`Discrete Dynamics in Nature and SocietyVolumeΒ 2008Β (2008), Article IDΒ 243291, 6 pageshttp://dx.doi.org/10.1155/2008/243291`
Research Article

On the Asymptotic Behavior of a Difference Equation with Maximum

College of Computer Science, Chongqing University, Chongqing 400044, China

Received 25 May 2008; Revised 6 June 2008; Accepted 18 June 2008

Copyright Β© 2008 Fangkuan Sun. 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 study the asymptotic behavior of positive solutions to the difference equation , where . We prove that every positive solution to this equation converges to .

1. Introduction

Recently, there has been a considerable interest in studying, the so-called, max-type difference equations, see for example, [1β21] and the references cited therein. The max-type operators arise naturally in certain models in automatic control theory (see [9, 11]). The investigation of the difference equationwhere , are real numbers such that at least one of them is different from zero and the initial values are different from zero was proposed in [6]. Some results about (1.1) and its generalizations can be found in [1, 3β5, 7, 8, 10, 12, 17, 18, 19] (see also the references therein). The study of max-type equations whose some terms contain nonconstant numerators was initiated by Stevi, see for example, [2, 14β16]. For some closely related papers, see also [20, 21].

Motivated by the aforementioned papers and by computer simulations, in this paper we study the asymptotic behavior of positive solutions to the difference equationwhere . We prove that every positive solution of this equation converges to .

2. Main Results

In this section, we will prove the following result concerning (1.2).

Theorem 2.1. Let be a positive solution to (1.2).
Then

In order to establish Theorem 2.1, we need the following lemma and its corollary which can be found in [13].

Lemma 2.2. Let be a sequence of positive numbers which satisfies the inequality where and are fixed. Then there exist such that

Corollary 2.3. Let be a sequence of positive numbers as in Lemma 2.2. Then there exists such that

Now, we are in a position to prove Theorem 2.1.

Proof. We proceed by distinguishing two possible cases.
Case (). We prove as .
Set , then (1.2) becomeswhere . Since , we have . To prove as , it suffices to prove as .
We proceed by two cases: and .

Case . In this case (2.5) is reduced towhere . Choose a number so that . Let . Then, is a solution to the difference equationTo prove as , it suffices to prove as .
It can be easily proved that there is a positive integer such that for all By simple computation, we get that, for all ,
Since , (2.10) implies as . From (2.9) and (2.11), it follows that as . This implies .
Case . Let , then is a solution to the difference equationTo prove as , it suffices to prove as . If , then we have for all . Next, we assume either or . Then the following four claims are obviously true.
Claim 1. If and for some , then
Claim 2. If and for some , then .
Claim 3. If and for some , then .
Claim 4. If and for some , then
In general, we havewhere . From (2.15) and Corollary 2.3, there exists such thatThis implies as .
Case (). We prove as .
Similar to the proof of Case 1, we set , then (1.2) becomeswhere . To prove as , it suffices to prove as . Let , then is a solution to the difference equationTo prove as , it suffices to prove as . If , then we have for all . Next, we assume either or , then the following four claims are obviously true.
Claim 1. If and for some , then
Claim 2. If and for some , then .
Claim 3. If and for some , then
Claim 4. If and for some , then .
In general, we havewhere . Then the rest of the proof is similar to the proof of Case 1 and will be omitted. The proof is complete.

Theorem 2.4. Every solution to the difference equation converges to .

Proof. Let , then the equation becomesFrom this and the condition , it follows that as which implies as .

3. Conclusions and Remarks

This paper examines the asymptotic behavior of positive solutions to the difference equation (1.2) with . The method used in this work may provide insight into the asymptotic behavior of positive solutions to the generic difference equationwhere . We close this work by proposing the following conjecture.

Conjecture 3.1.Assume that is a positive solution to (3.1). Then as .

Acknowledgments

The author is grateful to the anonymous referees for their huge number of valuable comments and suggestions, which considerably improved the paper. This work is supported by Natural Science Foundation of China (10771227).

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