Advances in Difference Equations Volume 2007, Article ID 16249,7pages doi:10.1155/2007/16249
Research Article
Global Asymptotic Stability in a Class of Difference Equations
Xiaofan Yang, Limin Cui, Yuan Yan Tang, and Jianqiu Cao
Received 29 April 2007; Accepted 5 November 2007
Recommended by John R. Graef
We study the difference equationxn=[(f ×g1+g2+h)/(g1+f ×g2+h)](xn−1,...,xn−r), n=1, 2,...,x1−r,...,x0>0, where f,g1,g2: (R+)r→R+ and h: (R+)r→[0, +∞) are all
continuous functions, and min1≤i≤r{ui, 1/ui} ≤ f(u1,...,ur) ≤ max1≤i≤r{ui, 1/ui},
(u1,...,ur)T∈(R+)r.We prove that this difference equation admitsc=1 as the globally
asymptotically stable equilibrium. This result extends and generalizes some previously known results.
Copyright © 2007 Xiaofan Yang et al. 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.
1. Introduction
Ladas [1] suggested investigating the nonlinear difference equation
xn=xxn−1+xn−2xn−3
n−1xn−2+xn−3, n=1, 2,..., x−2,x−1,x0>0. (1.1)
Since then, it has been proved that c=1 is the common globally asymptotically sta-ble equilibrium of this difference equation and all of the following difference equations (whereaandbare nonnegative constants):
xn=xxn−2+xn−1xn−3
n−1xn−2+xn−3, n=1, 2,..., x−2,x−1,x0>0 (see [1]), (1.2)
xn=xxn−1xn−2+xn−3+a
n−1+xn−2xn−3+a, n=1, 2,..., x−2,x−1,x0>0 (see [6, 12]), (1.3)
xn=xxn−2+xn−1xn−3+a
xn=xxn−1+xn−2xn−3+a
Motivated by the above work and the work by Sun and Xi [2], this article addresses the difference equation
We prove that (1.9) admitsc=1 as the globally asymptotically stable equilibrium. As a consequence, our result includes all of the above-mentioned results.
2. Preliminary knowledge
LetR+denote the whole set of positive real numbers. The part metric (or Thompson’s
Theorem2.1 (see [5, Theorem 2.2], see also [3]). LetT: (R+)r→(R+)rbe a continuous mapping with an equilibriumC∈(R+)r. Consider the following difference equation:
Xn=TXn−1
, n=1, 2,..., X0∈
R+ r
. (2.3)
Suppose there is a positive integer k such thatp(Tk(X),C)< p(X,C)holds for all X=C. Then C is globally asymptotically stable.
Theorem 2.2 (see [6, page 1]). Let a1,...,an, b1,...,bn, c1,...,cn be positive numbers. Then
min abi
i: 1≤i≤n
≤
n i=1ciai n
i=1cibi
≤max abi
i: 1≤i≤n
. (2.4)
Moreover, one of the two equalities holds if and only ifa1/b1=a2/b2= ··· =an/bn.
3. Main result
The main result of this article is the following.
Theorem3.1. Consider the difference equation
xn= f ×g
1+g2+h g1+f×g2+h
xn−1,...,xn−r, n=1, 2,..., x1−r,...,x0>0, (3.1)
where f,g1,g2: (R+)r→R+andh: (R+)r→[0, +∞)are all continuous functions, and
min
1≤i≤r
ui, 1/ui≤fu1,...,ur
≤max
1≤i≤r
ui, 1/ui, u1,...,ur T
∈R+ r
. (3.2)
Let{xn}be a solution of (3.1). Then the following assertions hold:
(i)for all n≥1 and j≥0, one has
min
1≤i≤r
xn−i, 1/xn−i≤xn+j≤max
1≤i≤r
xn−i, 1/xn−i; (3.3)
(ii)there exist n≥1 and j≥0 such that one of the two equalities in chain (3.3) holds if and only if(xn−1,...,xn−r)=(1,..., 1);
Proof. (i) For any givenn≥1, we prove the assertion by induction onj. ByTheorem 2.2
ByTheorem 2.2, chain (3.2), and the inductive hypothesis, we get
xn+j=
(ii) The sufficiency follows immediately from the first assertion of this theorem. Ne-cessity.Suppose there existn≥1 andj≥0 such thatxn+j=min1≤i≤r{xn−i, 1/xn−i}. Then
all of the equalities in chain (3.6) hold. This chain of equalities plusTheorem 2.2yield
f(xn+j−1,...,xn+j−r)=1 and, hence, (xn−1,...,xn−r)=(1,..., 1). Likewise, one can show
that (xn−1,...,xn−r)=(1,..., 1) ifxn+j=max1≤i≤r{xn−i, 1/xn−i}.
(iii) The system of first-order difference equations associated with (3.1) is
Yn=T(Yn−1), n=1, 2,..., (3.8) two assertions of this theorem, we induce
min
ByTheorem 2.1, we conclude thatCis the globally asymptotically stable equilibrium of system (3.8). This implies thatc=1 is the globally asymptotically stable equilibrium of
(3.1).
4. Applications
Example 4.1. Consider the difference equation
xn=
Example 4.2. Consider the difference equation
xn= f ×g
1+g2+h g1+f×g2+h
xn−1,...,xn−r, n=1, 2,..., x1−r,...,x0>0, (4.3)
whereg1,g2: (R+)r→R+andh: (R+)r→[0, +∞) are all continuous functions, 1≤p≤r,
1≤q≤r, 1≤s≤r, f(u1,...,ur)=(up+uq+ 1/us/3, andu1,...,urT∈R+ r
.
As f(u1,...,ur) is the arithmetic mean ofup,uq, and 1/us, we get
fu1,...,ur
≤max{up,uq, 1/us≤max
1≤i≤r
ui, 1/ui,
fu1,...,ur≥minup,uq, 1/us≥min 1≤i≤r
ui, 1/ui,
(4.4)
ByTheorem 3.1,c=1 is the globally asymptotically stable equilibrium of (4.3).
Example 4.3. Consider the difference equation
xn= f ×g
1+g2+h g1+f×g2+h
xn−1,...,xn−r, n=1, 2,..., x1−r,...,x0>0, (4.5)
whereg1,g2: (R+)r→R+andh: (R+)r→[0, +∞) are all continuous functions, 1≤p≤r,
1≤q≤r, 1≤s≤r, fu1,...,ur
=3upuq/us, andu 1,...,ur
T
∈R+ r
As f(u1,...,ur) is the geometric mean ofup,uq, and 1/us, we get
fu1,...,ur
≤maxup,uq, 1/us≤max 1≤i≤r
ui, 1/ui,
fu1,...,ur
≥minup,uq, 1/us≥min
1≤i≤r
ui, 1/ui.
(4.6)
ByTheorem 3.1,c=1 is the globally asymptotically stable equilibrium of (4.5).
5. Conclusions
This article has studied the global asymptotic stability of a class of difference equations. The result obtained extends and generalizes some previous results. We are attempting to apply the technique used in this article to deal with other generic difference equations which include some well-studied difference equations such as those in [7,8].
Acknowledgments
References
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Xiaofan Yang: College of Computer Science, Chongqing University, Chongqing 400044, China; School of Computer and Information, Chongqing Jiaotong University, Chongqing 400074, China Email address:xf [email protected]
Limin Cui: Department of Computer Science, Hong Kong Baptist University, Kowloon, Hong Kong Email address:[email protected]
Yuan Yan Tang: College of Computer Science, Chongqing University, Chongqing 400044, China Current address: Department of Computer Science, Hong Kong Baptist University,
Kowloon, Hong Kong
Email address:[email protected]
Jianqiu Cao: School of Computer and Information, Chongqing Jiaotong University, Chongqing 400074, China