Part-Metric and Its Applications to Cyclic Discrete Dynamic Systems

Liu, Wanping; Yang, Xiaofan; Stević, Stevo; Iricanin, Bratislav

doi:https://doi.org/10.1155/2011/534974

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2011 | Article ID 534974 | https://doi.org/10.1155/2011/534974

Part-Metric and Its Applications to Cyclic Discrete Dynamic Systems

Wanping Liu,¹Xiaofan Yang,¹Stevo Stević,²and Bratislav Iricanin³

Academic Editor: Allan C. Peterson

Received10 Nov 2010

Accepted06 Mar 2011

Published09 Jun 2011

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., [1–37]). 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 -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, 18–22, 34–37] (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 systems where , , , , are positive initial vectors and 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 . Denote by the set of all positive -dimensional vectors and by the set of all matrices with positive entries, that is, , .

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

Theorem 2.1. Let be a complete metric space, where denotes a metric and is an open subset of , and let be a continuous mapping with the unique equilibrium . Suppose that for the discrete dynamic system there is a such that for the th iterate of , the next inequality holds for all . Then, is globally asymptotically stable with respect to metric .

The part-metric (see [9, 32]) is a metric defined on by for arbitrary vectors and .

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

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 be a continuous mapping with a unique equilibrium . Suppose that for the discrete dynamic system (2.1) there is some such that for the part-metric inequality holds for all . Then, is globally asymptotically stable.

Our idea is to adapt the part-metric to matrices. For any two matrices with positive entries and , we define the part-metric in the following natural way: Note that an matrix is equivalent to a vector with elements, such as Thus, for the above matrices and , we have that where , , .

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

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

Theorem 2.3. Let be a continuous mapping with the unique equilibrium . Suppose that for the discrete dynamic system there is a such that for metric , the inequality holds for each . Then, is globally asymptotically stable.

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

3. Main Result

Let be a square matrix, where , , and is defined by where is a continuous mapping. Clearly, is a continuous mapping and our system becomes where , , .

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 is a continuous mapping. Let , , be an equilibrium of (3.2). If for , then is globally asymptotically stable.

Proof. Define a matrix mapping such that where , . Then, (3.2) can be converted into the first-order recursive matrix equation with initial matrix, with positive entries.
Clearly is an equilibrium of (3.5).
Let be an arbitrary positive solution to (3.2), and denote then we get a matrix sequence . Apparently, the matrix sequence is a solution to (3.5).
When , it is clear that holds for . Hence, in what follows, we assume that .
Let the relation “” be either “” or “<” for each . Since then By (3.2) and condition (3.3), we get that for each where with .
Let .
Case 1. If , then there exists at least one such that the relation “” in (3.9) is “<”. Thus, Case 2. If , then and “” is always “” for each , which implies that From relations (3.10) and (3.11), we obtain that From the following set of inequalities and since , it follows that there exists at least one index , such that the relation “” is “<”, which implies From the definition of the part-metric, we have that Then, we derive Because is arbitrary and , then by Theorem 2.3 (see also Remark 2.4) we have that is a globally asymptotically stable equilibrium of (3.5), which implies that the equilibrium 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 , , as follows: where is a real parameter belonging to the interval .

Many researchers have studied the symmetric difference equation where , , , and .

In the following, we mainly investigate the behaviour of positive solutions to the following class of cyclic difference equation systems where , , and .

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 .

Proof. Let be an arbitrary positive equilibrium of system (4.3). Since the mappings and are both symmetric, then we derive that from which it follows that , , and then By Lemma 2.1 in [13], we obtain , as desired.

Lemma 4.2. Let and be positive real numbers with , and . Then,

Proof. From the next identities it is easy to see that when the following inequalities hold: Because , then for the case , we easily obtain that The case follows immediately from the case due to the fact that .

Lemma 4.3. Let be an integer and . Let be positive real numbers with . Then,

Proof. For the case , the assertion follows from Lemma 4.2. Next, we argue by the induction and assume that the assertion is true for . Then, it suffices to prove that the assertion holds when . Now, let be positive real numbers with . Consider the following function in a variable where are arbitrary (but fixed) positive numbers. Clearly, The first derivative of the function regarding the variable is equal to In the following, we distinguish three possibilities.Case 1 (). Then, holds for all , which implies that the function is strictly increasing in variable . From this, we obtain Case 2 (). Then, holds for all , implying that the function is strictly decreasing in variable . From this, we obtain that Case 3 (). This implies for all . From this relation and the inspection that the condition implies and , we obtain that Hence, by induction, the assertion immediately holds for , and then the case follows directly from the case because

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 and of the system (4.3), resp.) where , , were studied in [12].

5. On a System of Difference Equations

Let , be a continuous mapping defined by Then, the following difference equation 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: where , for .

Lemma 5.1. System (5.3) has unique positive equilibrium .

Proof. Let be an arbitrary positive equilibrium of the system (5.3). Then, we get which imply Applying (5.5), the system in (5.4) is reduced to If , then it follows from the second identity that . Now, assume . By solving the first equation in (5.6) with respect to variable , we get that the discriminant which implies the first equation in (5.6) has no real roots. This contradicts . Hence, , which along with (5.5) implies , 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 , and be positive real numbers with . Then,

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

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

In the following, we consider the next -dimensional () generalization of system?(5.3) where , , .

It is easy to see that 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 be an integer with . Let be positive numbers with . Then,

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

Theorem 5.5. The positive equilibrium 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 Stevic 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 where is a continuous function with a unique equilibrium . Suppose that there is a such that for each solution of (6.1), with equality if and only if . Then, is globally asymptotically stable.

Motivated by [18], in recent paper [2], Berg and Stevic 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 , , let for , where we define and , and let or reversed, is the sum over the even, and is the sum over the odd .

Theorem 6.2. Suppose is a nonnegative continuous function on , , and . If a sequence satisfies the difference equation where with , then it converges to the unique positive equilibrium 1.

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

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 and with , and and satisfy the following two conditions: and .. Then, is the unique positive equilibrium of the difference equation which is globally asymptotically stable (here ).

Proof. Let We should determine the sign of the product of the next expressions
From (6.9) and (6.10), we see if we show that and have the same sign for , then will be nonpositive.
There are four cases to be considered.
Case 1 (, ). Clearly, in this case, . By and , we have that Hence, and consequently Case 2 (, ). Since we obtain . On the other hand, by and we have so that . Hence, (6.12), follows in this case.Case 3 (Case , ). Then we have that and consequently . On the other hand, we have so that . Hence, (6.12) follows in this case too.Case 4 (Case , ). Then . On the other hand, we have so that . Hence, (6.12) also holds in this case. Thus , for every .
Assume that , then, or . Using (6.9) or (6.10) along with (6.12) in any of these two cases, we have that Hence, , .
Finally, let be a solution of the equation where denotes the vector consisting of copies of . Then according to the considerations in Cases 1–4 it follows that , so that . Hence 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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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.

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