DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 750852 10.1155/2013/750852 750852 Research Article Global Asymptotic Stability of a Family of Nonlinear Difference Equations Liao Maoxin Çinar Cengiz School of Mathematics and Physics University of South China Hengyang Hunan 421001 China scut.edu.cn 2013 4 12 2013 2013 17 06 2013 09 11 2013 2013 Copyright © 2013 Maoxin Liao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this note, we consider global asymptotic stability of the following nonlinear difference equation 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(i=1,2,,v),v2, β1[-1,1], β2,β3,,βv(-,+), x-m,x-m+1,,x-1(0,), and m=max1iv{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).

1. Introduction

The study of dynamical properties of nonlinear difference equations has been an area of intense interest in recent years (e.g., see ).

In , 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.  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 1k<m  and proved that every solution of (3) converges to the unique equilibrium 1. Based on the above facts, Berenhaut et al.  put forward the following two conjectures.

Conjecture 1.

Suppose that 1k<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 that m is odd and 1k1<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 , the authors of  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(i=1,2,,v),v2, β1[-1,1], β2,β3,,βv(-,+), x-m,x-m+1,,x-1(0,), and m=max1iv{ki}.

It is noticed that, letting v=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, 1k1<k2<<km, and β1=β2==βm=1, (9) reduces to (5). Clearly, (5) is a special example of (9).

In 2007, Berenhaut and Stević  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, 49, 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 point x¯ 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  v2, (14)i=2v(x¯βi+1)-i=2v(x¯βi-1)>0,i=2v(x¯βi+1)+i=2v(x¯βi-1)>0.

If β1=-1,0,1, from (13) and (14), we can get that (9) has a unique positive equilibrium x¯=1.

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.

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 point x¯=1 of (9) is globally asymptotically stable.

In order to prove Theorem 4, we introduce the following lemma by Kruse and Nesemann  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 and f:(0,)k(0,) is a continuous function with some unique equilibrium x¯. Suppose that there is a p 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 Theorem <xref ref-type="statement" rid="thm3.1">4</xref>.

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+k120,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 -1β11, we always have (27)1xnβ1+xnβ11xn+xn. Further, from (23) and (27), we have (28)1-xn+k1(1xn+xn)+xn+k121-xn+k1(1xnβ1+xnβ1)+xn+k120,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, it follows that the unique positive equilibrium point x¯=1 of (9) is globally asymptotically stable. The proof is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the referees for their careful reading of the paper and many valuable comments and suggestions that greatly improved the presentation of this work. This paper is supported partly by Hunan Provincial Natural Science Foundation of China (no. 13JJ3075), Soft Science Fund of Science and Technology Department of Hunan Province (no. 2011ZK3066), Start-up Fund of University of South China (no. 2011XQD49), and the construct program in USC.

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