Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2008 / Article

Research Article | Open Access

Volume 2008 |Article ID 193872 | https://doi.org/10.1155/2008/193872

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
Received28 Sep 2007
Accepted06 Nov 2007
Published25 Feb 2008

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 min1â‰¤ğ‘–â‰¤ğ‘›î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡î‚€ğ‘Žâ‰¤ğ‘“1,…,ğ‘Žğ‘›î‚â‰¤max1â‰¤ğ‘–â‰¤ğ‘›î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡,(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: min1≤𝑖≤4î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡â‰¤ğ‘Ž1+ğ‘Ž2+ğ‘Ž3ğ‘Ž4ğ‘Ž1ğ‘Ž2+ğ‘Ž3+ğ‘Ž4≤max1≤𝑖≤4î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡(see[5]),(1.2)min1â‰¤ğ‘–â‰¤ğ‘˜î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡â‰¤ğ‘Ž1+⋯+ğ‘Žğ‘˜âˆ’2+ğ‘Žğ‘˜âˆ’1ğ‘Žğ‘˜ğ‘Ž1ğ‘Ž2+ğ‘Ž3+⋯+ğ‘Žğ‘˜â‰¤max1â‰¤ğ‘–â‰¤ğ‘˜î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡(see[11]),(1.3)min1≤𝑖≤5î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡â‰¤(1+𝑤)ğ‘Ž1ğ‘Ž2ğ‘Ž3+ğ‘Ž4+ğ‘Ž5ğ‘Ž1ğ‘Ž2+ğ‘Ž1ğ‘Ž3+ğ‘Ž2ğ‘Ž3+ğ‘¤ğ‘Ž4ğ‘Ž5≤max1≤𝑖≤5î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡,1≤𝑤≤2(see[13]).(1.4)min1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡â‰¤â„Žğ‘¤(ğ‘Ž1,…,ğ‘Ž2𝑝−1)≤max1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡,(1.5)â„Žğ‘¤ In this article, we establish the following PMR inequality chain: 𝑝−2≤𝑤≤𝑝−1 where 𝑝=3 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 ğ‘“î‚€ğ‘Ž1,…,ğ‘Ž2𝑝−1||𝑖1âˆ¼ğ‘–ğ‘Ÿî‚€ğ‘Ž=𝑓1,…,ğ‘Ž2𝑝−1||ğ‘Žğ‘–ğ‘—=𝑚,1≤𝑗≤𝑟.(2.1).

2. Main Results

This section establishes the main results of this paper. For a function ğ‘Ž1,…,ğ‘Žğ‘›,𝑏1,…,𝑏𝑛>0, let min1≤𝑖≤𝑛{ğ‘Žğ‘–/𝑏𝑖}≤(ğ‘Ž1+⋯+ğ‘Žğ‘›)/(𝑏1+

Lemma 2.1. Let ⋯+𝑏𝑛)≤max1≤𝑖≤𝑛{ğ‘Žğ‘–/𝑏𝑖}. Then ğ‘Ž1/𝑏1=⋯=ğ‘Žğ‘›/𝑏𝑛𝑝≥3. One equality in the chain holds if and only if 𝑤>0.

For â„Žğ‘¤âˆ¶(ℜ+)2𝑝−1→ℜ+ and â„Žğ‘¤î‚€ğ‘Ž1,…,ğ‘Ž2𝑝−1=∏(1+𝑤)𝑝𝑖=1ğ‘Žğ‘–+∏2𝑝−1𝑖=𝑝+1ğ‘Žğ‘–Ã—âˆ‘2𝑝−1𝑖=𝑝+11/ğ‘Žğ‘–î‚âˆğ‘ğ‘–=1ğ‘Žğ‘–Ã—âˆ‘ğ‘ğ‘–=11/ğ‘Žğ‘–î‚âˆ+𝑤2𝑝−1𝑖=𝑝+1ğ‘Žğ‘–.(2.2), define a function â„Žğ‘¤î‚€ğ‘Ž1,…,ğ‘Ž5=(1+𝑤)ğ‘Ž1ğ‘Ž2ğ‘Ž3+ğ‘Ž4+ğ‘Ž5ğ‘Ž1ğ‘Ž2+ğ‘Ž1ğ‘Ž3+ğ‘Ž2ğ‘Ž3+ğ‘¤ğ‘Ž4ğ‘Ž5,â„Žğ‘¤î‚€ğ‘Ž1,…,ğ‘Ž7=(1+𝑤)ğ‘Ž1ğ‘Ž2ğ‘Ž3ğ‘Ž4+ğ‘Ž5ğ‘Ž6+ğ‘Ž5ğ‘Ž7+ğ‘Ž6ğ‘Ž7ğ‘Ž1ğ‘Ž2ğ‘Ž3+ğ‘Ž1ğ‘Ž2ğ‘Ž4+ğ‘Ž1ğ‘Ž3ğ‘Ž4+ğ‘Ž2ğ‘Ž3ğ‘Ž4+ğ‘¤ğ‘Ž5ğ‘Ž6ğ‘Ž7.(2.3) as follows: â„Žğ‘¤=â„Žğ‘¤(ğ‘Ž1,…,ğ‘Ž2𝑝−1) Below are two examples of this function: ğ‘Žğ‘Ÿ

For brevity, let â„Žğ‘¤. Note that, for each ğ‘Žğ‘Ÿ, â„Žğ‘¤ is linear fractional in ğ‘Žğ‘Ÿ. As a consequence, 𝑝≥3 is monotone in ğ‘Ž1,…,ğ‘Ž2𝑝−1>0. Through simple calculations, we get the following two lemmas.

Lemma 2.2. Let 𝑚=min1≤𝑖≤2𝑝−1{ğ‘Žğ‘–}, 1≤𝑟≤𝑝, â„Žğ‘âˆ’2, â„Žğ‘âˆ’2∑≤(𝑝−1)/𝑝𝑖=1,𝑖≠𝑟(1/ğ‘Žğ‘–).
(1)If â„Žğ‘âˆ’2 is increasing in ar, then ğ‘Žğ‘Ÿ. The equality holds if and only if â„Žğ‘âˆ’2 is constant in ğ‘Žğ‘Ÿ.(2)If â„Žğ‘âˆ’2â‰¤â„Žğ‘âˆ’2|ğ‘Žğ‘Ÿ=𝑚 is strictly decreasing in ğ‘Žğ‘Ÿ=𝑚, then 𝑝≥3. The equality holds if and only if ğ‘Ž1,…,ğ‘Ž2𝑝−1>0.

Lemma 2.3. Let 𝑚=min1≤𝑖≤2𝑝−1{ğ‘Žğ‘–}, 𝑝+1≤𝑟≤2𝑝−1, â„Žğ‘âˆ’2, ğ‘Žğ‘Ÿ.
(1)If â„Žğ‘âˆ’2≤∑2𝑝−1𝑖=𝑝+1,𝑖≠𝑟(1/ğ‘Žğ‘–)/(𝑝−2) is increasing in â„Žğ‘âˆ’2, then ğ‘Žğ‘Ÿ. The equality holds if and only if â„Žğ‘âˆ’2 is constant in ğ‘Žğ‘Ÿ.(2)If â„Žğ‘âˆ’2â‰¤â„Žğ‘âˆ’2|ğ‘Žğ‘Ÿ=𝑚 is strictly decreasing in ğ‘Žğ‘Ÿ=𝑚, then 𝑝≥3. The equality holds if and only if ğ‘Ž1,…,ğ‘Ž2𝑝−1>0.

Theorem 2.4. Let min1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}â‰¤â„Žğ‘âˆ’2≤, max1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}. Then ğ‘Ž1=⋯=ğ‘Ž2𝑝−1=1𝑚=min1≤𝑖≤2𝑝−1{ğ‘Žğ‘–}. One of the two equalities holds if and only if 𝑀=max1≤𝑖≤2𝑝−1{ğ‘Žğ‘–}.

Proof. Let â„Žğ‘âˆ’2≤max{𝑀,1/𝑚}, min{𝑀,1/𝑚}â‰¤â„Žğ‘âˆ’2.
We prove only 𝑖1,…,𝑖2𝑝−1 because 1,2,…,2𝑝−1 can be proved similarly. We proceed by distinguishing two possible cases.
Case 1. There is a permutation 1≤𝑘≤2𝑝−1 of ğ‘Žğ‘–ğ‘˜=𝑚 such that for each â„Žğ‘âˆ’2|𝑖1∼𝑖𝑘−1, either ğ‘Žğ‘–ğ‘˜ or â„Žğ‘âˆ’2â‰¤â„Žğ‘âˆ’2||𝑖1â‰¤â‹¯â‰¤â„Žğ‘âˆ’2||𝑖1∼𝑖2𝑝−1=121𝑚+𝑚1≤max𝑚,𝑚1≤max𝑀,𝑚.(2.4) is strictly decreasing in 𝑖1,…,𝑖𝑟. Then1,2,…,2𝑝−1
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 1≤𝑘≤𝑟 (ğ‘Žğ‘–ğ‘˜=𝑚) such that (a) for each â„Žğ‘âˆ’2|𝑖1∼𝑖𝑘−1, either ğ‘Žğ‘–ğ‘˜ or 𝑡∈{1,…,2𝑝−1}−{𝑖1,…,𝑖𝑟} is strictly decreasing in ğ‘Žğ‘–ğ‘¡â‰ ğ‘š, and (b) for each â„Žğ‘âˆ’2|𝑖1∼𝑖𝑟, ğ‘Žğ‘¡ and 𝑡∈{1,…,2𝑝−1}−{𝑖1,…,𝑖𝑟} is increasing in â„Žğ‘âˆ’2|𝑖1∼𝑖𝑟. One of the following two cases must occur.
Case 1. There is ğ‘Žğ‘¡ such that 𝑡∈{1,…,𝑝}−{𝑖1,…,𝑖𝑟} is strictly increasing in â„Žğ‘âˆ’2â‰¤â„Žğ‘âˆ’2||𝑖1∼𝑖𝑟<(𝑝−1)∑𝑝𝑖=1,𝑖≠𝑡1/ğ‘Žğ‘–î‚||𝑖1∼𝑖𝑟≤max1≤𝑖≤𝑝,ğ‘–â‰ ğ‘¡î‚†ğ‘Žğ‘–î‚‡|𝑖1∼𝑖𝑟1≤max𝑀,𝑚.(2.8). If 𝑡∈{𝑝+1,…,2𝑝−1}−{𝑖1,…,𝑖𝑟}, it follows by (2.5), (2.6), and Lemma 2.2 that â„Žğ‘âˆ’2â‰¤â„Žğ‘âˆ’2||𝑖1∼𝑖𝑟<∑2𝑝−1𝑖=𝑝+1,𝑖≠𝑡1/ğ‘Žğ‘–î‚||(𝑝−2)𝑖1∼𝑖𝑟≤max𝑝+1≤𝑖≤2𝑝−1,𝑖≠𝑡1ğ‘Žğ‘–î‚‡|𝑖1∼𝑖𝑟1≤max𝑀,𝑚.(2.9) A contradiction occurs. Whereas if 𝑡∈{1,…,2𝑝−1}−{𝑖1,…,𝑖𝑟}, it follows by (2.5), (2.7), and Lemma 2.3 that â„Žğ‘âˆ’2|𝑖1∼𝑖𝑟 Again a contradiction occurs.
Case 2. For each ğ‘Žğ‘¡, {1,…,𝑝}⊆{𝑖1,…,𝑖𝑟} is constant in 𝑡∈{1,…,𝑝}−{𝑖1,.
First, let us show that …,𝑖𝑟}. Otherwise, there is â„Žğ‘âˆ’2||𝑖1∼𝑖𝑟=(𝑝−1)∑𝑝𝑖=1,𝑖≠𝑡1/ğ‘Žğ‘–î‚||𝑖1∼𝑖𝑟.(2.10)𝑠∈{1,…,𝑝}−{𝑖1,…,𝑖𝑟,𝑡}. By Lemma 2.2, we have â„Žğ‘âˆ’2|𝑖1∼𝑖𝑟If there is ğ‘Žğ‘ , it follows from (2.10) that {1,…,𝑝}−{𝑖1,…,𝑖𝑟}={𝑡} is strictly increasing in 1max𝑀,𝑚=â„Žğ‘âˆ’2â‰¤â„Žğ‘âˆ’2||𝑖1∼𝑖𝑟=â„Žğ‘âˆ’2(ğ‘Ž1,…,ğ‘Ž2𝑝−1)||ğ‘Žğ‘–1=⋯=ğ‘Žğ‘–ğ‘Ÿ=𝑚=𝑚<𝑀,(2.11), a contradiction occurs. So, {1,…,𝑝}⊆{𝑖1,…,𝑖𝑟} and thus 𝑡∈{𝑝+1,…,2𝑝−1}−{𝑖1,…,𝑖𝑟} from which a contradiction follows. So, â„Žğ‘âˆ’2||𝑖1∼𝑖𝑟=∑2𝑝−1𝑖=𝑝+1,𝑖≠𝑡1/ğ‘Žğ‘–î‚||(𝑝−2)𝑖1∼𝑖𝑟.(2.12).

According to the previous argument, there is 𝑠∈{𝑝+1,…,2𝑝−1}−{𝑖1,…,𝑖𝑟,𝑡}. By Lemma 2.3, we get â„Žğ‘âˆ’2|𝑖1∼𝑖𝑟 If there is ğ‘Žğ‘ , it follows from (2.12) that {𝑝+1,…,2𝑝−1}−{𝑖1,…,𝑖𝑟}={𝑡} is strictly decreasing in ğ‘Ž1=⋯=ğ‘Žğ‘¡âˆ’1=ğ‘Žğ‘¡+1=⋯=ğ‘Ž2𝑝−1=𝑚.(2.13), a contradiction. So, â„Žğ‘âˆ’2=â„Žğ‘âˆ’2||𝑖1∼𝑖𝑟=(𝑝−1)𝑚3+𝑚+(𝑝−2)ğ‘Žğ‘¡ğ‘ğ‘š2+(𝑝−2)ğ‘šğ‘Žğ‘¡.(2.14) and thus â„Žğ‘âˆ’2|𝑖1∼𝑖𝑟 By (2.13) and (2.2), we get ğ‘Žğ‘¡ Since (d/dğ‘Žğ‘¡)â„Žğ‘âˆ’2|𝑖1∼𝑖𝑟=((𝑝−1)(𝑝−2)𝑚2(1−𝑚2))/ is constant in [𝑝𝑚2+(𝑝−2)ğ‘šğ‘Žğ‘¡]2, and 𝑚=1â„Žğ‘âˆ’2|𝑖1∼𝑖𝑟=1/𝑚, we derive â„Žğ‘âˆ’2=max{𝑀,1/𝑚}. From (2.12) and (2.13), we get 𝑚=1. Since â„Žğ‘âˆ’2|𝑖1∼𝑖𝑟=1/𝑚=1≥𝑀, all equalities in chains (2.5) and (2.7) hold. These plus 𝑀=𝑚=1 yield ğ‘Žğ‘¡=1=𝑚, from which we derive â„Žğ‘âˆ’2=max{𝑀,1/𝑚}. So, ğ‘Ž1=⋯=ğ‘Ž2𝑝−1=𝑚. This is a contradiction. Claim 1 is proved.

By Claim 1 and â„Žğ‘âˆ’2(𝑚,…,𝑚)=(𝑚+1/𝑚)/2=𝑚, all equalities in (2.4) must hold. This plus Lemma 2.2 yields ğ‘Ž1=⋯=ğ‘Ž2𝑝−1=1 and 𝑝≥3. This implies ğ‘Ž1,…,ğ‘Ž2𝑝−1>0.

Theorem 2.5. Let min1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}â‰¤â„Žğ‘âˆ’1≤, max1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}. Then, ğ‘Ž1=⋯=ğ‘Žğ‘=1/ğ‘Žğ‘+1=⋯=1/ğ‘Ž2𝑝−1. One of the two equalities holds if and only if â„Žğ‘âˆ’1î‚†ğ‘Žâ‰¤max1,…,ğ‘Žğ‘,1ğ‘Žğ‘+11,…,ğ‘Ž2𝑝−1≤max1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡,â„Žğ‘âˆ’1î‚†ğ‘Žâ‰¥min1,…,ğ‘Žğ‘,1ğ‘Žğ‘+11,…,ğ‘Ž2𝑝−1≥min1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡.(2.15)𝑝≥3.

Proof. By Lemma 2.1 and (2.2), we get 𝑝−2≤𝑤≤𝑝−1 The second claim follows immediately from Lemma 2.1.
We are ready to present the main result of this paper.

Theorem 2.6. Let ğ‘Ž1,…,ğ‘Ž2𝑝−1>0, ğ‘Žğ‘˜=â„Žğ‘¤î‚€ğ‘Žğ‘˜âˆ’2𝑝+1,…,ğ‘Žğ‘˜âˆ’1,𝑘=2𝑝,2𝑝+1,….(2.16), min1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}â‰¤ğ‘Žğ‘˜â‰¤max1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}. Let 𝑘=2𝑝,2𝑝+1,… Then 𝑘≥2𝑝+1, ğ‘Ž1=⋯=ğ‘Ž2𝑝−1=1. If â„Žğ‘¤, then one of the two equalities holds if and only if ğ‘Ž2ğ‘î‚†â„Žâ‰¥min𝑝−2î‚€ğ‘Ž1,…,ğ‘Ž2𝑝−1,â„Žğ‘âˆ’1î‚€ğ‘Ž1,…,ğ‘Ž2𝑝−1≥min1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡,ğ‘Ž2ğ‘î‚†â„Žâ‰¤max𝑝−2î‚€ğ‘Ž1,…,ğ‘Ž2𝑝−1,â„Žğ‘âˆ’1î‚€ğ‘Ž1,…,ğ‘Ž2𝑝−1≤max1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡,ğ‘Ž2𝑝+1ℎ≥min𝑝−2î‚€ğ‘Ž2,…,ğ‘Ž2𝑝,â„Žğ‘âˆ’1î‚€ğ‘Ž2,…,ğ‘Ž2𝑝≥min2≤𝑖≤2ğ‘î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡â‰¥min1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡,ğ‘Ž2𝑝+1ℎ≤max𝑝−2î‚€ğ‘Ž2,…,ğ‘Ž2𝑝,â„Žğ‘âˆ’1î‚€ğ‘Ž2,…,ğ‘Ž2𝑝≤max2≤𝑖≤2ğ‘î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡â‰¤max1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡.(2.17).

Proof. Regard 𝑘=2𝑝 as a linear fractional function in w, which is monotone in w. By Theorems 2.4 and 2.5, we obtain 2𝑝+1,… Working inductively, we conclude that for ğ‘Žğ‘˜î‚†â„Žâ‰¥min𝑝−2î‚€ğ‘Žğ‘˜âˆ’2𝑝+1,…,ğ‘Žğ‘˜âˆ’1,â„Žğ‘âˆ’1î‚€ğ‘Žğ‘˜âˆ’2𝑝+1,…,ğ‘Žğ‘˜âˆ’1≥min1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡ğ‘Ž,(2.18)ğ‘˜î€½â„Žâ‰¤max𝑝−2(ğ‘Žğ‘˜âˆ’2𝑝+1,…,ğ‘Žğ‘˜âˆ’1),â„Žğ‘âˆ’1(ğ‘Žğ‘˜âˆ’2𝑝+1,…,ğ‘Žğ‘˜âˆ’1)≤max1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡.(2.19), ğ‘Ž2𝑝+1=max1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}, ğ‘Ž1=⋯=ğ‘Ž2𝑝−1=1ğ‘Ž2𝑝+1ℎ=max𝑝−2î‚€ğ‘Ž2,…,ğ‘Ž2𝑝,â„Žğ‘âˆ’1î‚€ğ‘Ž2,…,ğ‘Ž2𝑝=max1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡.(2.20)

Claim 2. If ğ‘Ž2𝑝+1=â„Žğ‘âˆ’2(ğ‘Ž2,…,ğ‘Ž2𝑝)=max1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–} , then ğ‘Ž2=⋯=ğ‘Ž2𝑝=1.

Proof of Claim 2. By (2.19), we get ğ‘Ž2𝑝+1=1 Here, we encounter two possible cases. Case 1. 1=ğ‘Ž2𝑝+1=max1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}=max{ğ‘Ž1,. By Theorem 2.4, we get 1/ğ‘Ž1} and, hence, ğ‘Ž1=1. Then ğ‘Ž2𝑝+1=â„Žğ‘âˆ’1(ğ‘Ž2,…,ğ‘Ž2𝑝)=max1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}ğ‘Ž2=⋯=ğ‘Žğ‘+1=1ğ‘Žğ‘+21=⋯=ğ‘Ž2𝑝,(2.21), implying ğ‘Ž2𝑝+1=â„Žğ‘âˆ’1(ğ‘Ž2,…,ğ‘Ž2𝑝)=ğ‘Ž2.(2.22).Case 2. max1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡=ğ‘Ž2𝑝+1=1ğ‘Ž2𝑝≤1ℎmin𝑝−2î‚€ğ‘Ž1,…,ğ‘Ž2𝑝−1,â„Žğ‘âˆ’1î‚€ğ‘Ž1,…,ğ‘Ž2𝑝−1≤max1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡.(2.23). By Theorem 2.5, we get ℎmin𝑝−2î‚€ğ‘Ž1,…,ğ‘Ž2𝑝−1,â„Žğ‘âˆ’1î‚€ğ‘Ž1,…,ğ‘Ž2𝑝−1=min1≤𝑖≤2𝑝−1î‚†ğ‘Žğ‘–,1ğ‘Žğ‘–î‚‡.(2.24) and consequently, â„Žğ‘âˆ’2(ğ‘Ž1,…,ğ‘Ž2𝑝−1)=min1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–} Then, ğ‘Ž1=⋯= Hence, all equalities in this chain hold. In particular, we have ğ‘Ž2𝑝−1=1 If â„Žğ‘âˆ’1(ğ‘Ž1,…,ğ‘Ž2𝑝−1)=min1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}, it follows from Theorem 2.4 that ğ‘Ž1=⋯=ğ‘Žğ‘=1ğ‘Žğ‘+11=⋯=ğ‘Ž2𝑝−1.(2.25)ğ‘Ž1=⋯=ğ‘Ž2𝑝−1=1. Now, assume that ğ‘Žğ‘˜=max1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}. By Theorem 2.5, we get 𝑘≥2𝑝+1 Equations (2.21) and (2.25) imply that ğ‘Ž1=⋯=ğ‘Ž2𝑝−1=1. Claim 2 is proven.

By Claim 2 and working inductively, we get that if ğ‘Ž1=⋯=ğ‘Ž2𝑝−1=1 for some

ğ‘Žğ‘˜=min1≤𝑖≤2𝑝−1{ğ‘Žğ‘–,1/ğ‘Žğ‘–}, then 𝑘≥2𝑝+1.

Similarly, we can show that 𝑝≥3 if 𝑝−2≤𝑤≤𝑝−1 holds for some 𝑥𝑛=â„Žğ‘¤î‚€ğ‘¥ğ‘›âˆ’2𝑝+1,…,𝑥𝑛−1,𝑛=1,2,...,(2.26).

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

Theorem 2.7. Let 𝑦𝑛=𝑦𝑛−𝑘+𝑦𝑛−𝑚/1+𝑦𝑛−𝑘𝑦𝑛−𝑚, . 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).

References

  1. A. C. Thompson, “On certain contraction mappings in a partially ordered vector space,” Proceedings of the American Mathematical Society, vol. 14, no. 3, pp. 438–443, 1963. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. A. M. Amleh, N. Kruse, and G. Ladas, “On a class of difference equations with strong negative feedback,” Journal of Difference Equations and Applications, vol. 5, no. 6, pp. 497–515, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. K. S. Berenhaut, J. D. Foley, and S. Stević, “The global attractivity of the rational difference equation yn=yn−k+yn−m/1+yn−kyn−m,” Applied Mathematics Letters, vol. 20, no. 1, pp. 54–58, 2007. View at: Publisher Site | Google Scholar | MathSciNet
  4. K. S. Berenhaut and S. Stević, “The global attractivity of a higher order rational difference equation,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 940–944, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. N. Kruse and T. Nesemann, “Global asymptotic stability in some discrete dynamical systems,” Journal of Mathematical Analysis and Applications, vol. 235, no. 1, pp. 151–158, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. T. Nesemann, “Positive nonlinear difference equations: some results and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4707–4717, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. G. Papaschinopoulos and C. J. Schinas, “Global asymptotic stability and oscillation of a family of difference equations,” Journal of Mathematical Analysis and Applications, vol. 294, no. 2, pp. 614–620, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. S. Stević, “Global stability and asymptotics of some classes of rational difference equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 60–68, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  9. S. Stević, “Existence of nontrivial solutions of a rational difference equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 28–31, 2007. View at: Google Scholar | MathSciNet
  10. T. Sun and H. Xi, “Global asymptotic stability of a family of difference equations,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 724–728, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. X. Yang, “Global asymptotic stability in a class of generalized Putnam equations,” Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 693–698, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. X. Yang, D. J. Evans, and G. M. Megson, “Global asymptotic stability in a class of Putnam-type equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 1, pp. 42–50, 2006. View at: Publisher Site | Google Scholar | MathSciNet
  13. X. Yang, M. Yang, and H. Liu, “A part-metric-related inequality chain and application to the stability analysis of difference equation,” Journal of Inequalities and Applications, vol. 2007, Article ID 19618, 9 pages, 2007. View at: Publisher Site | Google Scholar | MathSciNet

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.


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