Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2008, Article ID 193872, 7 pages
doi:10.1155/2008/193872
Research Article
A New Part-Metric-Related Inequality Chain and
an Application
Xiaofan Yang,1, 2 Fangkuan Sun,2 and Yuan Yan Tang2, 3
1
School of Computer and Information, Chongqing Jiaotong University, Chongqing 400074, China
College of Computer Science, Chongqing University, Chongqing 400044, China
3
Department of Computer Science, Hong Kong Baptist University, Kowloon, Hong Kong
2
Correspondence should be addressed to Xiaofan Yang, xf yang@163.com
Received 28 September 2007; Accepted 6 November 2007
Recommended by Stevo Stevic
Part-metric-related PMR inequality chains are elegant and are useful in the study of difference
equations. In this paper, we establish a new PMR inequality chain, which is then applied to show
the global asymptotic stability of a class of rational difference equations.
Copyright q 2008 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
A part-metric related PMR inequality chain is a chain of inequalities of the form
1
1
≤ f a1 , . . . , an ≤ max ai ,
,
min ai ,
1≤i≤n
1≤i≤n
ai
ai
1.1
which is closely related to the well-known part metric 1 and has important applications in
the study of difference equations 2–13. Below are three previously known PMR inequality
chains:
a1 a2 a3 a4
1
1
min ai ,
≤
see 5,
1.2
≤ max ai ,
1≤i≤4
ai
a1 a2 a3 a4 1≤i≤4
ai
a1 · · · ak−2 ak−1 ak
1
1
min ai ,
≤
≤ max ai ,
1≤i≤k
ai
a1 a2 a3 · · · ak
ai
1≤i≤k
1 wa1 a2 a3 a4 a5
1
1
min ai ,
≤
,
≤ max ai ,
1≤i≤5
ai
a1 a2 a1 a3 a2 a3 wa4 a5 1≤i≤5
ai
see 11,
1.3
1 ≤ w ≤ 2 see 13.
1.4
2
Discrete Dynamics in Nature and Society
In this article, we establish the following PMR inequality chain:
min
1≤i≤2p−1
ai ,
1
ai
≤ hw a1 , . . . , a2p−1 ≤ max
1≤i≤2p−1
ai ,
1
,
ai
1.5
where hw will be defined in the next section, p − 2 ≤ w ≤ p − 1. When p 3, chain 1.5 reduces
to chain 1.4. On this basis, we prove that the difference equation
xn hw xn−2p1 , . . . , xn−1 ,
n 1, 2, . . . ,
1.6
with positive initial conditions admits a globally asymptotically stable equilibrium c 1.
2. Main results
This section establishes the main results of this paper. For a function fa1 , . . . , a2p−1 , let
f a1 , . . . , a2p−1 |i1 ∼ir f a1 , . . . , a2p−1 |
aij m,1≤j≤r
.
2.1
Lemma 2.1. Let a1 , . . . , an , b1 , . . . , bn > 0. Then min1≤i≤n {ai /bi } ≤ a1 · · · an /b1 · · · bn ≤ max1≤i≤n {ai /bi }. One equality in the chain holds if and only if a1 /b1 · · · an /bn .
For p ≥ 3 and w > 0, define a function hw : R 2p−1 → R as follows:
hw a1 , . . . , a2p−1
2p−1
2p−1 ip1 ai × ip1 1/ai
.
p 2p−1
i1 ai ×
i1 1/ai w
ip1 ai
p
1 w
p
i1 ai
2.2
Below are two examples of this function:
hw a1 , . . . , a5 1 wa1 a2 a3 a4 a5
,
a1 a2 a1 a3 a2 a3 wa4 a5
1 wa1 a2 a3 a4 a5 a6 a5 a7 a6 a7
hw a1 , . . . , a7 .
a1 a2 a3 a1 a2 a4 a1 a3 a4 a2 a3 a4 wa5 a6 a7
2.3
For brevity, let hw hw a1 , . . . , a2p−1 . Note that, for each ar , hw is linear fractional in ar .
As a consequence, hw is monotone in ar . Through simple calculations, we get the following
two lemmas.
Xiaofan Yang et al.
3
Lemma 2.2. Let p ≥ 3, a1 , . . . , a2p−1 > 0, m min1≤i≤2p−1 {ai }, 1 ≤ r ≤ p.
p
1 If hp−2 is increasing in ar , then hp−2 ≤ p − 1/ i1,i/r 1/ai . The equality holds if and only
if hp−2 is constant in ar .
2 If hp−2 is strictly decreasing in ar , then hp−2 ≤ hp−2 |ar m . The equality holds if and only if
ar m.
Lemma 2.3. Let p ≥ 3, a1 , . . . , a2p−1 > 0, m min1≤i≤2p−1 {ai }, p 1 ≤ r ≤ 2p − 1.
2p−1
1 If hp−2 is increasing in ar , then hp−2 ≤ ip1,i/r 1/ai /p − 2. The equality holds if and
only if hp−2 is constant in ar .
2 If hp−2 is strictly decreasing in ar , then hp−2 ≤ hp−2 |ar m . The equality holds if and only if
ar m.
Theorem 2.4. Let p ≥ 3, a1 , . . . , a2p−1 > 0. Then min1≤i≤2p−1 {ai , 1/ai } ≤
max1≤i≤2p−1 {ai , 1/ai }. One of the two equalities holds if and only if a1 · · · a2p−1 1.
hp−2
≤
Proof. Let m min1≤i≤2p−1 {ai }, M max1≤i≤2p−1 {ai }.
We prove only hp−2 ≤ max{M, 1/m} because min{M, 1/m} ≤ hp−2 can be proved similarly. We proceed by distinguishing two possible cases.
Case 1. There is a permutation i1 , . . . , i2p−1 of 1, 2, . . . , 2p − 1 such that for each 1 ≤ k ≤ 2p − 1,
either aik m or hp−2 |i1 ∼ik−1 is strictly decreasing in aik . Then
1
1
1
1
m
≤ max m,
≤ max M,
.
2.4
hp−2 ≤ hp−2 |i1 ≤ · · · ≤ hp−2 |i1 ∼i2p−1 2
m
m
m
Case 2. There is a partial permutation i1 , . . . , ir of 1, 2, . . . , 2p − 1 1 ≤ r ≤ 2p − 2 such that a
for each 1 ≤ k ≤ r, either aik m or hp−2 |i1 ∼ik−1 is strictly decreasing in aik , and b for each
t ∈ {1, . . . , 2p − 1} − {i1 , . . . , ir }, ait /
m and hp−2 |i1 ∼ir is increasing in at . Then
hp−2 ≤ hp−2 |i1 ≤ hp−2 |i1 ∼i2 ≤ · · · ≤ hp−2 |i1 ∼ir .
m < M,
2.5
Since r ≤ 2p − 2, there is t ∈ {1, . . . , 2p − 1} − {i1 , . . . , ir }. If t ∈ {1, . . . , p} − {i1 , . . . , ir }, it
follows from 2.5 and Lemma 2.2 that
p − 1
1
hp−2 ≤ hp−2 |i1 ∼ir ≤ p
.
2.6
≤ max ai |i1 ∼ir ≤ M ≤ max M,
1≤i≤p,i/
t
m
i1,it 1/ai |i1 ∼ir
/
Whereas if t ∈ {p 1, . . . , 2p − 1} − {i1 , . . . , ir }, it follows from 2.5 and Lemma 2.3 that
2p−1
hp−2 ≤ hp−2 |i1 ∼ir ≤
ip1,i/
t
1/ai
p − 2|i1 ∼ir
≤
max
p1≤i≤2p−1,i/
t
1
1
1
.
|i1 ∼ir ≤
≤ max M,
ai
m
m
2.7
Hence, hp−2 ≤ max{M, 1/m} is proven.
Second, we prove that a1 · · · a2p−1 1 if hp−2 max{M, 1/m}. The claim of “a1 · · · a2p−1 1 if hp−2 min{M, 1/m}” can be treated similarly. To this end, we need to prove
the following.
Claim 1. If hp−2 max{M, 1/m}, then there is a permutation i1 , . . . , i2p−1 of 1, . . . , 2p − 1 such
that for each 1 ≤ k ≤ 2p − 1, either aik m or hp−2 |i1 ∼ik−1 is strictly decreasing in aik .
4
Discrete Dynamics in Nature and Society
Proof of Claim 1. On the contrary, assume that Claim 1 is not true. Then there is a partial permutation i1 , . . . , ir of 1, 2, . . . , 2p − 1 1 ≤ r ≤ 2p − 2 such that a for each 1 ≤ k ≤ r, either aik m
m
or hp−2 |i1 ∼ik−1 is strictly decreasing in aik , and b for each t ∈ {1, . . . , 2p − 1} − {i1 , . . . , ir }, ait /
and hp−2 |i1 ∼ir is increasing in at . One of the following two cases must occur.
Case 1. There is t ∈ {1, . . . , 2p − 1} − {i1 , . . . , ir } such that hp−2 |i1 ∼ir is strictly increasing in at . If
t ∈ {1, . . . , p} − {i1 , . . . , ir }, it follows by 2.5, 2.6, and Lemma 2.2 that
p − 1
i1,i/
t 1/ai |
hp−2 ≤ hp−2 |i1 ∼ir < p
i1 ∼ir
1
.
≤ max ai |i1 ∼ir ≤ max M,
1≤i≤p,i
m
/t
2.8
A contradiction occurs. Whereas if t ∈ {p 1, . . . , 2p − 1} − {i1 , . . . , ir }, it follows by 2.5, 2.7,
and Lemma 2.3 that
2p−1
hp−2 ≤ hp−2 |i1 ∼ir <
ip1,i/
t
1/ai
≤
p − 2|i1 ∼ir
max
p1≤i≤2p−1,i/
t
1
1
.
|i1 ∼ir ≤ max M,
ai
m
2.9
Again a contradiction occurs.
Case 2. For each t ∈ {1, . . . , 2p − 1} − {i1 , . . . , ir }, hp−2 |i1 ∼ir is constant in at .
First, let us show that {1, . . . , p} ⊆ {i1 , . . . , ir }. Otherwise, there is t ∈ {1, . . . , p} − {i1 ,
. . . , ir }. By Lemma 2.2, we have
p − 1
1/ai |
i1,i
/t
hp−2 |i1 ∼ir p
.
2.10
i1 ∼ir
If there is s ∈ {1, . . . , p} − {i1 , . . . , ir , t}, it follows from 2.10 that hp−2 |i1 ∼ir is strictly increasing
in as , a contradiction occurs. So, {1, . . . , p} − {i1 , . . . , ir } {t} and thus
1
max M,
hp−2 ≤ hp−2 |i1 ∼ir hp−2 a1 , . . . , a2p−1 |ai ···ai m m < M,
r
1
m
2.11
from which a contradiction follows. So, {1, . . . , p} ⊆ {i1 , . . . , ir }.
According to the previous argument, there is t ∈ {p 1, . . . , 2p − 1} − {i1 , . . . , ir }. By
Lemma 2.3, we get
2p−1
hp−2 |i1 ∼ir ip1,i
/t
1/ai
p − 2|i1 ∼ir
.
2.12
If there is s ∈ {p 1, . . . , 2p − 1} − {i1 , . . . , ir , t}, it follows from 2.12 that hp−2 |i1 ∼ir is strictly
decreasing in as , a contradiction. So, {p 1, . . . , 2p − 1} − {i1 , . . . , ir } {t} and thus
a1 · · · at−1 at1 · · · a2p−1 m.
2.13
By 2.13 and 2.2, we get
hp−2 hp−2 |i1 ∼ir p − 1m3 m p − 2at
.
pm2 p − 2mat
2.14
Xiaofan Yang et al.
Since hp−2 |i1 ∼ir is constant in at , and d/dat hp−2 |i1 ∼ir
5
p − 1p − 2m2 1 − m2 /
pm2 p − 2mat 2 , we derive m 1. From 2.12 and 2.13, we get hp−2 |i1 ∼ir 1/m. Since
hp−2 max{M, 1/m}, all equalities in chains 2.5 and 2.7 hold. These plus m 1 yield
hp−2 |i1 ∼ir 1/m 1 ≥ M, from which we derive M m 1. So, at 1 m. This is a
contradiction. Claim 1 is proved.
By Claim 1 and hp−2 max{M, 1/m}, all equalities in 2.4 must hold. This plus
Lemma 2.2 yields a1 · · · a2p−1 m and hp−2 m, . . . , m m 1/m/2 m. This implies
a1 · · · a2p−1 1.
Theorem 2.5. Let p ≥ 3, a1 , . . . , a2p−1 > 0. Then, min1≤i≤2p−1 {ai , 1/ai } ≤ hp−1 ≤
max1≤i≤2p−1 {ai , 1/ai }. One of the two equalities holds if and only if a1 · · · ap 1/
ap1 · · · 1/a2p−1 .
Proof. By Lemma 2.1 and 2.2, we get
1
1
1
hp−1 ≤ max a1 , . . . , ap ,
≤ max ai ,
,
,...,
1≤i≤2p−1
ap1
a2p−1
ai
1
1
1
≥ min ai ,
.
,...,
hp−1 ≥ min a1 , . . . , ap ,
1≤i≤2p−1
ap1
a2p−1
ai
2.15
The second claim follows immediately from Lemma 2.1.
We are ready to present the main result of this paper.
Theorem 2.6. Let p ≥ 3, p − 2 ≤ w ≤ p − 1, a1 , . . . , a2p−1 > 0. Let
ak hw ak−2p1 , . . . , ak−1 , k 2p, 2p 1, . . . .
2.16
Then min1≤i≤2p−1 {ai , 1/ai } ≤ ak ≤ max1≤i≤2p−1 {ai , 1/ai }, k 2p, 2p 1, . . .. If k ≥ 2p 1, then one
of the two equalities holds if and only if a1 · · · a2p−1 1.
Proof. Regard hw as a linear fractional function in w, which is monotone in w. By Theorems 2.4
and 2.5, we obtain
1
,
a2p ≥ min hp−2 a1 , . . . , a2p−1 , hp−1 a1 , . . . , a2p−1 ≥ min ai ,
1≤i≤2p−1
ai
1
a2p ≤ max hp−2 a1 , . . . , a2p−1 , hp−1 a1 , . . . , a2p−1 ≤ max ai ,
,
1≤i≤2p−1
ai
2.17
1
1
≥ min ai ,
,
a2p1 ≥ min hp−2 a2 , . . . , a2p , hp−1 a2 , . . . , a2p ≥ min ai ,
2≤i≤2p
1≤i≤2p−1
ai
ai
1
1
≤ max ai ,
.
a2p1 ≤ max hp−2 a2 , . . . , a2p , hp−1 a2 , . . . , a2p ≤ max ai ,
2≤i≤2p
1≤i≤2p−1
ai
ai
Working inductively, we conclude that for k 2p, 2p 1, . . .,
1
,
1≤i≤2p−1
ai
1
.
ak ≤ max{hp−2 ak−2p1 , . . . , ak−1 , hp−1 ak−2p1 , . . . , ak−1 } ≤ max ai ,
1≤i≤2p−1
ai
ak ≥ min hp−2 ak−2p1 , . . . , ak−1 , hp−1 ak−2p1 , . . . , ak−1 ≥ min
ai ,
2.18
2.19
6
Discrete Dynamics in Nature and Society
Claim 2. If a2p1 max {ai , 1/ai }, then a1 · · · a2p−1 1.
1≤i≤2p−1
Proof of Claim 2. By 2.19, we get
a2p1
1
ai ,
.
1≤i≤2p−1
ai
max hp−2 a2 , . . . , a2p , hp−1 a2 , . . . , a2p max
2.20
Here, we encounter two possible cases.
Case 1. a2p1 hp−2 a2 , . . . , a2p max1≤i≤2p−1 {ai , 1/ai }. By Theorem 2.4, we get a2 · · · a2p 1 and, hence, a2p1 1. Then 1 a2p1 max1≤i≤2p−1 {ai , 1/ai } max{a1 ,
1/a1 }, implying a1 1.
Case 2. a2p1 hp−1 a2 , . . . , a2p max1≤i≤2p−1 {ai , 1/ai }. By Theorem 2.5, we get
a2 · · · ap1 1
1
··· ,
ap2
a2p
2.21
and consequently,
a2p1 hp−1 a2 , . . . , a2p a2 .
Then,
max
1≤i≤2p−1
ai ,
1
ai
2.22
1
1
≤
a2p min hp−2 a1 , . . . , a2p−1 , hp−1 a1 , . . . , a2p−1
1
.
≤ max ai ,
1≤i≤2p−1
ai
a2p1 Hence, all equalities in this chain hold. In particular, we have
min hp−2 a1 , . . . , a2p−1 , hp−1 a1 , . . . , a2p−1 min
1≤i≤2p−1
1
ai ,
.
ai
2.23
2.24
If hp−2 a1 , . . . , a2p−1 min1≤i≤2p−1 {ai , 1/ai }, it follows from Theorem 2.4 that a1 · · · a2p−1 1. Now, assume that hp−1 a1 , . . . , a2p−1 min1≤i≤2p−1 {ai , 1/ai }. By Theorem 2.5, we
get
a1 · · · ap 1
1
··· .
ap1
a2p−1
2.25
Equations 2.21 and 2.25 imply that a1 · · · a2p−1 1. Claim 2 is proven.
By Claim 2 and working inductively, we get that if ak max1≤i≤2p−1 {ai , 1/ai } for some
k ≥ 2p 1, then a1 · · · a2p−1 1.
Similarly, we can show that a1 · · · a2p−1 1 if ak min1≤i≤2p−1 {ai , 1/ai } holds for
some k ≥ 2p 1.
As an application of Theorem 2.6, we have the following theorem.
Theorem 2.7. Let p ≥ 3, p − 2 ≤ w ≤ p − 1. The difference equation
xn hw xn−2p1 , . . . , xn−1 , n 1, 2, ...,
with positive initial conditions admits the globally asymptotically stable equilibrium c = 1.
The proof of this theorem is similar to those in 11, 13, and hence is omitted.
2.26
Xiaofan Yang et al.
7
Acknowledgments
The authors are grateful to the anonymous referees for their valuable comments and suggestions. This work is supported by Natural Science Foundation of China 10771227, Program
for New Century Excellent Talent of Educational Ministry of China NCET-05-0759, Doctorate
Foundation of Educational Ministry of China 20050611001, and Natural Science Foundation
of Chongqing CSTC 2006BB2231.
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