We prove that this difference equation admits c =1 as the globally asymptotically stable equilibrium.. We prove that 1.9 admitsc =1 as the globally asymptotically stable equilibrium.. 3.1
Trang 1Volume 2007, Article ID 16249, 7 pages
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 equation xn =[(f × g1+g2+h)/(g1+f × g2+h)](x n −1, ,x n − 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 { u i, 1/u i } ≤ f (u1, ,u r) ≤ max1≤ i ≤ r { u i, 1/u i }, (u1, ,u r)T ∈(R+)r We prove that this difference equation admits c =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
x n = x n −1+x n −2x n −3
x n −1x n −2+x n −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
(where a and b are nonnegative constants):
x n = x n −2+x n −1x n −3
x n −1x n −2+x n −3, n =1, 2, , x −2,x −1,x0> 0 (see [1]), (1.2)
x n = x n −1x n −2+x n −3+a
x n −1+x n −2x n −3+a, n =1, 2, , x −2,x −1,x0> 0 (see [6,12]), (1.3)
x n = x n −2+x n −1x n −3+a
x n −1x n −2+x n −3+a, n =1, 2, , x −2,x −1,x0> 0 (see [6]), (1.4)
Trang 2x n = x n −1+x n −2x n −3+a
x n −1x n −2+x n −3+a, n =1, 2, , x −2,x −1,x0> 0 (see [14]), (1.5)
x n = x n −1x n −2+x n −3+a
x n −2+x n −1x n −3+a, n =1, 2, , x −2,x −1,x0> 0 (see [14]), (1.6)
x n = x b
n −1x n −3+x b
n −4+a
x b
n −1+x n −3x b
n −4+a, n =1, 2, , x −3,x −2,x −1,x0> 0 (see [7]), (1.7)
x n = x b
n − k x n − +x b
n − l+a
x b
n − k+x n − x b
n − l+a, n =1, 2, , x1−max{ k,m,l }, ,x0> 0 (see [8]). (1.8) Motivated by the above work and the work by Sun and Xi [2], this article addresses the
difference equation
x n =
f × g1+g2+h
g1+f × g2+h
x n −1, ,x n − r
, n =1, 2, , x1− r, ,x0> 0, (1.9)
where f ,g1,g2: (R+)r → R+andh : (R+)r →[0, +∞) are all continuous functions, and
min
1≤ i ≤ r
u i, 1/u i
≤ fu1, ,u r
≤max
1≤ i ≤ r
u i, 1/u i
u1, ,u rT
∈R+
r (1.10)
It can be seen that (1.9) subsumes (1.1) and (1.8) For example, if we letr =max{ k,l,m },
f (x n −1, ,x n − r)= x n − ,g1(x n −1, ,x n − r)= x b
n − k,g2(x n −1, ,x n − r)= x b
n − l, andh(x1, ,
x r)≡ a, then (1.9) reduces to (1.8)
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
For two functions, f (x1, ,x n) andg(x1, ,x n), we adopt the following notations:
[f + g]x1, ,x n
:= fx1, ,x n
+gx1, ,x n
,
f × g]x1, ,x n
:= fx1, ,x n
× gx1, ,x n
,
f g
x1, ,x n
:= fx1, ,x n
gx1, ,x n ifgx1, ,x n
=0.
(2.1)
LetR+denote the whole set of positive real numbers The part metric (or Thompson’s metric) [3,4] is a metric defined on (R+)rin the following way: for anyX =(x1, ,x r)T ∈
(R+)rand
Y =y1, , y rT
∈R+
r
, p(X,Y) : = −log2min
1≤ i ≤ r
x i /y i,y i /x i
Trang 3Theorem 2.1 (see [5, Theorem 2.2], see also [3]) Let T : (R+)r →(R+)r be a continuous mapping with an equilibrium C ∈(R+)r Consider the following difference equation:
X n = TX n −1
, n =1, 2, , X0∈R+
r
Suppose there is a positive integer k such that p(T k(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, ,a n, b1, ,b n, c1, ,c n be positive numbers Then
min
a i
b i: 1≤ i ≤ n ≤
n
i =1c i a i
n
i =1c i b i ≤max
a i
b i: 1≤ i ≤ n (2.4)
Moreover, one of the two equalities holds if and only if a1/b1= a2/b2= ··· = a n /b n
3 Main result
The main result of this article is the following
Theorem 3.1 Consider the di fference equation
x n =
f × g1+g2+h
g1+f × g2+h
x n −1, ,x n − r
, n =1, 2, , x1− r, ,x0> 0, (3.1)
where f ,g1,g2: (R+)r → R+and h : (R+)r →[0, +∞ ) are all continuous functions, and
min
1≤ i ≤ r
u i, 1/u i
≤ fu1, ,u r
≤max
1≤ i ≤ r
u i, 1/u i
u1, ,u rT
∈R+
r (3.2)
Let { x n } 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
x n − i, 1/x n − i
≤ x n+j ≤max
1≤ i ≤ r
x n − i, 1/x n − i
(ii) there exist n ≥ 1 and j ≥ 0 such that one of the two equalities in chain ( 3.3 ) holds if and only if (x n −1, ,x n − r)=(1, ,1);
(iii)c = 1 is the globally asymptotically stable equilibrium of ( 3.1 ).
Trang 4Proof (i) For any given n ≥ 1, we prove the assertion by induction on j ByTheorem 2.2
and chain (3.3), we have
x n =
f × g1+g2+h
g1+f × g2+h
x n −1, ,x n − r
≥min
fx n −1, ,x n − r
,fx n −1,1 ,x n − r
≥min
min
1≤ i ≤ r
x n − i, 1/x n − i
max1≤ i ≤ rx n − i, 1/x n − i =min
1≤ i ≤ r
x n − i, 1/x n − i
,
x n =
f × g1+g2+h
g1+f × g2+h
x n −1, ,x n − r
≤max
fx n −1, ,x n − r
, fx n −1,1 ,x n − r
max
1≤ i ≤ r
x n − i, 1/x n − i
max1≤ i ≤ r
x n − i, 1/x n − i =max
1≤ i ≤ r
x n − i, 1/x n − i
.
(3.4)
So the assertion is true forj =0
Suppose the assertion is true for all integer k (0 ≤ k ≤ j −1), that is,
min
1≤ i ≤ r
x n − i, 1/x n − i
≤ x n+k ≤max
1≤ i ≤ r
x n − i, 1/x n − i
, 0≤ k ≤ j −1. (3.5)
ByTheorem 2.2, chain (3.2), and the inductive hypothesis, we get
x n+j =f × g1+g2+h
g1+f × g2+h
x n+j −1, ,x n+j − r
≥min
fx n+j −1, ,x n+j − r
, fx n+j −1,1 ,x n+j − r
≥min
min
1≤ i ≤ r
x n+j − i, 1/x n+j − i
max1≤ i ≤ r
x n+j − i, 1/x n+j − i =min
1≤ i ≤ r
x n+j − i, 1/x n+j − i
≥min
min
1≤ i ≤ r
x n − i, 1/x n − i
max1≤ i ≤ r
x n − i, 1/x n − i =min
1≤ i ≤ r
x n − i, 1/x n − i
; (3.6)
x n+j =f × g1+g2+h
g1+ f × g2+h
x n+j −1, ,x n+j − r
≤max
fx n+j −1, ,x n+j − r
,fx n+j −1,1 ,x n+j − r
≤max
max
1≤ i ≤ r
x n+j − i, 1/x n+j − i }, 1
min1≤ i ≤ r
x n+j − i, 1/x n+j − i } =max1≤ i ≤ r
x n+j − i, 1/x n+j − i
≤max
max
1≤ i ≤ r
x n − i, 1/x n − i
min1≤ i ≤ r
x n − i, 1/x n − i =max
1≤ i ≤ r
x n − i, 1/x n − i
.
(3.7)
Thus the assertion is true for j The inductive proof of this assertion is complete.
Trang 5(ii) The sufficiency follows immediately from the first assertion of this theorem
Ne-cessity Suppose there exist n ≥ 1 and j ≥0 such thatx n+j =min1≤ i ≤ r { x n − i, 1/x n − i } Then all of the equalities in chain (3.6) hold This chain of equalities plusTheorem 2.2yield
f (x n+j −1, ,x n+j − r)=1 and, hence, (x n −1, ,x n − r)=(1, ,1) Likewise, one can show
that (x n −1, ,x n − r)=(1, ,1) if x n+j =max1≤ i ≤ r { x n − i, 1/x n − i }
(iii) The system of first-order difference equations associated with (3.1) is
whereT : (R+)r →(R+)ris a mapping defined by
Ty1, , y rT
=
y2, , y r,
f × g1+g2+h
g1+ f × g2+h
y r, , y1
T
By chain (3.2), we have f (1, ,1) =1 Hence,C =(1, ,1) T is an equilibrium of sys-tem (3.8) Consider an arbitraryX =(x1, ,x r)T ∈(R+)r,X = C Then
T r
x1, ,x rT
=x r+1, ,x2r T
wherex j =[(f × g1+g2+h)/(g1+ f × g2+h)](x j −1, ,x j − r), r + 1 ≤ j ≤2r By the first
two assertions of this theorem, we induce
min
r+1 ≤ i ≤2r
x i, 1/x i
> min min
1≤ i ≤ r
x i, 1/x i
max1≤ i ≤ r
x i, 1/x i =min
1≤ i ≤ r
x i, 1/x i
. (3.11) Hence,
pT r
X,C= −log2 min
r+1 ≤ i ≤2r
x i, 1/x i
< −log2min
1≤ i ≤ r
x i, 1/x i
= pX,C. (3.12)
ByTheorem 2.1, we conclude that C is the globally asymptotically stable equilibrium of
system (3.8) This implies thatc =1 is the globally asymptotically stable equilibrium of
4 Applications
Example 4.1 Consider the difference equation
x n =f × g1+g2+h
g1+f × g2+h
x n −1, ,x n − r
, n =1, 2, , x1− r, ,x0> 0, (4.1) whereg1,g2: (R+)r → R+andh : (R+)r →[0, +∞) are all continuous functions, 1≤ p ≤ r,
1≤ q ≤ r, 1 ≤ s ≤ r, fu1, ,u r
=u p+u q+u s
/3, andu1, ,u rT
∈R+
r
.
As f (u1, ,u r ) is the arithmetic mean of u p , u q , and u s, we get
fu1, ,u r
≤max
u p,u q,u s
≤max
1≤ i ≤ r
u i, 1/u i
,
fu1, ,u r
≥minu p,u q,u s
≥min
1≤ i ≤ r
u i, 1/u i. (4.2)
ByTheorem 3.1,c =1 is the globally asymptotically stable equilibrium of (4.1)
Trang 6Example 4.2 Consider the difference equation
x n =
f × g1+g2+h
g1+f × g2+h
x n −1, ,x n − 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, ,u r)=(u p+u q+ 1/u s
/3, andu1, ,u rT
∈R+
r
As f (u1, ,u r ) is the arithmetic mean of u p , u q , and 1/ u s, we get
fu1, ,u r
≤max{ u p,u q, 1/u s
≤max
1≤ i ≤ r
u i, 1/u i
,
fu1, ,u r
≥min
u p,u q, 1/u s
≥min
1≤ i ≤ r
u i, 1/u i
ByTheorem 3.1,c =1 is the globally asymptotically stable equilibrium of (4.3)
Example 4.3 Consider the difference equation
x n =
f × g1+g2+h
g1+f × g2+h
x n −1, ,x n − 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, ,u r
= 3
u p u q /u s, and
u1, ,u rT
∈R+
r
As f (u1, ,u r ) is the geometric mean of u p , u q , and 1/u s, we get
fu1, ,u r
≤max
u p,u q, 1/u s
≤max
1≤ i ≤ r
u i, 1/u i
,
fu1, ,u r
≥min
u p,u q, 1/u s
≥min
1≤ i ≤ r
u i, 1/u i
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
The authors are grateful to the anonymous reviewers for their valuable comments and suggestions This work is supported by Natural Science Foundation of China (Grant no 10771227), Program for New Century Excellent Talent of Educational Ministry of China (Grant no NCET-05-0759), Doctorate Foundation of Educational Ministry of China (Grant no 20050611001), and Natural Science Foundation of Chongqing (Grants no CSTC 2006BB2231, 2005BB2191)
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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 yang1964@yahoo.com
Limin Cui: Department of Computer Science, Hong Kong Baptist University, Kowloon, Hong Kong
Email address:clm628@hotmail.com
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:yytang@cqu.edu.cn
Jianqiu Cao: School of Computer and Information, Chongqing Jiaotong University,
Chongqing 400074, China
Email address:caojq86@cquc.edu.cn
... 2007.[3] N Kruse and T Nesemann, ? ?Global asymptotic stability in some discrete dynamical systems,”
Journal of Mathematical Analysis and Applications,... 4707–4717, 2001.
[5] X Yang, D J Evans, and G M Megson, ? ?Global asymptotic stability in a class of Putnam-type
equations,” Nonlinear Analysis: Theory,... class= "text_page_counter">Trang 7
[1] G Ladas, “Open problems and conjectures,” Journal of Di fference Equations and Applications,
vol