# Global Asymptotic Stability of a Family of Nonlinear Difference Equations

Authors: Maoxin Liao
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

In this note, we consider global asymptotic stability of the following nonlinear difference equationxn=(∏i=1v(xn-kiβi+1)+∏i=1v(xn-kiβi-1))/(∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1)),n=0,1,…, where ki∈ℕ(i=1,2,…,v),v≥2, β1∈[-1,1], β2,β3,…,βv∈(-∞,+∞), x-m,x-m+1,…,x-1∈(0,∞), and m=max1≤i≤v{ki}. Our result generalizes the corresponding results in the recent literature and simultaneously conforms to a conjecture in the work by Berenhaut et al. (2007).


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## Body

## 1. Introduction

The study of dynamical properties of nonlinear difference equations has been an area of intense interest in recent years (e.g., see [1–13]).In [4], by analysis of semicycle structure, the authors discussed the global asymptotic stability of rational difference equation
(1)xn+1=xnxn-1+1xn+xn-1,n=0,1,…,
where the initial values x-1,x0∈(0,+∞).Li [5, 6] investigated the qualitative behavior of the rational difference equations
(2)xn=xn-1+xn-2+xn-4+xn-1xn-2xn-4+a1+xn-1xn-2+xn-2xn-4+xn-1xn-4+a,n=0,1,2,…,xn=xn-2+xn-3+xn-4+xn-2xn-3xn-4+a1+xn-2xn-3+xn-3xn-4+xn-2xn-4+a,n=0,1,2,…,
with x-4,x-3,…,x-1∈(0,∞)  and  a∈[0,∞) via analysis of semicycle structure and verified that every solution of (2) converges to equilibrium 1.By using the transformation method, Berenhaut et al. [1] studied the behavior of positive solutions to the rational difference equation
(3)xn=xn-k+xn-m1+xn-kxn-m,n=0,1,2,…,
with x-m,x-m+1,…,x-1∈(0,∞) and 1≤k<m  and proved that every solution of (3) converges to the unique equilibrium 1. Based on the above facts, Berenhaut et al. [1] put forward the following two conjectures.Conjecture 1.
Suppose that1≤k<l<m and that {xn} satisfies
(4)xn=xn-k+xn-l+xn-m+xn-kxn-lxn-m1+xn-kxn-l+xn-lxn-m+xn-mxn-k,n=0,1,2,…
with x-m,x-m+1,…,x-1∈(0,∞). Then, the sequence {xn} converges to the unique equilibrium 1.Conjecture 2.
Suppose thatm is odd and 1≤k1<k2<⋯<km,  and define S={1,2,…,m}. If {xn} satisfies
(5)xn=f1(xn-k1,xn-k2,…,xn-km)f2(xn-k1,xn-k2,…,xn-km),n=0,1,2,…,
with x-km,x-km+1,…,x-1∈(0,∞), where
(6)f1(x1,x2,…,xm)=∑j∈{1,3,…,m}‍∑{t1,t2,…,tj}⊂S;t1<t2<⋯<tj‍xt1,xt2,…,xtj,f2(x1,x2,…,xm)=1+∑j∈{2,4,…,m-1}‍∑{t1,t2,…,tj}⊂S;t1<t2<⋯<tj‍xt1,xt2,…,xtj,
then the sequence {xn} converges to the unique equilibrium 1.Recently, by method used in [4–6], the authors of [12] studied the global asymptotic stability of the following nonlinear difference equation. 						(7)xn+1=F(xn,xn-1,xn-2,xn-3)G(xn,xn-1,xn-2,xn-3),n=0,1,…,
where
(8)F(x,y,z,w)=xα1yα2+xα1zα3+xα1wα4+yα2zα3+yα2wα4+zα3wα4+xα1yα2zα3wα4+1,G(x,y,z,w)=xα1+yα2+zα3+wα4+xα1yα2zα3+xα1yα2wα4+xα1zα3wα4+yα2zα3wα4,
the parameter α1∈(0,1],α2,α3,α4∈(0,+∞),  and the initial values x-3,x-2,x-1,x0∈(0,+∞).Motivated by the above studies, in this note, we propose and consider the following nonlinear difference equation.(9)xn=∏i=1v(xn-kiβi+1)+∏i=1v(xn-kiβi-1)∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1),n=0,1,…,
where ki∈ℕ(i=1,2,…,v),v≥2, β1∈[-1,1], β2,β3,…,βv∈(-∞,+∞), x-m,x-m+1,…,x-1∈(0,∞), and m=max1≤i≤v{ki}.It is noticed that, lettingv=2, β1=β2=1, k1=1,  and  k2=2, (9) reduces to (1); letting v=3,  β1=β2=β3=1,  k1=1,k2=2,  and  k3=4 and v=3,   β1=β2=β3=1,   k1=2,   k2=3,   and  k3=4, (9) reduces to (2); letting v=2,  β1=β2=1, k1=k,  and  k2=m, (9) reduces to (3); letting v=3, β1=k,   β2=l,  and  β3=m, (9) reduces to (4); letting v=4,  β1∈(0,1],  βi=αi(i=1,2,3,4),  k1=1,  k2=2, k3=3, and k4=4, (9) reduces to (7); letting v=m be odd, 1≤k1<k2<⋯<km, and β1=β2=⋯=βm=1, (9) reduces to (5). Clearly, (5) is a special example of (9).In 2007, Berenhaut and Stević [2] had proved Conjecture 1. In this paper, by making full use of analytical techniques, we mainly prove that the unique positive equilibrium point of (9) is globally asymptotically stable. It is clear that our result generalizes the corresponding works in [1, 2, 4–9, 12] and simultaneously conforms to Conjecture 2.


## 2. Existence of a Unique Positive Equilibrium

In this section, we mainly show the existence of a unique positive equilibrium of (9).Theorem 3.
In (9) there exists a unique positive equilibrium point x¯=1.Proof.
A positive equilibrium pointx¯ of (9) satisfies the next equation:
(10)x¯=∏i=1v(x¯βi+1)+∏i=1v(x¯βi-1)∏i=1v(x¯βi+1)-∏i=1v(x¯βi-1),
from which we may get
(11)(x¯-1)∏i=1v‍(x¯βi+1)=(x¯+1)∏i=1v‍(x¯βi-1);
that is,
(12)(x¯-1)(x¯β1+1)∏i=2v‍(x¯βi+1)=(x¯+1)(x¯β1-1)∏i=2v‍(x¯βi-1).
From the above equation, we can get
(13)(x¯β1+1-1)(∏i=2v‍(x¯βi+1)-∏i=2v‍(x¯βi-1))+(x¯-x¯β1)(∏i=2v‍(x¯βi+1)+∏i=2v‍(x¯βi-1))=0.
One can see that for any x¯>0  and  v≥2,
(14)∏i=2v‍(x¯βi+1)-∏i=2v‍(x¯βi-1)>0,∏i=2v‍(x¯βi+1)+∏i=2v‍(x¯βi-1)>0.(i)
Ifβ1=-1,0,1, from (13) and (14), we can get that (9) has a unique positive equilibrium x¯=1.(ii)
If-1<β1<0 or 0<β1<1 and 0<x¯<1, we have
(15)x¯<x¯β1,x¯β1+1<1.Further, we have(16)(x¯β1+1-1)(∏i=2v‍(x¯βi+1)-∏i=2v‍(x¯βi-1))+(x¯-x¯β1)(∏i=2v‍(x¯βi+1)+∏i=2v‍(x¯βi-1))<0.(iii)
If-1<β1<0 or 0<β1<1 and  x¯>1, we have
(17)(x¯β1+1-1)(∏i=2v‍(x¯βi+1)-∏i=2v‍(x¯βi-1))+(x¯-x¯β1)(∏i=2v‍(x¯βi+1)+∏i=2v‍(x¯βi-1))>0.It is clear that (9) has a unique positive equilibrium x¯=1. The proof is complete.


## 3. Global Asymptotic Stability for the Unique Positive Equilibrium Point

In this section, we give our main result.Theorem 4.
The unique positive equilibrium pointx¯=1 of (9) is globally asymptotically stable.In order to prove Theorem4, we introduce the following lemma by Kruse and Nesemann [3] and make full use of analytical techniques.Lemma 5.
Consider the difference equation(18)xn+k=f(xn+k-1,…,xn),n=0,1,2,…,
where k∈ℕ and f:(0,∞)k→(0,∞) is a continuous function with some unique equilibrium x¯. Suppose that there is a p∈ℕ such that for all solutions {xn} of (18)
(19)(xn-xn+p)(x¯2xn-xn+p)≤0,
where equality holds if and only if xn=x¯. Then x¯ is globally asymptotically stable.Proof of Theorem4.
Let{xn}n=-m∞ be any solution of (9). We have
(20)xn-xn-k1β1=∏i=1v(xn-kiβi+1)+∏i=1v(xn-kiβi-1)∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1)-xn-k1β1=(1-xn-k1β1)∏i=1v(xn-kiβi+1)+(1+xn-k1β1)∏i=1v(xn-kiβi-1)∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1)=(1-xn-k1β1)(1+xn-k1β1)(∏i=2v(xn-kiβi+1)-∏i=2v(xn-kiβi-1))∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1),n=0,1,…,1xn-k1β1-xn=1xn-k1β1-∏i=1v(xn-kiβi+1)+∏i=1v(xn-kiβi-1)∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1)=(1-xn-k1β1)∏i=1v(xn-kiβi+1)-(1+xn-k1β1)∏i=1v(xn-kiβi-1)∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1),=(1-xn-k1β1)(1+xn-k1β1)(∏i=2v(xn-kiβi+1)+∏i=2v(xn-kiβi-1))∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1),n=0,1,….
It follows from (20) that
(21)xn+k1-xnβ1=(1-xnβ1)(1+xnβ1)(∏i=2v(xn+k1-kiβi+1)-∏i=2v(xn+k1-kiβi-1))∏i=1v(xn+k1-kiβi+1)-∏i=1v(xn+k1-kiβi-1),n=-k1,-k1+1,…,1xnβ1-xn+k1=(1-xnβ1)(1+xnβ1)(∏i=2v(xn+k1-kiβi+1)+∏i=2v(xn+k1-kiβi-1))∏i=1v(xn+k1-kiβi+1)-∏i=1v(xn+k1-kiβi-1),n=-k1,-k1+1,….
Clearly, from (21), we have
(22)(xnβ1-xn+k1)(1xnβ1-xn+k1)≤0,n=-k1,-k1+1,….
From (22), we have
(23)1-xn+k1(1xnβ1+xnβ1)+xn+k12≤0,n=-k1,-k1+1,….
If β1=±1, it is clear that
(24)1xnβ1+xnβ1=1xn+xn.
If 0<xn<1 and -1<β1<1, we have xn<xnβ1 and 0<xnβ1+1<1, so that
(25)(xn-xnβ1)(1-1xnxnβ1)>0.
Similarly, if xn>1 and -1<β1<1, we have xn>xnβ1 and xnβ1+1>1, so that
(26)(xn-xnβ1)(1-1xnxnβ1)>0.
Hence, for -1≤β1≤1, we always have
(27)1xnβ1+xnβ1≤1xn+xn.
Further, from (23) and (27), we have
(28)1-xn+k1(1xn+xn)+xn+k12≤1-xn+k1(1xnβ1+xnβ1)+xn+k12≤0,n=-k1,-k1+1,….
Therefore,
(29)(xn-xn+k1)(1xn-xn+k1)≤0,n=-k1,-k1+1,…,
where equality holds if and only if xn=x¯=1. By Lemma 5 and (29), with p=k1∈ℕ, it follows that the unique positive equilibrium point x¯=1 of (9) is globally asymptotically stable. The proof is complete.



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