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Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 506373, 12 pages
Global Attractivity of a Family of Max-Type Difference Equations
1College of Computer Science, Chongqing University, Chongqing 400044, China
2Department of Computer Science, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Received 20 October 2010; Accepted 13 January 2011
Academic Editor: Ibrahim Yalcinkaya
Copyright © 2011 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.
We propose to study a generalized family of max-type difference equations and then prove the global attractivity of a particular case of it under some parameter conditions. Through some numerical results of other cases, we finally pose a generic conjecture.
The study of max-type difference equations is a hotspot in the area of discrete dynamics because such equations are often closely related to automatic control theory and competitive dynamics. For recent advances in this direction see [1–8] and the references therein.
Motivated by , Liu et al.  studied the following nonautonomous max-type difference equation: where , , , and , are mappings satisfying the condition , for some fixed . When , , they proved that every positive solution to (1.1) converges to zero if , while if . If and , then each positive solution to (1.1) converges to , for some , except for the case , , where . Note that the behavior of positive solutions to (1.1) for the case , , is still an unsolved open problem as was mentioned in .
Here, we propose to investigate the asymptotic behavior of positive solutions to the generalized family of max-type difference equations where , , , and the functions , satisfy the condition for some fixed .
In this paper, we mainly consider the particular case that all are zero, and then obviously (1.2) reduces to the following form: Let be a nonnegative equilibrium point of (1.4), then we have It follows directly from (1.5) that if for all , then (1.4) has the unique nonnegative equilibrium , while if there exists at least one such that , then (1.4) has a zero equilibrium and a unique positive equilibrium .
Finally, the following two beautiful theorems are derived.
Theorem 1.2. Consider (1.4) with positive initial values and positive and . Let be functions such that for some fixed , there hold If there exists at least one such that , then the unique positive equilibrium of (1.4) is a global attractor.
2. Preliminary Lemmas
For the purpose of establishing the main results, some auxiliary lemmas are essential.
Lemma 2.1. Consider the first-order difference equation with positive initial value and positive and . If there exists at least one such that , then
Proof. Suppose that , which is positive, for some . By making the variable change into (2.1) and then canceling the positive term from the resulting equation, we can derive Note that for , . Then it follows from (2.3) that In addition, the following two inequalities hold: In the following, we are confronted with three possibilities.Case 1. If there exists such that , then it follows from (2.4) and (2.5) that holds for all .Case 2. If there exists such that , then it follows from (2.5) and (2.6) that Thus there is a finite limit . By taking the limits on both sides of (2.3) and canceling the positive factor from the resulting equation, we obtain which implies . Because if , then leading to a contradiction.Case 3. If for all , then it follows from (2.5) and (2.6) that Therefore, the limit of exists, denoted by . By taking the limits on both sides of (2.3) and canceling the nonzero factor from the resulting equation, there hold which implies . Because if , then which is a contradiction.In either of the above three cases, we get , implying .
From Lemma 2.1, we have the following result.
Lemma 2.2. Consider the -order difference equation with positive initial values and . If there exists at least one such that , then
Proof. Let be an arbitrary positive solution to (2.13). Apparently we know that the sequence can be divided into subsequences , , which are, respectively, positive solutions to the first-order equation (2.1) with positive initial values . According to Lemma 2.1, we derive for all , which directly lead to .
Lemma 2.3. Let , , and . Define two sequences and in the following way: Then .
Proof. Observe that It follows by induction that is increasing and is decreasing. Again by induction we derive and , . Hence there are two finite limits and . By taking limits on both sides of (2.15), we derive which imply . Therefore .
3. Proofs of Main Theorems
In this section, we are in a position to prove the main theorems presented in Section 1.
Proof of Theorem 1.1. Note that for the case , , the behavior of positive solutions to (1.4) is quite simple. In this case, we have that
where . Easily the subsequences , converge to zero, hence the sequence also converges to zero.
For the case , with at least one such that , we can obtain that In this case, the subsequences , are all positive and nonincreasing, thus they converge, respectively, to some nonnegative limits , .
If we replace in (1.4) by , for an arbitrary fixed and let , we can get where , . Without loss of generality, assume that , then we obtain that with some fixed number . Because , then it follows from (3.4) that Therefore we have leading to , which is a contradiction. Hence we have that , , and every positive solution to (1.4) converges to zero, if for all .
Let be an arbitrary positive solution to (1.4). Next, we proceed by proving two claims.
Claim 1. There exists such that for all .
Proof. Note that
Consider the following difference equation:
Let be a positive solution to (3.7) with the initial values , .
Note that the mapping is strictly increasing on the interval . It follows by induction that for all . By Lemma 2.2, we have . Hence there is an integer such that for .
Let . Note that Consider the difference equation with , . Note the monotonicity of , it follows by induction that for all . By Lemma 2.2, we get that . Thus there exists an integer such that for all .
Working inductively, we will reach the following claim.
Claim 2. For every , there exists such that for all .
Proof. Obviously, the case follows directly from Claim 1. In the following, we proceed by induction. Assume that the assertion is true for . Then it suffices to prove the assertion is also true for .
Note that Consider the difference equation with , . Note the monotonicity of , it follows by induction that for all . By Lemma 2.2, we have that . So there is an integer such that for all . Then note that Consider the following difference equation with , . By the monotonicity of , it follows by induction that for all . By Lemma 2.2, we have that . So there is an integer such that for all .
4. Simulations and Future Work
In the previous section, we proved the global attractivity of (1.2) when all are zero. In this section, we investigate the dynamic behavior of (1.2) provided that all are not zero. First, it is trivial to confirm that when all are not zero, (1.2) has the following unique positive equilibrium point . In the following, some numerical results are presented.
Experiment 3. Consider the third-order difference equation where the initial values . (See Figure 5).
Inspired by this work and the results of , here we pose the following conjecture.
Conjecture 4.1. Consider (1.2) with nonnegative and positive and . Let , be functions such that for some fixed , there hold If for all , then every positive solution to (1.2) converges to the equilibrium point
This work was financially supported by the Fundamental Research Funds for the Central Universities (no. CDJXS10181130), the National Natural Science Foundation of China (no. 10771227), and the New Century Excellent Talent Project of China (no. NCET-05-0759).
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