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

Authors: Wen-Xiu Ma
Journal: Complexity
 (2019)

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## Abstract

Letk be a nonnegative integer and c a real number greater than or equal to 1. We present qualitative global behavior of solutions to a rational nonlinear higher-order difference equation zn+1=(czn+zn-k+c-1znzn-k)/(znzn-k+c),n≥0, with positive initial values z-k,z-k+1,⋯,z0, and show the global asymptotic stability of its positive equilibrium solution.


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## Body

## 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 functionf reads (1)xn+1=xn-1fxn-xnfxn-1fxn-fxn-1,n≥0,and an application of this to a quadratic function f(x)=x2-a,a>0, gives (2)xn+1=xnxn-1+axn+xn-1,n≥0.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 (3)zn+1=czn+zn-k+c-1znzn-kznzn-k+c,n≥0,with positive initial values z-k,z-k+1,⋯,z0, which engender positive solutions. We take a transformation (4)zn=cyn,n≥-k,and then obtain another difference equation (5)yn+1=ynyn-k+cyn+yn-k+c-1,n≥0,Obviously, the equilibrium solution of the rational nonlinear difference equation (3), z¯=c, becomes the equilibrium solution of the transformed difference equation (5), y¯=1.If we further takec=1, then we obtain the nonlinear difference equation discussed in [6, 7]: (6)yn+1=ynyn-k+1yn+yn-k,n≥0.Introducing xn=ayn into (6) yields (7)xn+1=xnxn-k+axn+xn-k,n≥0,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 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: 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 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 (8)zn+1-c=c-zn-kzn-cznzn-k+c,n≥0,(9)zn+1-zn=zn-kzn+1c-znznzn-k+c,n≥0,(10)zn+1-zn-k=znzn-k+1c-zn-kznzn-k+c,n≥0.Further from (9) and (10), we can easily derive the following solution properties.Theorem 1.
If{zn}n=-k∞ is a solution to the rational nonlinear difference equation (3), then one has (11)zn+1<znifzn>c,zn+1>znifzn<c,(12)zn+1<zn-kifzn-k>c,zn+1>zn-kifzn-k<c, where n≥0.Ifk=0, the rational difference equation (3) becomes a first-order difference equation (13)zn+1=2czn+c-1zn2zn2+c,n≥0.Then for n≥0, one has zn+1≤c, because -zn2+2czn≤c2. For n≥1, one has zn+1≥zn, since (c-1)zn+c≥zn2, due to zn≤c. Therefore, every solution zn decays to c, when n→∞.Generally, there are three types of solutions to the rational nonlinear higher-order difference equation (3).Theorem 2.
Letk≥1. If {zn}n=-k∞ is a solution to the rational nonlinear higher-order difference equation (3), and then
(a) it is eventually equal toc, more precisely zn=c,n≥m, which occurs when zm=c for some m≥0;
(b) it is eventually less thanc, more precisely zn<zn+1<c, n≥m+k, which occurs when zm,zm+1,⋯,zm+k<c for some m≥-k; or
(c) it oscillates aboutc, 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 thatn1,n2≥0 are two integers satisfying n1<n2. We express (14)zn2-zn1=zn2-zn2-1+zn2-1-zn2-2+⋯+zn1+1-zn1=D, where D can be written as (15)D=∑j=n1n2-1zj-kc-zjzj+1zjzj-k+c, by (9).
Ifzn>c for n1≤n≤n2, then each term in D is less than zero, and so zn2<zn1, due to (14). If zn<c for n1≤n≤n2, then each term in D is greater than zero, and so zn2>zn1, 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

Whenk=0, the equilibrium solution 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 generalk≥1, we can show the same global asymptotic stability of the positive equilibrium solution 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 z¯=c (see [14] for details on the strong negative feedback property).Theorem 3.
The positive equilibrium solutionz¯=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 (16)c2zn-k-zn+1=c-zn-kc-1znzn-k+czn-k+c2zn-kznzn-k+c,n≥0. From this equality and the equality in (10), we can obtain (17)zn-k-zn+1c2zn-k-zn+1=-znzn-k+1c-zn-k2c-1znzn-k+czn-k+c2zn-kznzn-k+c2,n≥0, which leads to a strong negative feedback property: (18)zn-k-zn+1c2zn-k-zn+1≤0,n≥0, with equality for all n≥0 if and only if zn=c,n≥-k. It, therefore, follows from a stability theorem (Corollary 3 of [14]) that the equilibrium solution z¯=c of the rational nonlinear difference equation (3) is globally asymptotically stable. Thus, the proof is finished.The above theorems withc=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 Theorems2 and 3, here we present two illustrative examples associated with two special cases:(19)c=32,k=3,z-3=65,z-2=169,z-1=107,z0=1;c=2,k=5,z-5=54,z-4=73,z-3=53,z-2=94,z-1=32,z0=73;in Figure 1. From the plot pictures, we see that the convergence is achieved very fast in both cases.Figure 1
Profiles of{zn}n=-k∞ with c=3/2,k=3 (left) and c=2,k=5 (right).Finally, letk≥1. For an oscillatory solution {zn}n=-k∞ of the rational nonlinear difference equation (3), we define (20)Ng=n∣zn>candn≥0,Nl=n∣zn<candn≥0.Since {zn}n=-k∞ is oscillatory, Theorem 2 guarantees that both Ng and Nl have infinitely many numbers. A basic open question that we are very interested in is if zn is decreasing on Ng and increasing on Nl. 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 zn-1,zn+1<c but zn>c or zn-1,zn+1>c but zn<c for some n>1.


## 2.1. Classification of Solutions

First of all, based on the rational difference equation (3), one can have (8)zn+1-c=c-zn-kzn-cznzn-k+c,n≥0,(9)zn+1-zn=zn-kzn+1c-znznzn-k+c,n≥0,(10)zn+1-zn-k=znzn-k+1c-zn-kznzn-k+c,n≥0.Further from (9) and (10), we can easily derive the following solution properties.Theorem 1.
If{zn}n=-k∞ is a solution to the rational nonlinear difference equation (3), then one has (11)zn+1<znifzn>c,zn+1>znifzn<c,(12)zn+1<zn-kifzn-k>c,zn+1>zn-kifzn-k<c, where n≥0.Ifk=0, the rational difference equation (3) becomes a first-order difference equation (13)zn+1=2czn+c-1zn2zn2+c,n≥0.Then for n≥0, one has zn+1≤c, because -zn2+2czn≤c2. For n≥1, one has zn+1≥zn, since (c-1)zn+c≥zn2, due to zn≤c. Therefore, every solution zn decays to c, when n→∞.Generally, there are three types of solutions to the rational nonlinear higher-order difference equation (3).Theorem 2.
Letk≥1. If {zn}n=-k∞ is a solution to the rational nonlinear higher-order difference equation (3), and then
(a) it is eventually equal toc, more precisely zn=c,n≥m, which occurs when zm=c for some m≥0;
(b) it is eventually less thanc, more precisely zn<zn+1<c, n≥m+k, which occurs when zm,zm+1,⋯,zm+k<c for some m≥-k; or
(c) it oscillates aboutc, 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 thatn1,n2≥0 are two integers satisfying n1<n2. We express (14)zn2-zn1=zn2-zn2-1+zn2-1-zn2-2+⋯+zn1+1-zn1=D, where D can be written as (15)D=∑j=n1n2-1zj-kc-zjzj+1zjzj-k+c, by (9).
Ifzn>c for n1≤n≤n2, then each term in D is less than zero, and so zn2<zn1, due to (14). If zn<c for n1≤n≤n2, then each term in D is greater than zero, and so zn2>zn1, 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

Whenk=0, the equilibrium solution 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 generalk≥1, we can show the same global asymptotic stability of the positive equilibrium solution 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 z¯=c (see [14] for details on the strong negative feedback property).Theorem 3.
The positive equilibrium solutionz¯=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 (16)c2zn-k-zn+1=c-zn-kc-1znzn-k+czn-k+c2zn-kznzn-k+c,n≥0. From this equality and the equality in (10), we can obtain (17)zn-k-zn+1c2zn-k-zn+1=-znzn-k+1c-zn-k2c-1znzn-k+czn-k+c2zn-kznzn-k+c2,n≥0, which leads to a strong negative feedback property: (18)zn-k-zn+1c2zn-k-zn+1≤0,n≥0, with equality for all n≥0 if and only if zn=c,n≥-k. It, therefore, follows from a stability theorem (Corollary 3 of [14]) that the equilibrium solution z¯=c of the rational nonlinear difference equation (3) is globally asymptotically stable. Thus, the proof is finished.The above theorems withc=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 Theorems2 and 3, here we present two illustrative examples associated with two special cases:(19)c=32,k=3,z-3=65,z-2=169,z-1=107,z0=1;c=2,k=5,z-5=54,z-4=73,z-3=53,z-2=94,z-1=32,z0=73;in Figure 1. From the plot pictures, we see that the convergence is achieved very fast in both cases.Figure 1
Profiles of{zn}n=-k∞ with c=3/2,k=3 (left) and c=2,k=5 (right).Finally, letk≥1. For an oscillatory solution {zn}n=-k∞ of the rational nonlinear difference equation (3), we define (20)Ng=n∣zn>candn≥0,Nl=n∣zn<candn≥0.Since {zn}n=-k∞ is oscillatory, Theorem 2 guarantees that both Ng and Nl have infinitely many numbers. A basic open question that we are very interested in is if zn is decreasing on Ng and increasing on Nl. 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 zn-1,zn+1<c but zn>c or zn-1,zn+1>c but zn<c for some n>1.



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