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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 534974, 16 pages
http://dx.doi.org/10.1155/2011/534974
Research Article

Part-Metric and Its Applications to Cyclic Discrete Dynamic Systems

1College of Computer Science, Chongqing University, Chongqing 400044, China
2Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36/III, 11000 Beograd, Serbia
3Faculty of Electrical Engineering, Bulevar Kralja Aleksandra 73, 11120 Beograd, Serbia

Received 10 November 2010; Accepted 6 March 2011

Academic Editor: Allan C. Peterson

Copyright © 2011 Wanping Liu 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.

Abstract

We adapt the part metric and use it in studying positive solutions of a certain family of discrete dynamic systems. Some examples are presented, and we also compare some results in the literature.

1. Introduction

There has been an increasing interest in studying discrete dynamic systems recently (see, e.g., [137]). For some recent papers on the systems of difference equations which are not derived from differential equations, see, for example, [4, 12, 14, 15], and the related references therein. In particular, in [4], were considered some cyclic systems of difference equations for the first time. Motivated by [4], in [12], the global attractivity of four 𝑘-dimensional systems of higher-order difference equations with two or three delays was investigated. The results in [12] can be easily extended to the corresponding systems with arbitrary number of delays by using the main results in [28].

In [9], the authors used Thompson's part-metric [32] to investigate the behaviour of positive solutions to a difference equation from the William Lowell Putman Mathematical Competition [33] by applying a result on discrete dynamic systems in finite dimensional complete metric spaces. Further investigations devoted to applying various part-metric-related inequalities and some asymptotic methods in order to study (scalar) difference equations related to the equation in [33] can be found, for example, in [1, 3, 5, 1822, 3437] (see also the related references therein).

In this paper, we adapt the part-metric and apply it in studying of the behaviour of positive solutions to the following family of discrete dynamic systems𝑌𝑛=Φ𝑌𝑛𝑘1,𝑌𝑛𝑘2,,𝑌𝑛𝑘𝑞,𝑛0,(1.1) where 𝑞{1}, 1𝑘1<𝑘2<<𝑘𝑞, 𝑘1,,𝑘𝑞, 𝑌𝑛=(𝑦(1)𝑛,𝑦(2)𝑛,,𝑦(𝑞)𝑛)𝑇, 𝑌𝑘𝑞,𝑌𝑘𝑞+1,,𝑌1 are positive initial vectors and Φ𝑞×𝑞+𝑞+ is a continuous mapping which will be specified later.

In Section 2, we present some preliminary results which will be applied in the proofs of main results, given in Section 3. Some applications of the main result are given in Sections 4 and 5. In Section 6, we show that some recent results follow from a result in [9].

2. Auxiliary Results

Let be the whole set of reals and let +=(0,+). Denote by 𝑛+ the set of all positive 𝑛-dimensional vectors and by 𝑚×𝑛+ the set of all 𝑚×𝑛 matrices with positive entries, that is, 𝑚×𝑛+={(𝑎𝑖𝑗)𝑚×𝑛𝑎𝑖𝑗+}, 𝑚,𝑛.

The following theorem was proved in [9, Theorem 1].

Theorem 2.1. Let (𝑀,𝑑) be a complete metric space, where 𝑑 denotes a metric and 𝑀 is an open subset of 𝑛, and let 𝒯𝑀𝑀 be a continuous mapping with the unique equilibrium 𝑥𝑀. Suppose that for the discrete dynamic system 𝑥𝑛+1=𝒯𝑥𝑛,𝑛0,(2.1) there is a 𝑘 such that for the 𝑘th iterate of 𝒯, the next inequality holds 𝑑𝒯𝑘𝑥,𝑥<𝑑𝑥,𝑥(2.2) for all 𝑥𝑥. Then, 𝑥 is globally asymptotically stable with respect to metric 𝑑.

The part-metric (see [9, 32]) is a metric defined on 𝑛+ by 𝑝(𝑋,𝑌)=log2max1𝑖𝑛𝑥𝑖𝑦𝑖,𝑦𝑖𝑥𝑖,(2.3) for arbitrary vectors 𝑋=(𝑥1,𝑥2,,𝑥𝑛)𝑇𝑛+ and 𝑌=(𝑦1,𝑦2,,𝑦𝑛)𝑇𝑛+.

Recall that the part-metric 𝑝 has the following properties [9, 10]:(1)𝑝 is a continuous metric on 𝑛+,(2)(𝑛+,𝑝) is a complete metric space,(3)the distances induced by the part-metric and by the Euclidean norm are equivalent on 𝑛+.

Based on these properties and Theorem 2.1, Kruse and Nesemann in [9] obtained the following result.

Lemma 2.2 (see [9, Corollary 2]). Let 𝒯𝑛+𝑛+ be a continuous mapping with a unique equilibrium 𝑥𝑛+. Suppose that for the discrete dynamic system (2.1) there is some 𝑘 such that for the part-metric 𝑝 inequality 𝑝(𝒯𝑘𝑥,𝑥)<𝑝(𝑥,𝑥) holds for all 𝑥𝑥. Then, 𝑥 is globally asymptotically stable.

Our idea is to adapt the part-metric to matrices. For any two matrices with positive entries 𝐀=(𝑎𝑖𝑗)𝑚×𝑛𝑚×𝑛+ and 𝐁=(𝑏𝑖𝑗)𝑚×𝑛𝑚×𝑛+, we define the part-metric in the following natural way:𝒫(𝐀,𝐁)=log2max1𝑖𝑚,1𝑗𝑛𝑎𝑖𝑗𝑏𝑖𝑗,𝑏𝑖𝑗𝑎𝑖𝑗.(2.4) Note that an 𝑚×𝑛 matrix (𝑎𝑖𝑗)𝑚×𝑛 is equivalent to a vector with 𝑚𝑛 elements, such as 𝑎𝑖𝑗𝑚×𝑛𝑎1,1,𝑎2,1,,𝑎𝑚,1,𝑎1,2,𝑎2,2,,𝑎𝑚,2,,𝑎1,𝑛,𝑎2,𝑛,,𝑎𝑚,𝑛𝑇.(2.5) Thus, for the above matrices 𝐀 and 𝐁, we have that𝒫(𝐀,𝐁)=𝑝𝐴𝑇1,𝐴𝑇2,,𝐴𝑇𝑛𝑇,𝐵𝑇1,𝐵𝑇2,,𝐵𝑇𝑛𝑇=max1𝑗𝑛𝑝𝐴𝑗,𝐵𝑗,(2.6) where 𝐴𝑗=(𝑎1𝑗,𝑎2𝑗,,𝑎𝑚𝑗)𝑇, 𝐵𝑗=(𝑏1𝑗,𝑏2𝑗,,𝑏𝑚𝑗)𝑇, 𝑗=1,2,,𝑛.

From this and the above-mentioned properties for the part-metric we have the following:(1)the part-metric 𝒫 is a continuous metric on 𝑚×𝑛+,(2)(𝑚×𝑛+,𝒫) is a complete metric space,(3)the distances induced by the part-metric 𝒫 and the Euclidean norm are equivalent on 𝑚×𝑛+.

From this and by Lemma 2.2, we have that the next result holds.

Theorem 2.3. Let 𝒯𝑚×𝑛+𝑚×𝑛+ be a continuous mapping with the unique equilibrium 𝐂𝑚×𝑛+. Suppose that for the discrete dynamic system 𝐗𝑛+1=𝒯𝐗𝑛,𝑛0,(2.7) there is a 𝑘 such that for metric 𝒫, the inequality 𝒫(𝒯𝑘𝐗,𝐂)<𝒫(𝐗,𝐂) holds for each 𝐗𝐂. Then, 𝐂 is globally asymptotically stable.

Remark 2.4. Note that if we do not assume in Theorem 2.3 that 𝐂 is the unique equilibrium of (2.7), then if 𝐂 is another equilibrium it must be 𝒫𝐂,𝐂=𝒫𝒯𝑘𝐂,𝐂<𝒫𝐂,𝐂,(2.8) which is impossible. Hence, there is only one equilibrium of (2.7).

3. Main Result

Let 𝐘=(𝑌1,𝑌2,,𝑌𝑞) be a square 𝑞×𝑞 matrix, where 𝑌𝑖=(𝑦(1)𝑖,𝑦(2)𝑖,,𝑦(𝑞)𝑖)𝑇, 𝑖=1,2,,𝑞, and Φ is defined by Φ(𝐘)=Φ𝑌1,𝑌2,,𝑌𝑞=𝜙𝑦(1)1,𝑦(2)2,,𝑦(𝑞)𝑞𝜙𝑦(2)1,𝑦(3)2,,𝑦(1)𝑞𝜙𝑦(𝑞)1,𝑦(1)2,,𝑦(𝑞1)𝑞,(3.1) where 𝜙𝑞++ is a continuous mapping. Clearly, Φ is a continuous mapping and our system becomes𝑌𝑛=𝑦(1)𝑛𝑦(2)𝑛𝑦(𝑞)𝑛=𝜙𝑦(1)𝑛𝑘1,𝑦(2)𝑛𝑘2,,𝑦(𝑞)𝑛𝑘𝑞𝜙𝑦(2)𝑛𝑘1,𝑦(3)𝑛𝑘2,,𝑦(1)𝑛𝑘𝑞𝜙𝑦(𝑞)𝑛𝑘1,𝑦(1)𝑛𝑘2,,𝑦(𝑞1)𝑛𝑘𝑞,𝑛0,(3.2) where 𝑞{1}, 1𝑘1<𝑘2<<𝑘𝑞, 𝑘1,𝑘2,,𝑘𝑞.

As an application of Theorem 2.3, we will establish a theorem regarding the global asymptotic stability of cyclic system of difference equations in (3.2), as follows.

Theorem 3.1. Consider system (3.2), where 𝜙𝑞++,𝑞2 is a continuous mapping. Let 𝐶=(𝑐,𝑐,,𝑐)𝑇, 𝑐>0, be an equilibrium of (3.2). If for (𝑥1,𝑥2,,𝑥𝑞)𝑇𝐶, min1𝑖𝑞𝑥𝑖,𝑐2𝑥𝑖<𝜙𝑥1,𝑥2,,𝑥𝑞<max1𝑖𝑞𝑥𝑖,𝑐2𝑥𝑖,(3.3) then 𝐶 is globally asymptotically stable.

Proof. Define a matrix mapping 𝒯𝑞×𝑘𝑞+𝑞×𝑘𝑞+ such that 𝒯𝑋1,𝑋2,,𝑋𝑘𝑞=Φ𝑋𝑘1,𝑋𝑘2,,𝑋𝑘𝑞,𝑋1,𝑋2,,𝑋𝑘𝑞1,(3.4) where 𝑋𝑖=(𝑥1𝑖,𝑥2𝑖,,𝑥𝑞𝑖)𝑇, 𝑖=1,2,,𝑘𝑞. Then, (3.2) can be converted into the first-order recursive 𝑞×𝑘𝑞 matrix equation 𝐌𝑛=𝒯𝐌𝑛1,𝑛,(3.5) with 𝐌0 initial matrix, with positive entries.
Clearly 𝐂=(𝐶,𝐶,,𝐶) is an equilibrium of (3.5).
Let (𝑌𝑛)+𝑛=𝑘𝑞 be an arbitrary positive solution to (3.2), and denote𝐌𝑛=𝑌𝑛1,𝑌𝑛2,,𝑌𝑛𝑘𝑞,𝑛0,(3.6) then we get a matrix sequence (𝐌𝑛)𝑛=0. Apparently, the matrix sequence (𝐌𝑛)𝑛=0 is a solution to (3.5).
When 𝐌0=𝐂, it is clear that 𝐌𝑛=𝐂 holds for 𝑛0. Hence, in what follows, we assume that 𝐌0𝐂.
Let the relation “𝑗” be either “=” or “<” for each 𝑗0. Since𝒯𝐌𝑛=𝐌𝑛+1=𝑌𝑛,𝑌𝑛1,,𝑌𝑛𝑘𝑞+1,𝑛0,(3.7) then 𝑌𝑛=Φ𝑌𝑛𝑘1,𝑌𝑛𝑘2,,𝑌𝑛𝑘𝑞.(3.8) By (3.2) and condition (3.3), we get that for each 𝑗{1,2,,𝑞}𝑦(𝑗)𝑛𝑐=𝜙𝑦(𝑗)𝑛𝑘1,,𝑦(𝜃(𝑗+𝑞1))𝑛𝑘𝑞𝑐𝑗max1𝑖𝑞𝑦(𝜃(𝑗+𝑖1))𝑛𝑘𝑖𝑐,𝑐𝑦(𝜃(𝑗+𝑖1))𝑛𝑘𝑖,𝑐𝑦(𝑗)𝑛=𝑐𝜙𝑦(𝑗)𝑛𝑘1,,𝑦(𝜃(𝑗+𝑞1))𝑛𝑘𝑞𝑗max1𝑖𝑞𝑦(𝜃(𝑗+𝑖1))𝑛𝑘𝑖𝑐,𝑐𝑦(𝜃(𝑗+𝑖1))𝑛𝑘𝑖,(3.9) where 𝜃(𝑛)𝑛(mod𝑞) with 𝜃(𝑞)=𝑞.
Let 𝐁𝑛=(𝑌𝑛𝑘1,𝑌𝑛𝑘2,,𝑌𝑛𝑘𝑞).
Case 1. If 𝐁𝑛(𝐶,𝐶,,𝐶), then there exists at least one 𝑗{1,2,,𝑞} such that the relation “𝑗” in (3.9) is “<”. Thus, 𝑝𝑌𝑛,𝐶<max𝑝𝑌𝑛𝑘1,𝐶,𝑝𝑌𝑛𝑘2,𝐶,,𝑝𝑌𝑛𝑘𝑞,𝐶,𝑛0.(3.10)Case 2. If 𝐁𝑛=(𝐶,𝐶,,𝐶), then 𝑌𝑛=𝐶 and “𝑗” is always “=” for each 𝑗{1,2,,𝑞}, which implies that 𝑝𝑌𝑛,𝐶=max𝑝𝑌𝑛𝑘1,𝐶,𝑝𝑌𝑛𝑘2,𝐶,,𝑝𝑌𝑛𝑘𝑞,𝐶=0,𝑛0.(3.11)From relations (3.10) and (3.11), we obtain that 𝑝𝑌𝑛,𝐶𝑛max𝑝𝑌𝑛𝑘1,𝐶,𝑝𝑌𝑛𝑘2,𝐶,,𝑝𝑌𝑛𝑘𝑞,𝐶,𝑛0.(3.12) From the following set of inequalities 𝑝𝑌0,𝐶0max𝑝𝑌𝑘1,𝐶,𝑝𝑌𝑘2,𝐶,,𝑝𝑌𝑘𝑞,𝐶,𝑝𝑌1,𝐶1max𝑝𝑌1𝑘1,𝐶,𝑝𝑌1𝑘2,𝐶,,𝑝𝑌1𝑘𝑞,𝐶,𝑝𝑌𝑘𝑞1,𝐶𝑘𝑞1max𝑝𝑌𝑘𝑞1𝑘1,𝐶,𝑝𝑌𝑘𝑞1𝑘2,𝐶,,𝑝𝑌𝑘𝑞1𝑘𝑞,𝐶,(3.13) and since 𝐌0𝐂, it follows that there exists at least one index 𝑗{0,1,,𝑘𝑞1} such that the relation “𝑗” is “<”, which implies max0𝑖𝑘𝑞1𝑝𝑌𝑖,𝐶<max𝑘𝑞𝑖1𝑝𝑌𝑖,𝐶.(3.14) From the definition of the part-metric, we have that 𝒫𝐌𝑛,𝐂=log2max1𝑖𝑘𝑞,1𝑗𝑞𝑐𝑦(𝑗)𝑛𝑖,𝑦(𝑗)𝑛𝑖𝑐=𝒫𝑌𝑛1,,𝑌𝑛𝑘𝑞,𝐂.(3.15) Then, we derive 𝒫𝑇𝑘𝑞𝐌0,𝐂=𝒫𝐌𝑘𝑞,𝐂=𝒫𝑌𝑘𝑞1,𝑌𝑘𝑞2,,𝑌𝑘𝑞𝑘𝑞,𝐂=max0𝑖𝑘𝑞1𝑝𝑌𝑖,𝐶<max𝑘𝑞𝑖1𝑝𝑌𝑖,𝐶=𝒫𝐌0,𝐂.(3.16) Because 𝐌0 is arbitrary and 𝐌0𝐂, then by Theorem 2.3 (see also Remark 2.4) we have that 𝐂 is a globally asymptotically stable equilibrium of (3.5), which implies that the equilibrium 𝐶 of system (3.2) is globally asymptotically stable, as desired.

4. On Some Symmetric Discrete Dynamic Systems

For the sake of convenience, first we define two continuous mappings 𝑓,𝑔𝑞++, 𝑞2, as follows:𝑓𝑥1,𝑥2,,𝑥𝑞=𝑞𝑗=1𝑥𝑟𝑗+1𝑞𝑗=1𝑥𝑟𝑗1𝑞𝑗=1𝑥𝑟𝑗+1+𝑞𝑗=1𝑥𝑟𝑗1,𝑔𝑥1,𝑥2,,𝑥𝑞=𝑞𝑗=1𝑥𝑟𝑗+1+𝑞𝑗=1𝑥𝑟𝑗1𝑞𝑗=1𝑥𝑟𝑗+1𝑞𝑗=1𝑥𝑟𝑗1,(4.1) where 𝑟 is a real parameter belonging to the interval (0,1].

Many researchers have studied the symmetric difference equation 𝑥𝑛=𝜙𝑥𝑛𝑘1,𝑥𝑛𝑘2,,𝑥𝑛𝑘𝑞,𝑛0,(4.2) where 𝑞{1}, 1𝑘1<𝑘2<<𝑘𝑞, 𝑘1,𝑘2,,𝑘𝑞, and 𝜙{𝑓,𝑔}.

In the following, we mainly investigate the behaviour of positive solutions to the following class of cyclic difference equation systems𝑦(1)𝑛𝑦(2)𝑛𝑦(𝑞)𝑛=𝜙𝑦(1)𝑛𝑘1,𝑦(2)𝑛𝑘2,,𝑦(𝑞)𝑛𝑘𝑞𝜙𝑦(2)𝑛𝑘1,𝑦(3)𝑛𝑘2,,𝑦(1)𝑛𝑘𝑞𝜙𝑦(𝑞)𝑛𝑘1,𝑦(1)𝑛𝑘2,,𝑦(𝑞1)𝑛𝑘𝑞,𝑛0,(4.3) where 𝑞{1},1𝑘1<𝑘2<<𝑘𝑞, 𝑘1,𝑘2,,𝑘𝑞, and 𝜙{𝑓,𝑔}.

In order to establish the main result concerning (4.3), we need some preliminary lemmas.

Lemma 4.1. System (4.3) has unique positive equilibrium (1,1,,1𝑞)𝑇.

Proof. Let (𝑐1,𝑐2,,𝑐𝑞)𝑇 be an arbitrary positive equilibrium of system (4.3). Since the mappings 𝑓 and 𝑔 are both symmetric, then we derive that 𝑐𝑖=𝜙𝑐1,𝑐2,,𝑐𝑞,𝑖=1,2,,𝑞,(4.4) from which it follows that 𝑐𝑖=𝑐>0, 𝑖=1,2,,𝑞, and then 𝑐=𝜙(𝑐,𝑐,,𝑐).(4.5) By Lemma 2.1 in [13], we obtain 𝑐=1, as desired.

Lemma 4.2. Let 𝑎1 and 𝑎2 be positive real numbers with (𝑎1,𝑎2)(1,1), and 𝜙{𝑓,𝑔}. Then, min𝑎1,𝑎2,1𝑎1,1𝑎2<𝜙𝑎1,𝑎2<max𝑎1,𝑎2,1𝑎1,1𝑎2.(4.6)

Proof. From the next identities 𝑥𝑥+𝑦1+𝑥𝑦=𝑦𝑥211+𝑥𝑦,𝑦𝑥+𝑦1+𝑥𝑦=𝑥𝑦211+𝑥𝑦,1𝑥𝑥+𝑦1+𝑥𝑦=1𝑥2𝑥(1+𝑥𝑦),1𝑦𝑥+𝑦1+𝑥𝑦=1𝑦2𝑦(1+𝑥𝑦),(4.7) it is easy to see that when (𝑎1,𝑎2)(1,1) the following inequalities hold: min𝑎1,𝑎2,1𝑎1,1𝑎2<𝑎1+𝑎21+𝑎1𝑎2<max𝑎1,𝑎2,1𝑎1,1𝑎2.(4.8) Because 𝑟(0,1], then for the case 𝜙=𝑓, we easily obtain that min𝑎1,𝑎2,1𝑎1,1𝑎2min𝑎𝑟1,𝑎𝑟2,1𝑎𝑟1,1𝑎𝑟2<𝑓𝑎1,𝑎2=𝑎𝑟1+𝑎𝑟21+𝑎𝑟1𝑎𝑟2<max𝑎𝑟1,𝑎𝑟2,1𝑎𝑟1,1𝑎𝑟2max𝑎1,𝑎2,1𝑎1,1𝑎2.(4.9) The case 𝜙=𝑔 follows immediately from the case 𝜙=𝑓 due to the fact that 𝑓𝑔1.

Lemma 4.3. Let 𝑞2 be an integer and 𝜙{𝑓,𝑔}. Let 𝑎1,𝑎2,,𝑎𝑞 be positive real numbers with (𝑎1,𝑎2,,𝑎𝑞)(1,1,,1). Then, min1𝑖𝑞𝑎𝑖,1𝑎𝑖<𝜙𝑎1,𝑎2,,𝑎𝑞<max1𝑖𝑞𝑎𝑖,1𝑎𝑖.(4.10)

Proof. For the case 𝑞=2, the assertion follows from Lemma 4.2. Next, we argue by the induction and assume that the assertion is true for 𝑞=𝑘(𝑘2). Then, it suffices to prove that the assertion holds when 𝑞=𝑘+1. Now, let 𝑎1,𝑎2,,𝑎𝑘+1 be positive real numbers with (𝑎1,𝑎2,,𝑎𝑘+1)(1,1,,1). Consider the following function in a variable 𝑥𝑥;𝑎1,𝑎2,,𝑎𝑘=(𝑥𝑟+1)𝑘𝑗=1𝑎𝑟𝑗+1(𝑥𝑟1)𝑘𝑗=1𝑎𝑟𝑗1(𝑥𝑟+1)𝑘𝑗=1𝑎𝑟𝑗+1+(𝑥𝑟1)𝑘𝑗=1𝑎𝑟𝑗1,(4.11) where 𝑎1,𝑎2,,𝑎𝑘 are arbitrary (but fixed) positive numbers. Clearly, 𝑎𝑘+1;𝑎1,𝑎2,,𝑎𝑘=𝑓𝑎1,𝑎2,,𝑎𝑘+1.(4.12) The first derivative of the function regarding the variable 𝑥 is equal to 𝑥;𝑎1,𝑎2,,𝑎𝑘=4𝑟𝑥𝑟1𝑘𝑗=1𝑎2𝑟𝑗1(𝑥𝑟+1)𝑘𝑗=1𝑎𝑟𝑗+1+(𝑥𝑟1)𝑘𝑗=1𝑎𝑟𝑗12.(4.13) In the following, we distinguish three possibilities.Case 1 (𝑘𝑗=1(𝑎2𝑟𝑗1)<0). Then, (𝑥;𝑎1,𝑎2,,𝑎𝑘)>0 holds for all 𝑥>0, which implies that the function (𝑥;𝑎1,𝑎2,,𝑎𝑘) is strictly increasing in variable 𝑥. From this, we obtain 𝑎𝑘+1;𝑎1,𝑎2,,𝑎𝑘<lim𝑥+𝑥;𝑎1,𝑎2,,𝑎𝑘=𝑘𝑗=1𝑎𝑟𝑗+1𝑘𝑗=1𝑎𝑟𝑗1𝑘𝑗=1𝑎𝑟𝑗+1+𝑘𝑗=1𝑎𝑟𝑗1=𝑓𝑎1,𝑎2,,𝑎𝑘max1𝑖𝑘𝑎𝑖,1𝑎𝑖max1𝑖𝑘+1𝑎𝑖,1𝑎𝑖,𝑎𝑘+1;𝑎1,𝑎2,,𝑎𝑘>lim𝑥0+𝑥;𝑎1,𝑎2,,𝑎𝑘=𝑘𝑗=1𝑎𝑟𝑗+1+𝑘𝑗=1𝑎𝑟𝑗1𝑘𝑗=1𝑎𝑟𝑗+1𝑘𝑗=1𝑎𝑟𝑗1=𝑔𝑎1,𝑎2,,𝑎𝑘min1𝑖𝑘𝑎𝑖,1𝑎𝑖min1𝑖𝑘+1𝑎𝑖,1𝑎𝑖.(4.14)Case 2 (𝑘𝑗=1(𝑎2𝑟𝑗1)>0). Then, (𝑥;𝑎1,𝑎2,,𝑎𝑘)<0 holds for all 𝑥>0, implying that the function (𝑥;𝑎1,𝑎2,,𝑎𝑘) is strictly decreasing in variable 𝑥. From this, we obtain that 𝑎𝑘+1;𝑎1,𝑎2,,𝑎𝑘<lim𝑥0+𝑥;𝑎1,𝑎2,,𝑎𝑘=𝑘𝑗=1𝑎𝑟𝑗+1+𝑘𝑗=1𝑎𝑟𝑗1𝑘𝑗=1𝑎𝑟𝑗+1𝑘𝑗=1𝑎𝑟𝑗1=𝑔𝑎1,𝑎2,,𝑎𝑘max1𝑖𝑘𝑎𝑖,1𝑎𝑖max1𝑖𝑘+1𝑎𝑖,1𝑎𝑖,𝑎𝑘+1;𝑎1,𝑎2,,𝑎𝑘>lim𝑥+𝑥;𝑎1,𝑎2,,𝑎𝑘=𝑘𝑗=1𝑎𝑟𝑗+1𝑘𝑗=1𝑎𝑟𝑗1𝑘𝑗=1𝑎𝑟𝑗+1+𝑘𝑗=1𝑎𝑟𝑗1=𝑓𝑎1,𝑎2,,𝑎𝑘min1𝑖𝑘𝑎𝑖,1𝑎𝑖min1𝑖𝑘+1𝑎𝑖,1𝑎𝑖.(4.15)Case 3 (𝑘𝑗=1(𝑎2𝑟𝑗1)=0). This implies (𝑥;𝑎1,𝑎2,,𝑎𝑘)=1 for all 𝑥>0. From this relation and the inspection that the condition (𝑎1,𝑎2,,𝑎𝑘+1)(1,1,,1) implies max1𝑖𝑘+1{𝑎𝑖,1/𝑎𝑖}>1 and min1𝑖𝑘+1{𝑎𝑖,1/𝑎𝑖}<1, we obtain that min1𝑖𝑘+1𝑎𝑖,1𝑎𝑖<𝑎𝑘+1;𝑎1,𝑎2,,𝑎𝑘<max1𝑖𝑘+1𝑎𝑖,1𝑎𝑖.(4.16) Hence, by induction, the assertion immediately holds for 𝜙=𝑓, and then the case 𝜙=𝑔 follows directly from the case 𝜙=𝑓 because 𝑓𝑔1

By Lemmas 4.1 and 4.3 and Theorem 3.1, we obtain the following theorem.

Theorem 4.4. The unique equilibrium of system (4.3) is globally asymptotically stable.

Remark 4.5. Note that the following two systems (particular cases 𝑞=2 and 𝑞=3 of the system (4.3), resp.) 𝑢𝑛=𝜙𝑢𝑛𝑘,𝑣𝑛𝑙,𝑣𝑛=𝜙𝑣𝑛𝑘,𝑢𝑛𝑙,𝑛0,𝑢𝑛=𝜙𝑢𝑛𝑘,𝑣𝑛𝑙,𝑤𝑛𝑚,𝑣𝑛=𝜙𝑣𝑛𝑘,𝑤𝑛𝑙,𝑢𝑛𝑚,𝑤𝑛=𝜙𝑤𝑛𝑘,𝑢𝑛𝑙,𝑣𝑛𝑚,𝑛0,(4.17) where 1𝑘<𝑙<𝑚, 𝜙{𝑓,𝑔}, were studied in [12].

5. On a System of Difference Equations

Let 𝜇𝑞++, 𝑞4 be a continuous mapping defined by𝜇𝑡1,𝑡2,,𝑡𝑞=𝑞2𝑖=1𝑡𝑖+𝑡𝑞1𝑡𝑞𝑡1𝑡2+𝑞𝑖=3𝑡𝑖.(5.1) Then, the following difference equation𝑥𝑛+1=𝜇𝑥𝑛,𝑥𝑛1,𝑥𝑛2,𝑥𝑛3,𝑛0,(5.2) is an extension of the difference equation in [33], which was studied in [9].

First, we consider the next four-dimensional system of difference equations:𝑤𝑛𝑥𝑛𝑦𝑛𝑧𝑛=𝜇𝑤𝑛𝑘1,𝑥𝑛𝑘2,𝑦𝑛𝑘3,𝑧𝑛𝑘4𝜇𝑥𝑛𝑘1,𝑦𝑛𝑘2,𝑧𝑛𝑘3,𝑤𝑛𝑘4𝜇𝑦𝑛𝑘1,𝑧𝑛𝑘2,𝑤𝑛𝑘3,𝑥𝑛𝑘4𝜇𝑧𝑛𝑘1,𝑤𝑛𝑘2,𝑥𝑛𝑘3,𝑦𝑛𝑘4,𝑛0,(5.3) where 1𝑘1<𝑘2<𝑘3<𝑘4, 𝑘𝑖 for 𝑖{1,2,3,4}.

Lemma 5.1. System (5.3) has unique positive equilibrium (1,1,1,1)𝑇.

Proof. Let (𝑎,𝑏,𝑐,𝑑)𝑇 be an arbitrary positive equilibrium of the system (5.3). Then, we get 𝑎=𝑎+𝑏+𝑐𝑑𝑎𝑏+𝑐+𝑑,𝑏=𝑏+𝑐+𝑑𝑎𝑏𝑐+𝑑+𝑎,𝑐=𝑐+𝑑+𝑎𝑏𝑐𝑑+𝑎+𝑏,𝑑=𝑑+𝑎+𝑏𝑐𝑑𝑎+𝑏+𝑐,(5.4) which imply 𝑎𝑐=1,𝑏𝑑=1.(5.5) Applying (5.5), the system in (5.4) is reduced to 𝑎𝑎21𝑏2+𝑎(1𝑎)𝑏+𝑎21=0,𝑏3+𝑎(𝑎1)𝑏2+(𝑎1)𝑏𝑎2=0.(5.6) If 𝑎=1, then it follows from the second identity that 𝑏=1. Now, assume 𝑎+{1}. By solving the first equation in (5.6) with respect to variable 𝑏, we get that the discriminant Δ=𝑎2(1𝑎)24𝑎𝑎212=𝑎(𝑎1)24𝑎27𝑎4<0,(5.7) which implies the first equation in (5.6) has no real roots. This contradicts 𝑏>0. Hence, 𝑎=𝑏=1, which along with (5.5) implies 𝑐=𝑑=1, finishing the proof.

The following lemma follows directly from Lemma 3.3 in [34] or the proof of Lemma 4 in [35].

Lemma 5.2. Let 𝑎1,𝑎2,𝑎3, and 𝑎4 be positive real numbers with (𝑎1,𝑎2,𝑎3,𝑎4)(1,1,1,1). Then, min1𝑖4𝑎𝑖,1𝑎𝑖<𝜇𝑎1,𝑎2,𝑎3,𝑎4<max1𝑖4𝑎𝑖,1𝑎𝑖.(5.8)

By Lemmas 5.1 and 5.2, and Theorem 3.1, we easily derive the next theorem.

Theorem 5.3. Unique positive equilibrium (1,1,1,1)𝑇 of system (5.3) is globally asymptotically stable.

In the following, we consider the next 𝑞-dimensional (𝑞5) generalization of system (5.3)𝑦(1)𝑛𝑦(2)𝑛𝑦(𝑞)𝑛=𝜇𝑦(1)𝑛𝑘1,𝑦(2)𝑛𝑘2,,𝑦(𝑞)𝑛𝑘𝑞𝜇𝑦(2)𝑛𝑘1,𝑦(3)𝑛𝑘2,,𝑦(1)𝑛𝑘𝑞𝜇𝑦(𝑞)𝑛𝑘1,𝑦(1)𝑛𝑘2,,𝑦(𝑞1)𝑛𝑘𝑞,𝑛0,(5.9) where 5𝑞, 1𝑘1<𝑘2<<𝑘𝑞, 𝑘1,𝑘2,,𝑘𝑞.

It is easy to see that (1,1,,1)𝑇 is a positive equilibrium of system (5.9), but it is not so easy to confirm its uniqueness as in the proof of Lemma 5.1. However, we have the following lemma which follows directly from Lemmas 3.4 and 3.5 in [34].

Lemma 5.4. Let 𝑟 be an integer with 𝑟4. Let 𝑎1,𝑎2,,𝑎𝑟 be positive numbers with (𝑎1,𝑎2,,𝑎𝑟)(1,1,,1). Then, min1𝑖𝑟𝑎𝑖,1𝑎𝑖<𝑟2𝑖=1𝑎𝑖+𝑎𝑟1𝑎𝑟𝑎1𝑎2+𝑟𝑖=3𝑎𝑖<max1𝑖𝑟𝑎𝑖,1a𝑖.(5.10)

Hence, we can apply Theorem 3.1 to establish the following theorem.

Theorem 5.5. The positive equilibrium (1,1,,1)𝑇 of system (5.9) is globally asymptotically stable.

6. An Application of a Kruse-Nesseman Result

Numerous papers studied particular cases of (4.2) by using semi-cycle analysis of their solutions. It was shown by Berg and Stević in [1] that this analysis is unnecessarily complicated and useful only for lower-order difference equations. They also described some methods for determining rules of semi-cycles which can be used in many classes of difference equations. On the other hand, it has been noticed in several papers (see, e.g., [18]) that the stability results in many of these papers follow from the following result by Kruse and Nesemann in [9].

Theorem 6.1. Consider the difference equation 𝑦𝑛=𝐹𝑦𝑛1,,𝑦𝑛𝑚,𝑛0,(6.1) where 𝐹𝑚++ is a continuous function with a unique equilibrium 𝑥+. Suppose that there is a 𝑘0 such that for each solution (𝑦𝑛) of (6.1), 𝑦𝑛𝑦𝑛𝑘0𝑦𝑛𝑥2𝑦𝑛𝑘00(6.2) with equality if and only if 𝑦𝑛=𝑥. Then, 𝑥 is globally asymptotically stable.

Motivated by [18], in recent paper [2], Berg and Stević also applied Theorem 6.1 by proving the next result, which covers numerous particular cases appearing in the literature. We formulate the proposition here as a useful information to the reader. Before we formulate it we need some notation. Let 𝑆𝑗={1,2,,𝑗}, 𝑗=1,,𝑘, let 𝑇𝑘𝑟={𝑡1,𝑡2,,𝑡𝑟}𝑆𝑘𝑡1<𝑡2<<𝑡𝑟𝑥𝑡1𝑥𝑡2𝑥𝑡𝑟,(6.3) for 𝑟=0,1,,𝑘, where we define 𝑇𝑘0=1 and 𝑇𝑘1=𝑇𝑘1𝑘=0, and let 𝑇𝑘𝑟=𝑘𝑟=1𝑟odd𝑇𝑘𝑟,𝑇𝑘𝑟=𝑘𝑟=0𝑟even𝑇𝑘𝑟,(6.4) or reversed, 𝑇𝑘𝑟 is the sum over the even, and 𝑇𝑘𝑟 is the sum over the odd 𝑟.

Theorem 6.2. Suppose 𝜒 is a nonnegative continuous function on 𝑘+, 𝑘, and 1𝑖1<𝑖2<<𝑖𝑘. If a sequence (𝑦𝑖) satisfies the difference equation 𝑦𝑛=𝑓𝑦𝑛𝑖1,𝑦𝑛𝑖2,,𝑦𝑛𝑖𝑘𝑔𝑦𝑛𝑖1,𝑦𝑛𝑖2,,𝑦𝑛𝑖𝑘,𝑛0,(6.5) where 𝑓𝑥1,𝑥2,,𝑥𝑘=𝜒+𝑇𝑘𝑟,𝑔𝑥1,𝑥2,,𝑥𝑘=𝜒+𝑇𝑘𝑟,(6.6) with 𝑦𝑖𝑘,,𝑦1+, then it converges to the unique positive equilibrium 1.

Another proof of the previous result, in the case 𝜒=0, can be also find in recent paper [28] by Stević.

Recently Sun and Xi in [31] gave an interesting proof of the following result. At first sight their result looked new and not so closely related to Theorem 6.1. However, we prove here that it is also a consequence of Theorem 6.1.

Theorem 6.3. Let 𝑓𝐶(𝑅𝑘+,𝑅+) and 𝑔𝐶(𝑅𝑙+,𝑅+) with 𝑘,𝑙, 0𝑟1<<𝑟𝑘 and 0𝑚1<<𝑚𝑙 and satisfy the following two conditions: (𝐻1)[𝑓(𝑢1,𝑢2,,𝑢𝑘)]=𝑓(𝑢1,𝑢2,,𝑢𝑘) and [𝑔(𝑢1,𝑢2,,𝑢𝑙)]=𝑔(𝑢1,𝑢2,,𝑢𝑙).(𝐻2)𝑓(𝑢1,𝑢2,,𝑢𝑘)𝑢1. Then, 𝑥=1 is the unique positive equilibrium of the difference equation 𝑥𝑛=𝑓𝑥𝑛𝑟11,,𝑥𝑛𝑟𝑘1𝑔𝑥𝑛𝑚11,,𝑥𝑛𝑚𝑙1+1𝑓𝑥𝑛𝑟11,,𝑥n𝑟𝑘1+𝑔𝑥𝑛𝑚11,,𝑥𝑛𝑚𝑙1,𝑛,(6.7) which is globally asymptotically stable (here 𝑢=max{𝑢,1/𝑢}).

Proof. Let 𝑓𝑛=𝑓𝑥𝑛𝑟11,,𝑥𝑛𝑟𝑘1,𝑔𝑛=𝑔𝑥𝑛𝑚11,,𝑥𝑛𝑚𝑙1.(6.8) We should determine the sign of the product of the next expressions 𝑃𝑛=𝑓𝑛𝑔𝑛+1𝑓𝑛+𝑔𝑛𝑥𝑛𝑟11=1𝑓𝑛+𝑔𝑛𝑓𝑛𝑔𝑛1𝑥𝑛𝑟11𝑓𝑛+1𝑥𝑛𝑟11𝑓𝑛,(6.9)𝑄𝑛=𝑓𝑛𝑔𝑛+1𝑓𝑛+𝑔𝑛1𝑥𝑛𝑟11=1𝑥𝑛𝑟11𝑓𝑛+𝑔𝑛𝑔𝑛𝑥𝑛𝑟11𝑓𝑛1+𝑓𝑛𝑥𝑛𝑟11𝑓𝑛1.(6.10)
From (6.9) and (6.10), we see if we show that 𝑥𝑛𝑟11𝑓𝑛1 and (𝑥𝑛𝑟11/𝑓𝑛)1 have the same sign for 𝑛, then 𝑃𝑛𝑄𝑛 will be nonpositive.
There are four cases to be considered.
Case 1 (𝑥𝑛𝑟111, 𝑓𝑛1). Clearly, in this case, 𝑥𝑛𝑟11𝑓𝑛10. By (𝐻1) and (𝐻2), we have that 1𝑓𝑛=𝑓𝑛=𝑓𝑥𝑛𝑟11,,𝑥𝑛𝑟𝑘1𝑥𝑛𝑟11=𝑥𝑛𝑟11.(6.11) Hence, (𝑥𝑛𝑟11/𝑓𝑛)10 and consequently 𝑥𝑛𝑟11𝑓𝑛1𝑥𝑛𝑟11𝑓𝑛10.(6.12)Case 2 (𝑥𝑛𝑟111, 𝑓𝑛1). Since 1/𝑓𝑛1 we obtain (𝑥𝑛𝑟11/𝑓𝑛)10. On the other hand, by (𝐻1) and (𝐻2) we have 1𝑓𝑛=𝑓𝑛=𝑓𝑥𝑛𝑟11,,𝑥𝑛𝑟𝑘1𝑥𝑛𝑟11=𝑥𝑛𝑟11,(6.13) so that 𝑥𝑛𝑟11𝑓𝑛10. Hence, (6.12), follows in this case.Case 3 (Case 𝑥𝑛𝑟111, 𝑓𝑛1). Then we have that 1/𝑓𝑛1 and consequently (𝑥𝑛𝑟11/𝑓𝑛)10. On the other hand, we have 𝑓𝑛=𝑓𝑛=𝑓𝑥𝑛𝑟11,,𝑥𝑛𝑟𝑘1𝑥𝑛𝑟11=1𝑥𝑛𝑟11,(6.14) so that 𝑥𝑛𝑟11𝑓𝑛10. Hence, (6.12) follows in this case too.Case 4 (Case 𝑥𝑛𝑟111, 𝑓𝑛1). Then 𝑥𝑛𝑟11𝑓𝑛10. On the other hand, we have 1𝑓𝑛=𝑓𝑛=𝑓𝑥𝑛𝑟11,,𝑥𝑛𝑟𝑘1𝑥𝑛𝑟11=1𝑥𝑛𝑟11,(6.15) so that (𝑥𝑛𝑟11/𝑓𝑛)10. Hence, (6.12) also holds in this case. Thus 𝑃𝑛𝑄𝑛0, for every 𝑛.
Assume that 𝑃𝑛𝑄𝑛=0, then, 𝑃𝑛=0 or 𝑄𝑛=0. Using (6.9) or (6.10) along with (6.12) in any of these two cases, we have that𝑓𝑛=1𝑥𝑛𝑟11=𝑥𝑛𝑟11,𝑛.(6.16) Hence, 𝑥𝑛𝑟11=1, 𝑛.
Finally, let 𝑦 be a solution of the equation 0=𝑓𝑦𝑘𝑔𝑦𝑙+1𝑓𝑦𝑘+𝑔𝑦𝑙𝑦=1𝑓𝑦𝑘+𝑔𝑦𝑙𝑓𝑦𝑘𝑔𝑦𝑙1𝑦𝑓𝑦𝑘+1𝑦𝑓𝑦𝑘,(6.17) where 𝑦𝑗=(𝑦,,𝑦) denotes the vector consisting of 𝑗 copies of 𝑦. Then according to the considerations in Cases 14 it follows that 𝑓(𝑦𝑘)=𝑦=1/𝑦, so that 𝑦=1. Hence 𝑦=1 is a unique positive equilibrium of (6.7).
From all above mentioned and by Theorem 6.1, we get the result.

Acknowledgments

This paper is financially supported by the Fundamental Research Funds for the Central Universities (no. CDJXS10180017), the New Century Excellent Talent Project of China (no. NCET-05-0759), the National Natural Science Foundation of China (no. 10771227) and by the Serbian Ministry of Science, project OI 171007, project III 41025 and project III 44006.

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