++}^{n}\). If f is homogeneous of degree k, then \(D_{j}f=\partial f/\partial x_{j}\) is homogeneous of degree \(k-1\).
The rest of the paper is organized as follows. In Sect. 2, we study the stability behavior and boundedness of the solutions of equation (1.1) and give an illustrative example in support of our analysis. In Sect. 3, we present a technique to investigate the periodic behavior of the solutions of equation (1.1). A distinguishing feature of our criteria is that the coefficients l and k of equation (1.1) can be odd or even. Two examples are provided to illustrate the new method for studying periodic solutions. In Sect. 4, the practicability, maneuverability, and efficiency of the results obtained are illustrated via two applications.
2 Dynamics of equation (1.1)
2.1 Local stability
Theorem 2.1
Proof
Example 2.1
2.2 Boundedness
In this section, we study the boundedness of the solutions of equation (1.1).
Theorem 2.2
If \(a+b < 1\) and there exists a positive constant L such that \(f ( u,v ) < L\) for all \(u,v\in ( 0,\infty )\), then every solution of equation (1.1) is bounded.
Proof
Remark 2.3
As fairly noticed by the referees, the global asymptotic stability of equation (1.1) remains an open problem for further research.
3 Periodic solutions
Theorem 3.1
If l and k are either odd or even, then equation (1.1) has no solutions of prime period two.
Proof
Theorem 3.2
Proof
Theorem 3.3
Proof
The proof is similar to that of Theorem 3.2 and thus is omitted. □
Example 3.1
Remark 3.4
4 Applications
Here, two test cases are given to validate the asymptotic behavior of the proposed new class of difference equations.
4.1 Application 1
Conjecture 1
Note that, if \(\sum_{i=1}^{\beta} ( 2i-1 ) c_{i}< c_{0}\), then every solution of equation (4.1) converges either to the equilibrium point or to a periodic solution having period two.
4.2 Application 2
Declarations
Acknowledgements
The authors express their sincere gratitude to the editors and two anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Funding
This research is supported by NNSF of P.R. China (Grant No. 61503171), CPSF (Grant No. 2015M582091), NSF of Shandong Province (Grant No. ZR2016JL021), KRDP of Shandong Province (Grant No. 2017CXGC0701), DSRF of Linyi University (Grant No. LYDX2015BS001), and the AMEP of Linyi University, P.R. China.
Authors’ contributions
All four authors contributed equally to this work. They all read and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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