Research Article | Open Access

Xiaofan Yang, Fangkuan Sun, Yuan Yan Tang, "A New Part-Metric-Related Inequality Chain and an Application", *Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 193872, 7 pages, 2008. https://doi.org/10.1155/2008/193872

# A New Part-Metric-Related Inequality Chain and an Application

**Academic Editor:**Stevo Stevic

#### Abstract

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.

#### 1. Introduction

A part-metric related (PMR) inequality chain is a chain of inequalities of the form 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: In this article, we establish the following PMR inequality chain: where will be defined in the next section, . When , chain (1.5) reduces to chain (1.4). On this basis, we prove that the difference equation with positive initial conditions admits a globally asymptotically stable equilibrium .

#### 2. Main Results

This section establishes the main results of this paper. For a function , let

Lemma 2.1. *Let . Then .
One equality in the chain holds if and only if . *

For and , define a function as follows: Below are two examples of this function:

For brevity, let . Note that, for each , is linear fractional in . As a consequence, is monotone in . Through simple calculations, we get the following two lemmas.

Lemma 2.2. *Let , , , . *

(1)*If is
increasing in a _{r}, then .
The equality holds if and only if is constant
in .*(2)

*If is strictly decreasing in , then . The equality holds if and only if .*

Lemma 2.3. *Let , , , . *

(1)*If is
increasing in , then .
The equality holds if and only if is constant
in .*(2)*If is
strictly decreasing in ,
then .
The equality holds if and only if .*

Theorem 2.4. *Let , .
Then .
One of the two equalities holds if and only if .*

*Proof. *Let , .

We prove only because can be proved similarly. We proceed by
distinguishing two possible cases.*Case 1. *There is a permutation of such that for each , either or is strictly decreasing in .
Then*Case 2. *There is a partial permutation of () such that (a) for each ,
either or is strictly decreasing in ,
and (b) for each , and is increasing in .
Then

Since ,
there is .
If ,
it follows from (2.5) and Lemma 2.2 that
Whereas if ,
it follows from (2.5) and Lemma 2.3 that
Hence, is proven.

Second, we prove that if .
The claim of “ if ” can be treated similarly. To this end, we need
to prove the following.

*Claim 1. *If ,
then there is a permutation of such that for each , either or is strictly decreasing in .

*Proof of Claim 1. *On the
contrary, assume that
Claim 1 is not true. Then there is a partial permutation of () such that (a) for each ,
either or is strictly decreasing in ,
and (b) for each , and is increasing in .
One of the following two cases must occur.*Case 1. *There is such that is strictly increasing in . If ,
it follows by (2.5), (2.6), and Lemma 2.2 that
A contradiction occurs. Whereas if ,
it follows by (2.5), (2.7), and Lemma 2.3 that
Again a contradiction occurs.*Case 2. *For each , is constant in .

First, let us show that .
Otherwise, there is .
By Lemma 2.2, we have
If there is ,
it follows from (2.10) that is strictly increasing in ,
a contradiction occurs. So, and thus
from which a contradiction follows.
So, .

According to the previous argument, there is . By Lemma 2.3, we get If there is , it follows from (2.12) that is strictly decreasing in , a contradiction. So, and thus By (2.13) and (2.2), we get Since is constant in , and , we derive . From (2.12) and (2.13), we get . Since , all equalities in chains (2.5) and (2.7) hold. These plus yield , from which we derive . So, . This is a contradiction. Claim 1 is proved.

By Claim 1 and , all equalities in (2.4) must hold. This plus Lemma 2.2 yields and . This implies .

Theorem 2.5. *Let , .
Then, .
One of the two equalities holds if and only if .*

*Proof. *By Lemma 2.1 and (2.2), we get
The second claim follows immediately
from Lemma 2.1.

We are ready to present the main result of this paper.

Theorem 2.6. *Let , , .
Let
**
Then , .
If , then one of the two equalities holds if and only if . *

*Proof. *Regard
as a linear fractional function in *w*,
which is monotone in *w*. By Theorems 2.4
and 2.5, we obtain
Working inductively, we conclude that
for , ,

*Claim 2. *If
,
then .

*Proof of Claim 2. *By (2.19),
we get
Here, we encounter two possible cases.
*Case 1. *.
By Theorem 2.4, we get and,
hence, .
Then ,
implying .*Case 2. *.
By Theorem 2.5, we get
and consequently,
Then,
Hence, all equalities in this chain
hold. In particular, we have
If ,
it follows from Theorem 2.4 that .
Now, assume that .
By Theorem 2.5, we get
Equations (2.21) and (2.25) imply that .
Claim 2 is proven.

By Claim 2 and working inductively, we get that if for some

, then .

Similarly, we can show that if holds for some .

As an application of Theorem 2.6, we have the following theorem.

Theorem 2.7. *Let , . The difference equation
**
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.

#### 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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#### Copyright

Copyright © 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.