- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Discrete Dynamics in Nature and Society

VolumeΒ 2008Β (2008), Article IDΒ 243291, 6 pages

http://dx.doi.org/10.1155/2008/243291

## 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

Academic Editor: StevoΒ Stevic

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, [1β21] 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, 3β5, 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, 14β16]. 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).

#### References

- J. C. Bidwell,
*On the periodic nature of solutions to the reciprocal delay difference equation with maximum*, Ph.D. dissertation, North Carolina State University, Raleigh, NC, USA, 2005. - K. S. Berenhaut, J. D. Foley, and S. SteviΔ, βBoundedness character of positive solutions of a max difference equation,β
*Journal of Difference Equations and Applications*, vol. 12, no. 12, pp. 1193β1199, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - C. Γinar, S. SteviΔ, and I. YalΓ§inkaya, βOn positive solutions of a reciprocal difference equation with minimum,β
*Journal of Applied Mathematics & Computing*, vol. 17, no. 1-2, pp. 307β314, 2005. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - J. Feuer, βOn the eventual periodicity of ${x}_{n+1}=\mathrm{max}\beta \x81\u2018\{1/{x}_{n},{A}_{n}/{x}_{n\beta \x88\x921}\}$ with a period-four parameter,β
*Journal of Difference Equations and Applications*, vol. 12, no. 5, pp. 467β486, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - C. M. Kent and M. A. Radin, βOn the boundedness nature of positive solutions of the difference equation ${x}_{n+1}=\mathrm{max}\beta \x81\u2018\{{A}_{n}/{x}_{n},{B}_{n}/{x}_{n\beta \x88\x921}\}$ with periodic parameters,β
*Dynamics of Continuous, Discrete & Impulsive Systems. Series B*, vol. 2003, supplement, pp. 11β15, 2003. View at Google Scholar Β· View at MathSciNet - G. Ladas, βOn the recursive sequence ${x}_{n+1}=\text{max}\{{\text{A}}_{\text{0}}/{x}_{n},\beta \x80\xa6,{A}_{k}/{x}_{n\beta \x88\x92k}\}$,β
*Journal of Difference Equations and Applications*, vol. 2, no. 3, pp. 339β341, 1996. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - D. P. Mishev, W. T. Patula, and H. D. Voulov, βA reciprocal difference equation with maximum,β
*Computers & Mathematics with Applications*, vol. 43, no. 8-9, pp. 1021β1026, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - D. P. Mishev, W. T. Patula, and H. D. Voulov, βPeriodic coefficients in a reciprocal difference equation with maximum,β
*Panamerican Mathematical Journal*, vol. 13, no. 3, pp. 43β57, 2003. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - A. D. MyΕ‘kis, βSome problems in the theory of differential equations with deviating argument,β
*Uspekhi Matematicheskikh Nauk*, vol. 32, no. 2(194), pp. 173β202, 1977. View at Google Scholar Β· View at MathSciNet - W. T. Patula and H. D. Voulov, βOn a max type recurrence relation with periodic coefficients,β
*Journal of Difference Equations and Applications*, vol. 10, no. 3, pp. 329β338, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - E. P. Popov,
*Automatic Regulation and Control*, Nauka, Moscow, Russia, 1966. - I. Szalkai, βOn the periodicity of the sequence ${x}_{n+1}=\mathrm{max}\beta \x81\u2018\{{A}_{0}/{x}_{n},{A}_{1}/{x}_{n\beta \x88\x921},\beta \x80\xa6,{A}_{k}/{x}_{n\beta \x88\x92k}\}$,β
*Journal of Difference Equations and Applications*, vol. 5, no. 1, pp. 25β29, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S. SteviΔ, βBehavior of the positive solutions of the generalized Beddington-Holt equation,β
*Panamerican Mathematical Journal*, vol. 10, no. 4, pp. 77β85, 2000. View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - S. SteviΔ, βOn the recursive sequence ${x}_{n+1}=A+({x}_{n}^{p}/{x}_{n\beta \x88\x921}^{r})$,β
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 40963, 9 pages, 2007. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - S. SteviΔ, βBoundedness character of a class of difference equations,β
*Nonlinear Analysis: Theory, Methods & Applications*. In press. View at Publisher Β· View at Google Scholar - S. SteviΔ, βOn the recursive sequence ${x}_{n+1}=\mathrm{max}\beta \x81\u2018\{c,{x}_{n}^{p}/{x}_{n\beta \x88\x921}^{p}\}$,β
*Applied Mathematics Letters*, vol. 21, no. 8, pp. 791β796, 2008. View at Publisher Β· View at Google Scholar - H. D. Voulov, βPeriodic solutions to a difference equation with maximum,β
*Proceedings of the American Mathematical Society*, vol. 131, no. 7, pp. 2155β2160, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - H. D. Voulov, βOn the periodic nature of the solutions of the reciprocal difference equation with maximum,β
*Journal of Mathematical Analysis and Applications*, vol. 296, no. 1, pp. 32β43, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - H. D. Voulov, βOn a difference equation with periodic coefficients,β
*Journal of Difference Equations and Applications*, vol. 13, no. 5, pp. 443β452, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - I. YalΓ§inkaya, B. D. IriΔanin, and C. Γinar, βOn a max-type difference equation,β
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 47264, 10 pages, 2007. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - X. Yang, X. Liao, and C. Li, βOn a difference equation wtih maximum,β
*Applied Mathematics and Computation*, vol. 181, no. 1, pp. 1β5, 2006. View at Publisher Β· View at Google Scholar Β· View at MathSciNet