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Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 364083, 13 pages
On Global Attractivity of a Class of Nonautonomous Difference Equations
1College of Computer Science, Chongqing University, Chongqing 400044, China
2School of Computer and Information, Chongqing Jiaotong University, Chongqing 400074, China
Received 28 February 2010; Accepted 28 June 2010
Academic Editor: Guang Zhang
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.
We mainly investigate the global behavior to the family of higher-order nonautonomous recursive equations given by , , with and positive initial values, and present some sufficient conditions for the parameters and maps , under which every positive solution to the equation converges to zero or a unique positive equilibrium. Our main result in the paper extends some related results from the work of Gibbons et al. (2000), Iričanin (2007), and Stević (vol. 33, no. 12, pages 1767–1774, 2002; vol. 6, no. 3, pages 405–414, 2002; vol. 9, no. 4, pages 583–593, 2005). Besides, several examples and open problems are presented in the end.
There has been a great interest in studying classes of nonlinear difference equations and systems, particularly those which model real situations in engineering and science, for example, [1–15]. On the other hand, non-autonomous difference equations also have a ubiquitous presence in applications from automatic controlling, ecology, economics, biology, population dynamics and so forth. Thus the main task when dealing them is to know the asymptotical behaviour of their solutions. For some recent advances in this area see [1, 16–24] and the references cited therein.
Gibbons et al.  discussed the behavior of nonnegative solutions to the rational recursive equation with and also proposed an open problem, which had been solved by Stević in , concerning the particular case in (1.1) (see also [26, 27] for the case of some related higher-order difference equations, as well as [28–30]).
In , Stević studied the behavior of nonnegative solutions of the following second-order difference equation where → is a nonnegative increasing mapping. Obviously (1.2) is a generalization of (1.1).
Later, Stević  extended (1.1) and (1.2) to the following more general equation where and is a continuous function nondecreasing in each variable such that and investigated the oscillatory behavior, the boundedness character and the global stability of nonnegative solutions to the equation.
Recently, Iričanin  studied the asymptotic behavior of the following class of autonomous difference equations: where and is a continuous mapping satisfying the condition for certain . In  he adopted the approach of frame sequences (a discrete analog of the method of frame curves used in the theory of differential equations), which has been used in the literature for many times, for example, [26–28, 30–38]; and showed that all positive solutions converge to zero if and converge to the unique positive equilibrium if .
Motivated by the above works, especially [2, 5], our aim in this paper is to study the global attractivity in the following family of non-autonomous difference equations: where , and are mappings satisfying the following condition for some fixed .
Through careful analysis, we find that the results in  also persist if the function in (1.4) is replaced by variable functions such as satisfying condition (1.7). If , then (1.6) can be transformed into the following form where , by setting . Then according to the results in , we have that if , then ; and if , then , for some . Thus it suffices to consider the case when in the following.
2. Auxiliary Results
Before proving the main result of this paper, in this section we first confirm two preliminary lemmas.
Let be the mapping where and , so as is decreasing in the first variable and increasing in the second one. Then (1.6) can be simplified to the following form:
Lemma 2.1. Consider the following higher-order rational difference equation: where , the parameter and initial values are arbitrary nonnegative numbers. Then every positive solution to (2.2) converges to the unique positive equilibrium point
Proof. First we show that (2.2) has a unique positive equilibrium. Assume that is an equilibrium point of (2.2), then which implies only one positive root
If , then (2.2) can be separated into analogous first-order difference equations of the form
with different initial values , where . Note that the equation is Riccati, so it can be solved and the convergence of its solutions can be proved (see, e.g.,  or a recent comment in ).
Let the symbol symbolize the greatest integer function and define a sequence . Obviously, for each positive solution to (2.2) we have
From the above analysis, it suffices to prove the case when . Suppose that for (2.2), then for all we haveCase 1. If , then by (2.7) is either nonincreasing or nondecreasing. On the other hand, we have that for all . Therefore, the limit of exists, and through simple calculations, we get Case 2. If , then by (2.8) and (2.9) and inductively we have that is nonincreasing and nondecreasing, or is nondecreasing and nonincreasing. Again by (2.10), the limits of and exist, denoted by and . From (2.2) we have which imply that . Hence .
The proof of Lemma 2.1 is complete.
Lemma 2.2. Suppose that the parameters, in (2.3), satisfy with . Define two sequences and as follows: where the initial value , and , If or , where , then
Proof. By simple calculations, we have
Obviously, Claim 1. .Proof. Define a function . It suffices to prove that for all . The derivative of is
Since and , thus for Therefore, it follows from (2.15) and Claim 1 that
Simply, we obtain that and .
Observe that With (2.17) and (2.19), it follows by induction that are strictly increasing and decreasing, respectively. In addition, , hence possesses a finite limit denoted by . From (2.12), we know that the limit of (denoted by also exists. Therefore, taking limits on both sides of (2.12), we have which imply that Claim 2. If or , then .Proof. Suppose that , then it follows from (2.22) that . By substituting into the second identity of (2.21), we get (i)If , then which is a contradiction to ,(ii)If , then the unique positive root of (2.23) is However, since is strictly decreasing.(iii)If , then (2.23) reduces to .(iv)If , then for (2.23), which implies that (2.23) has no real roots.(v)For , we have . So, which is contradictive to .(vi)For , (2.23) has two negative roots.(vii)For . Solving (2.23), we get implying . Hence , which contradicts the assumption.
Obviously Claim 2 follows directly from (i)–(vii).
Applying Claim 2 and (2.21), we conclude that Hence the lemma is complete.
3. Main Results
First, we present a proposition concerning the boundedness of all positive solutions to (1.6).
Proof. Let be a solution to (1.6) with positive initial values. Then, we have Thus we have , for all .
Proof of Theorem 1.1. Let be an arbitrary fixed number satisfying ( defined by (2.13) in Lemma 2.2). Define two sequences as shown by (2.12) in Lemma 2.2. Let be any positive solution to (1.6). In the following, we proceed by presenting two claims.Claim 1. There exists , such that for all .Proof. From (2.1), we have that
Suppose that is a solution to the following difference equation
with initial values . From this and in view of the monotonicity of the function , by induction we can easily get that for .
By Lemma 2.1, . Hence, there exists such that for , then for all .
From (2.1), (1.7), and (3.4), it follows that for all .
Suppose that is a solution to the following difference equation: with initial values .
Since the function is increasing on the interval , we can easily get by induction that for , and by Lemma 2.1, . Hence there exists a natural number such that for , then for .
Working inductively, we will eventually reach the following claim.
Claim 2. For each , there exists such that for all .Proof. By Claim 1, if , we have such that for all . Then by the method of induction, we can assume that for fixed, there exists such that for all . Thus, it suffices to show that there exists such that for all .
Let . Define a sequence as follows with for
By reasoning inductively on , one has By Lemma 2.1, . Therefore, there is such that
Define the other sequence as follows: where
Once more, by induction on , By Lemma 2.1, . Thus, let be greater enough so as for all .
Therefore, we get that there exists such that for all .By Claim 2, we have This plus Lemma 2.2 leads to The proof is complete.
4. Applications and Future Work
Next, several examples are presented.
Example 4.1. Let for all , and for some . If and or , where , then by Theorem 1.1 we conclude that every positive solution to the following non-autonomous difference equation: converges to the unique positive equilibrium point .
Example 4.2. Let for all , then under the conditions of Theorem 1.1, all positive solutions to the recursive equation converge to the unique positive equilibrium .
In this paper, the behavior of positive solutions to the case when where isn't investigated, since we have no further new ideas for the particular case. Through certain calculations, easily we know that the equation has two different positive roots, if , which implies From this we propose the following open problem.
Furthermore, the case for (1.6) is also of extreme value to study.
The authors are grateful to the referees for their huge number of valuable suggestions, which considerably improved the presentation in the paper. Besides, the authors thank Professor. Iričanin for very valuable comments regarding this subject. This work was financially supported by National Natural Science Foundation of China (no. 10771227).
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