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Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 179589, 21 pages
http://dx.doi.org/10.1155/2008/179589
Research Article

Eventually Periodic Solutions for Difference Equations with Periodic Coefficients and Nonlinear Control Functions

1Department of Mathematics, Yanbian University, Yanji 133002, China
2Department of Mathematics, Tsing Hua University, Hsinchu 30043, Taiwan

Received 13 August 2008; Accepted 24 October 2008

Academic Editor: Guang Zhang

Copyright © 2008 Chengmin Hou and Sui Sun Cheng. 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

For nonlinear difference equations of the form , it is usually difficult to find periodic solutions. In this paper, we consider a class of difference equations of the form , where are periodic sequences and is a nonlinear filtering function, and show how periodic solutions can be constructed. Several examples are also included to illustrate our results.

1. Introduction

There are good reasons to find “eventually periodic solutions” of difference equations of the form For instance, the well-known logistic population model is of the above form, and the study of the existence of its periodic solutions leads to chaotic solutions. As another example in [1], Chen considers the equation where is a nonnegative integer, and is a McCulloch-Pitts type function in which is a constant which acts as a threshold. Chen showed that all solutions of (1.3) are eventually periodic and pointed out that such a result may lead to more complicated dynamical behavior of a more general neural network. Recently, Zhu and Huang [2] discussed the periodic solutions of the following difference equation: where is a positive integer, and is a signal transmission function of the form (1.9). In particular, they obtained the following theorem.

Theorem A. Let If then (1.5) has an eventually -periodic solution -periodic solution

In this paper, we consider the following delay difference equation: where and are positive -periodic sequences such that for .

The integer is assumed to satisfy for some nonnegative integer The function can be chosen in a number of ways. Here, is a filtering function of the form where the positive number can be regarded as a threshold term. Therefore, if then , and so that (1.7) reduces to which includes (1.5) as a special case.

When , we have which will also be included in the following discussions.

Let denote the set of real finite sequences of the form Given , if we let then we may compute successively from (1.7) in a unique manner. Such a sequence is called a solution of (1.7) determined by Recall that a positive integer is a period of the sequence if for all and that is the least period of if is the least among all periods of The sequence is said to be -periodic if is the least period of In case is not periodic, it may happen that for some the subsequence is -periodic. Such a sequence is said to be eventually -periodic. In other words, let us call a translate of if for where is some integer greater than or equal to Then, is eventually -periodic if one of its translates is -periodic.

We will seek eventually periodic solutions of (1.7). This is a rather difficult question since the existence question depends on the sequences , the “delay” and the control term .

Throughout this paper, empty sums are taken to be and empty products to be We will also need the following elementary facts. If the real sequence satisfies the recurrence relation then and by induction, where Since and are positive -periodic sequences, we see further that that for and and that for and where

2. Convergence of Solutions

The filtering function will return for inputs that fall below or above the threshold constant . For this reason, we will single out some subsets of as follows:

Let be the solution of (1.7) determined by By (1.7), By induction, we may see that Since we see that

Next, let be the solution of (1.7) determined by If then by (1.7), By induction, we see that By (1.7), we see that In view of (1.18), we see further that where

Next, let be the solution of (1.7) determined by Then, by (1.7), and by induction, Although since (2.4) holds, we see that is a strictly decreasing sequence tending to Hence, there is a nonnegative integer such that but Then, If we let , then If then by what we have shown above, the solution of (1.7) determined by satisfies for By uniqueness, for In other words, the translate of the solution satisfies for .

We summarize the above discussions by means of the following result.

Lemma 2.1. A solution determined by will tend to and if then a solution determined by will satisfy (2.8) or one of its translates will satisfy it.

Lemma 2.2. If then for any solution of (1.7) determined by a , there exists an integer such that and .

Proof. First let be the solution of (1.7) determined by a . If for all , then so that by (1.18), we see that But, this is contrary to our assumption that . Hence, there is some nonnegative integer such that for but . Note that which implies that . Moreover, since we then have so that .
Next, let be the solution of (1.7) determined by a . As seen in the discussions immediately preceding Lemma 2.1, there is a nonnegative integer such that . If for all , then as we have just explained, a translate of will satisfy This is again a contradiction. Hence, we may conclude our proof in a manner similar to the above discussions. The proof is complete.

From the proof of Lemma 2.2, we see that if then to study the limiting behavior of a solution determined by in , we may assume without loss of generality that and . As an example, let us consider (1.11), where we recall that and .

Example 2.3. Let . Then, (1.11) has a -periodic solution with and . Indeed, let us choose (and hence, ). Then, Furthermore, so that and and .

3. Existence of Eventually Periodic Solutions

Recall that are the zeroth, first, second, and so forth and the th iterate of the function . Also, recall the fact that if is a sequence that satisfies then is a -periodic sequence if and only if For convenience, denote Since we see that

Theorem 3.1. Let and where Let where If and then (1.7) has an eventually -periodic solution (which can be explicitly generated).

Proof. From the condition that we have . By (3.5), we see that Hence, Thus, By Lemmas 2.1 and 2.2, we may look for our desired eventually periodic solution determined by such that .
Define and the mapping by We will show that and that maps into with a fixed point , where The first assertion is easy to show. Indeed, since we see that
We now show the second assertion. Note that the linear maps and satisfy Let for . Since and it is clear that the solution of (1.7) satisfies Moreover, it is easy to prove that Indeed, we have That is, holds for all . Let be the largest integer such that for . Then, from (1.7), we can obtain which implies that for where Since , we have that is, which shows that for . Thus, and In fact, from we have and, by induction, Then, it is easy to see that .
Taking in (3.20), we have Let be the largest integer such that for . Then, it follows from (1.7) that for . This implies that for , where .
Substituting (3.21) with into (3.28), we have for . Since we have From (3.29), we further have By (3.8), (3.24), (3.28), and (3.31) as well as , we have
Hence, which implies that In particular, taking in (3.33) and (3.28), we have, respectively,
Since is a linear map sending into then it is easy to see that it has a unique fixed point in which satisfies (3.13).
Next, we assert that there is a such that the solution determined by satisfies , and that is a periodic solution of (1.7) with minimal period . To see this, we choose and arbitrary . Then, clearly, the solution of (1.7) determined by will satisfy . Furthermore, we may show that . Indeed, from we have hence, Thus, Next, from we get so that On the other hand, by (3.5), we have so that In view of our assumption that , we may now see that .
In view of the above discussions, we see that for for and for . Since is the unique fixed point of in we have , and so forth, and hence, and so forth. Thus, and so forth.
By induction, we may see that for for where and . This shows that is an eventually periodic solution of (1.7), whose minimal period is . The proof is complete.

We remark that in the above result, cannot be We may, however, show the following by similar considerations.

Theorem 3.2. Let where If and then (1.7) has an eventually -periodic solution (which can be generated explicitly).

Proof. Similar to the proof of the Theorem 3.1, set (3.10) and define the mapping by We may show that and that maps into with a unique fixed point where Let us choose By we have and hence, Since then and hence, so that is an eventually -periodic solution of the system (1.7).

4. Examples and Remarks

Let be -periodic sequences, and Suppose and Consider the following “delay” difference equation: We can check that (4.3) has an eventually -periodic solution with .

In fact, as in the proof of Theorem 3.1, let and be arbitrary numbers in . Then, as shown in the proof of Theorem 3.1, the solution of (4.3) determined by satisfies and .

Since