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Global Behavior of a New Rational Nonlinear Higher-Order Difference Equation.

1. Introduction

Difference equations have wide applications in biology, computer science, digital signal processing, and economics. A general solution structure exists for linear difference equations [1]. However, in various situations of nonlinear higher-order difference equations, solution properties can only be observed by numerical simulation, and it is often exceedingly difficult to give a full mathematical proof for the properties predicted by numerical simulation and the conclusions formed on the basis of guesswork [2]. It is, therefore, important to make qualitative analysis on nonlinear higher-order difference equations, which is the topic of the current study. There have been some related studies on rational nonlinear difference equations in the literature (see, e.g., [3-8]). Global asymptotic properties of spectral functions are also crucial in determining algebro-geometric solutions of soliton equations (see, e.g., [9, 10]) and scattering data in matrix spectral problems (see, e.g., [11]).

An iterative algorithm to approximate a zero of a given function f reads

[mathematical expression not reproducible] (1)

and an application of this to a quadratic function f(x) = [x.sup.2] - a, a > 0, gives

[mathematical expression not reproducible]. (2)

Let k be a nonnegative integer and c a real number greater than or equal to 1. We would like to consider a more general rational nonlinear higher-order difference equation

[mathematical expression not reproducible], (3)

with positive initial values [z.sub.-k], [z.sub.-k+1], ..., [z.sub.0], which engender positive solutions. We take a transformation

[z.sub.n] = c/[y.sub.n], n [greater than or equal to] -k, (4)

and then obtain another difference equation

[y.sub.n+1] = [y.sub.n] [y.sub.n-k] + c/[y.sub.n] [y.sub.n-k] + c-1, n [greater than or equal to] 0, (5)

Obviously, the equilibrium solution of the rational nonlinear difference equation (3), [bar.z] = c, becomes the equilibrium solution of the transformed difference equation (5), [bar.y] = 1.

If we further take c = 1, then we obtain the nonlinear difference equation discussed in [6, 7]:

[y.sub.n+1] = [y.sub.n] [y.sub.n-k] + 1/[y.sub.n] [y.sub.n-k], n [greater than or equal to] 0, (6)

Introducing [x.sub.n] = [square root of a] [y.sub.n] into (6) yields

[x.sub.n+1] = [x.sub.n] [x.sub.n-k] + a/[x.sub.n] [x.sub.n-k], n [greater than or equal to] 0, (7)

where a > 0. When k =1, this gives the nonlinear difference equation in (2). The equation (7) in the case of k = 2 was studied in [5] and its closed-form solution was presented in [6]. In the general case of k, the asymptotic stability of the positive equilibrium solution [bar.x] = 1 of the equation (7) was proved in [7].

It is direct to see that the rational nonlinear higher-order difference equation, defined by (3), possesses three equilibria: [bar.z] = -1,0, c. In this article, we would like to explore global behavior of solutions to the rational nonlinear higher-order difference equation (3), show the global asymptotic stability of its positive equilibrium solution [bar.z] = c, and present two illustrative examples of positive solutions.

2. Global Behavior

2.1. Classification of Solutions. First of all, based on the rational difference equation (3), one can have

[mathematical expression not reproducible], (8)

[mathematical expression not reproducible], (9)

[mathematical expression not reproducible]. (10)

Further from (9) and (10), we can easily derive the following solution properties.

Theorem 1. If {[z.sub.n]}.sup.[infinity].sub.n=-k] is a solution to the rational nonlinear difference equation (3), then one has

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

where n [greater than or equal to] 0.

If k = 0, the rational difference equation (3) becomes a first-order difference equation

[mathematical expression not reproducible]. (13)

Then for n [greater than or equal to] 0, one has [z.sub.n+1] [less than or equal to] c, because -[z.sup.2.sub.n] + 2c[z.sub.n] [less than or equal to] [c.sup.2]. For n [greater than or equal to] 1,one has [z.sub.n+1] [greater than or equal to] [z.sub.n], since (c-1) [z.sub.n] + c [greater than or equal to] [z.sup.2.sub.n], due to [z.sub.n] [less than or equal to] c. Therefore, every solution [z.sub.n] decays to c, when n [right arrow] [infinity].

Generally, there are three types of solutions to the rational nonlinear higher-order difference equation (3).

Theorem 2. Let k [greater than or equal to] 1. If {[z.sub.n]}.sup.[infinity].sub.n=-k] is a solution to the rational nonlinear higher-order difference equation (3), and then

(a) it is eventually equal to c, more precisely [z.sub.n] = c,n[greater than or equal to] m, which occurs when [z.sub.m] = c for some m [greater than or equal to] 0;

(b) it is eventually less than c, more precisely [z.sub.n] < [z.sub.n+l] < c, n[greater than or equal to]m + k, which occurs when [z.sub.m], [z.sub.m+1],..., [z.sub.m+k] < cfor some m [greater than or equal to] -k; or

(c) it oscillates about c, possessing at most k consecutive increasing terms less than c and at most k +1 consecutive decreasing terms greater than c.

Proof. Equality (8) and property (11) directly tell that we have three types of solutions to the rational nonlinear higher-order difference equation (3).

The decreasing and increasing characteristics of oscillatory solutions in the third solution situation (c) can be proved as follows.

Suppose that [n.sub.1], [n.sub.2] [greater than or equal to] 0 are two integers satisfying [n.sub.1] < [n.sub.2]. We express

[mathematical expression not reproducible] (14)

where D can be written as

[mathematical expression not reproducible] (15)

by (9).

If [z.sub.n] > c for [n.sub.1] [less than or equal to] n [less than or equal to] [n.sub.2], then each term in D is less than zero, and so [mathematical expression not reproducible], then each term in D is greater than zero, and [mathematical expression not reproducible], due to (14). This completes the proof.

Note that based on (8), we can see that there is no solution situation that a solution of (3) is eventually greater than c.

2.2. Global Asymptotic Stability. When k = 0, the equilibrium solution [bar.z] = c of the first-order rational difference equation (13) is globally asymptotically stable, since it is a globally attractive equilibrium solution of a first-order difference equation (see [12] for a general theory).

For a general k [greater than or equal to] 1, we can show the same global asymptotic stability of the positive equilibrium solution [bar.z] = c of the rational nonlinear difference equation (3), by establishing the local asymptotic stability and the global attractivity, which imply the global asymptotic stability [2]. Instead, we establish a strong negative feedback property [13] to guarantee the global asymptotic stability of [bar.z] = c (see [14] for details on the strong negative feedback property).

Theorem 3. The positive equilibrium solution [bar.z] = c of the rational nonlinear higher-order difference equation (3) is globally asymptotically stable.

Proof. Based on the rational nonlinear difference equation (3), one can have

[mathematical expression not reproducible] (16)

From this equality and the equality in (10), we can obtain

[mathematical expression not reproducible] (17)

which leads to a strong negative feedback property:

[mathematical expression not reproducible], (18)

with equality for all n [greater than or equal to] 0 if and only if [z.sub.n] = c,n [greater than or equal to] -k. It, therefore, follows from a stability theorem (Corollary 3 of [14]) that the equilibrium solution [bar.z] = c of the rational nonlinear difference equation (3) is globally asymptotically stable. Thus, the proof is finished.

The above theorems with c = 1 gives the results in [4] (k = 1), [5] (k = 2) and [7] (a general k). There have also been similar studies on polynomial difference equations (see, for example, [15]) and other studies on positive rational function solutions, called lumps, to partial differential equations (see, e.g., [16]).

2.3. Illustrative Examples and an Open Question. To illustrate the global properties stated in Theorems 2 and 3, here we present two illustrative examples associated with two special cases:

[mathematical expression not reproducible] (19)

in Figure 1. From the plot pictures, we see that the convergence is achieved very fast in both cases.

Finally, let k [greater than or equal to] 1. For an oscillatory solution [[z.sub.n]}.sup.[infinity].sub.n=-k] of the rational nonlinear difference equation (3), we define

[N.sub.g] = {n | [z.sub.n] > c and n [greater than or equal to] 0}, [N.sub.l] = {n| [z.sub.n] < c and n >0}. (20)

Since {[z.sub.n]}.sup.[infinity].sub.n=-k] oscillatory, Theorem 2 guarantees that both [N.sub.g] and [N.sub.l] have infinitely many numbers. A basic open question that we are very interested in is if [z.sub.n] is decreasing on [N.sub.g] and increasing on [N.sub.l]. We point out that through the above two examples, we failed to find any counterexample to this statement, but found that two cases could occur: either [z.sub.n-1], [z.sub.n+1] < c but [z.sub.n] > c or [z.sub.n-1], [z.sub.n+1] > c but [z.sub.n] < c for some n > 1.

https://doi.org/10.1155/2019/2048941

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This study was in part supported by NNSFC under Grants 11371326, 11371086, and 11571079, NSF under Grant DMS1664561, and the distinguished professorships of King Abdulaziz University, Saudi Arabia and North-West University, South Africa.

References

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[16] W. X. Ma and Y. Zhou, "Lump solutions to nonlinear partial differential equations via Hirota bilinear forms," Journal of Differential Equations, vol. 264, no. 4, pp. 2633-2659, 2018.

Wen-Xiu Ma [ID] (1, 2, 3, 4, 5, 6)

(1) Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China

(2) Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

(3) Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA

(4) College of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090, China

(5) College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China

(6) Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mafikeng, Mmabatho 2735, South Africa

Correspondence should be addressed to Wen-Xiu Ma; mawx@cas.usf.edu

Received 10 April 2019; Revised 22 June 2019; Accepted 14 July 2019; Published 7 August 2019

Academic Editor: Mondher Farza

Caption: Figure 1: Profiles of [{[z.sub.n]}.sup.[infinity].sub.n=-k] with c = 3/2, fc = 3 (left) and c = 2, k = 5 (right).
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Title Annotation:Research Article
Author:Ma, Wen-Xiu
Publication:Complexity
Geographic Code:9CHIN
Date:Aug 31, 2019
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