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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 367492, 15 pages
http://dx.doi.org/10.1155/2010/367492
Research Article

Global Behavior of Two Families of Nonlinear Symmetric Difference Equations

College of Computer Science, Chongqing University, Chongqing 400044, China
School of Computer and Information, Chongqing Jiaotong University, Chongqing 400074, China
College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

Received 14 May 2010; Revised 3 June 2010; Accepted 16 June 2010

Academic Editor: Josef Diblik

Copyright © 2010 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 mainly investigate the global asymptotic stability and exponential convergence of positive solutions to two families of higher-order difference equations, one of which was recently studied in Stević's paper (2010). A new concise proof is given to a quite recent result by Stević and analogous parallel result of the other inverse equation, which extend related results of Aloqeili (2009), Berenhaut and Stević (2007), and Liao et al. (2009).

1. Introduction

The interest in investigating rational difference equations has a long history; for instance, see [124] and the references cited therein. More generally, it is meaningful to study not only rational recursive equations but also those with powers of arbitrary positive degrees.

For instance, at many conferences, Stević proposed to study the behavior of positive solutions of the following generic difference equation (see also [25]): where , and . For some recent results in this area, see [2632] and the references therein.

By a useful transformation method from [4], the authors of [3] confirmed that the unique positive equilibrium of the rational recursive equation where , is globally asymptotically stable for all solutions with positive initial values. In the meantime, they also remarked that the global asymptotic stability for the unique equilibrium of the difference equation can be shown through analogous calculations. Some particular cases of (1.3) had already been considered in [12, 13].

In [3] were proposed the following two conjectures.

Conjecture 1.1. Suppose that and that satisfies with positive initial values. Then, the sequence converges to the unique positive equilibrium point .

Some special cases of (1.4) had been studied by Li [9, 10] with a semicycle analysis method, which is useful for lower-order difference equations but tedious and complicated to some extent(see the explanation in [33]). Finally, Conjecture 1.1 was also confirmed in [2] with the similar transformation method used in [3, 4]. However, it is somewhat harder to prove the following conjecture in the same way.

Conjecture 1.2. Assume that is odd and , and define . If satisfies with , where Then the sequence converges to the unique positive equilibrium point .

Next, we present two definitions as defined in [1].

Definition 1.3. A function of variables is symmetric if it is invariant under any permutation of its variables. That is, a function is called symmetric if where is any permutation of the numbers .

Definition 1.4. The th elementary symmetric function of variables , where is defined by where the sum is taken over all choices of the indices from the set of integers .

Obviously, the functions defined by (1.6) and (1.7) are symmetric and can be rewritten as

In this paper, we give a new proof of a quite recent result by Stević in [34] where he, among others, studied the stability of one of the following two difference equations, which are dual: where is odd, and

Apparently, Equation (1.11) is the generic form of (1.2), (1.4), and (1.5).

In [6, 18] the authors proved that the main results in some of papers [912] are direct consequences of a result confirmed by Kruse and Nesemann [35]. For example, in [6] was showed that the main result in [13] is also a consequence of Corollary in [35]. On basis of these works, in 2008, Aloqeili [1] confirmed Conjecture 1.2 in the same way.

Later, Liao et al. [14] proved Conjecture 1.2 by using a new approach. They used a sort of “frame sequences” method(the notion suggested by Stević), which has been widely used in [5, 7, 18, 3641]. Through careful analysis, we find that the method used in [14] can be further simplified and applied in proving Stević’s result in a more concise and interesting way. Namely, we give a new proof of the following result, which generalizes related results in [1, 2, 9, 10, 14].

Theorem 1.5. Assume that is odd and positive integers are satisfying . Then(1)the unique positive equilibrium point of (1.10) is globally asymptotically stable;(2)the unique positive equilibrium point of (1.11) is globally asymptotically stable.

2. Auxiliary Results and Notation

In this section, we will introduce some useful notation and lemmas. Consider the following notation(for similar ones see [14]), which play an important role in the paper: Employing and , define a mapping as follows: Then (1.10) can be rewritten as or with being odd, and .

By the notation defined by (2.1), define the other function such that: Then (1.11) can be rewritten as or with being odd, and .

Lemma 2.1. If , then both (2.4) and (2.7) have the unique positive equilibrium point

Proof. Suppose that is an equilibrium of (2.4), then which implies Obviously , due to the different signs of both sides of the last equality for the case Likewise, let be an equilibrium of (2.7); then which indicates Assume that . If , then by the monotonicity of the map , we have that which contradicts (2.11).Similarly, if , then by the monotonicity of the function , we have that which also contradicts (2.11). Thus .
The proof is complete.

Lemma 2.2. (1) Let be defined by (2.2); then is monotonically increasing in if and only if and monotonically decreasing in if and only if for
(2) Let be defined by (2.5), then is monotonically decreasing in if and only if and monotonically increasing in if and only if for

Proof. The results follow directly from the facts below:

Remark 2.3. The second statement (i.e., (2)) in Lemma 2.2 can also be found in Stević’s paper [34] (see Lemma and Corollary ).

For odd, define a map such that which has the following simple property:

Lemma 2.4. Suppose that and let If then

Proof. Since is symmetric in without loss of generality, we suppose that If there exists such that , then by (2.2) and (2.5) we can easily get that . Thus, assume for all
Then we have the following cases to consider: By Lemma 2.2, for the above cases, we have that Obviously, follows directly from the above inequalities.
The proof of the case is analogous and hence omitted.

Lemma 2.5. Suppose that is fixed and let Then we have

Proof. By the monotonicity of the function we have that which implies Therefore, , that is,
The proof is complete.

The following corollary follows directly from Lemma 2.4 and Lemma 2.5.

Corollary 2.6. Assume that If any positive solution to (2.4) or (2.7) has the initial values then we have for

Define two sequences and as follows: with initial values .

Lemma 2.7. For the sequences defined by (2.24), if and , then

Proof. Inductively, we can simply obtain that , Through simple calculations, by (2.16), we have that Therefore by Lemma 2.4 and Lemma 2.5, we get that which implies that the sequences converge to some limits (denoted by and , resp.), that is, By taking limits on both sides of the first identity of (2.24), we get which implies Suppose that ; then by the monotonicity of the function , we have that which contradicts (2.29). Hence, we have that and then obviously it follows by (2.26) and (2.28) that .
The proof is complete.

3. Stability

In this section, we give a new, concise and clear proof of Stević’s Theorem 1.5, by the lemmas in Section 2.

Proof of Theorem 1.5. Employing Lemma 2.2, the linearized equations of (2.4) and (2.7) about the equilibrium are both Then by the Linearized Stability Theorem, is locally stable.
Thus it suffices to confirm that is also a global attractor for all positive solutions of (2.4) and (2.7).
Let be a positive solution to (2.4) or (2.7) with initial values We need to prove that
Apparently, there exists such that where Employing Corollary 2.6, we have Let sequences and be defined by (2.24). Let then in light of Lemma 2.4, (3.4), and (2.26), we get That is, In view of (3.6), (2.26) and Lemma 2.4, we have that That is, Reasoning inductively, we can get By Lemma 2.7 and (3.9), we obtain which implies
The proof is complete.

4. Exponential Convergence

In this section, we will prove that all positive solutions to (2.4) and (2.7) with are exponentially convergent, by using an approach from paper [42].

Theorem 4.1. If then every positive solution to (2.4) and (2.7) exponentially converges to 1.

Proof. Let be a positive solution to (2.4) or (2.7); then by Theorem 1.5, there exists a sufficiently large natural number such that for arbitrary fixed we have for all .
Denote ; then for all .
(1). For (2.4).
Let then by (2.4), we have
(2). For (2.7).
Let be fixed; then by (2.7), we get

From this inequality and Lemma in [43] (see also Corollary therein), the result directly follows.

5. Other Simple Results

In this section, we will present some elementary results of (2.3) and (2.6) with .

Proposition 5.1. If , then there is no positive solution to (2.3) such that .

Proof. Suppose is a positive solution to (2.3) such that Then for some fixed , there exists such that Employing (2.3) and (5.2), we can simply get that which contradicts (5.2). The proof is complete.

Proposition 5.2. We have the following simple statements:(1)if , then (2.6) has nonoscillatory positive solutions with all initial values , or (2)let and denote by the cardinality of the set . If , then for any positive solution to (2.3) or (2.6), we get for all .

6. Conclusions

In the following, let for any as defined in [20] and firstly we present [20, Theorem ].

Theorem 6.1 (see [20]). Let satisfy the following two conditions: (H1) and (H2)Then is the unique positive equilibrium for equation (1) which is globally asymptotically stable.
The Equation () mentioned in Theorem is the following difference equation: where and with , and , and the initial values are positive real numbers.

Remark 6.2. Equation (1.10) is a special case of equation (1) in [20].

Proof. Let , and define a recursive equation for all Then the following difference equation: where and the initial values , is the very Equation (1.10) in this paper.

Remark 6.3. Let , , and . Then through simple calculations, we have(H1);(H2)Thus the conditions and of [20, Theorem ] hold. By [20, Theorem ], we know that the unique positive equilibrium of (1.10) (also (6.3)) is globally asymptotically stable.

Remark 6.4. Although the stability of (1.10) can be also obtained as a corollary from Theorem of the paper by Sun and Xi [20], the method of proof of Theorem 1.5 in this paper is distinct.

Acknowledgments

The authors are grateful to the referees for their huge number of valuable suggestions, which considerably improved the presentation of this paper. This work was financially supported by National Natural Science Foundation of China (no. 10771227).

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