Research Article | Open Access

# On the Max-Type Equation with a Period-Two Parameter

**Academic Editor:**Cengiz Γinar

#### Abstract

We study the behavior of the well-defined solutions of the max type difference equation , , where the initial conditions are arbitrary nonzero real numbers and is a period-two sequence of real numbers with .

#### 1. Introduction and Preliminaries

Recently, the study of max-type difference equations attracted a considerable attention. Although max-type difference equations are relatively simple in form, it is, unfortunately, extremely difficult to understand thoroughly the behavior of their solutions; see, for example [1β39] and the relevant references cited therein. Max-type difference equations stem from certain models in automatic control theory (see [1, 24]). For some papers on periodicity of difference equation, see, for example, [15, 16, 19, 22] and the relevant references cited therein.

In [9], Simsek et al. studied the behavior of the solutions of the following max-type difference equation: where the initial conditions are nonzero real numbers.

In [10], Simsek studied the behavior of the solutions of the following max-type difference equation: where the initial conditions are negative real numbers.

In [18], Elabbasy and Elsayed studied the behavior of the solutions of (1.2) where the initial conditions are nonzero real numbers.

In [20], Elsayed and SteviΔ showed that every well-defined solution of the difference equation where , is eventually periodic with period three.

In [21], Elsayed and IriΔanin showed that every positive solution to the following third-order nonautonomous max-type difference equation: where is a three-periodic sequence of positive numbers and is periodic with period three.

In [29], YalΓ§inkaya et al. studied the behavior of the solutions of the following max-type difference equation: where and initial conditions are nonzero real numbers.

In this paper, we study the behavior of the well-defined solutions of the max type difference equation where the initial conditions are arbitrary nonzero real numbers and is a period-two sequence of real numbers with .

We need the following definitions and lemmas.

*Definition 1.1. *A sequence is said to be eventually periodic with period if there is such that for all . If , then we say that the sequence periodic with period .

We make two definitions regarding (1.6).

*Definition 1.2. *A right semicycle is a string of terms with , such that for all . Furthermore, if , and if .

*Definition 1.3. *A left semicycle is a string of terms with , such that for all . Furthermore, if , , and if ,.

We give the following lemmas which show us the periodic behavior of the solutions of (1.6).

Lemma 1.4. *Assume that is a well-defined solution of (1.6). If and such that , then the solution is eventually periodic with period two.*

*Proof. *We prove that
by induction. For , this is, assumption. Assume that (1.7) holds for all . We may assume that is odd. Then, by the inductive hypothesis, we have
from this and the inductive hypothesis, we have
which completes the proof (the case is even similar, so it will be omitted).

We omit the proof of the following lemma, since it can easily be obtained by induction.

Lemma 1.5. *Assume that is a well-defined solution of (1.6). If such that , then for all .*

Lemma 1.6. *Assume that is a well-defined solution of (1.6) and . If this solution is eventually positive, then it is eventually periodic with period two.*

*Proof. *Assume that is the smallest index such that for all . Then, we have
Using this, we have
then we get
Observe that there exists a positive integer such that
From this directly follows that is eventually periodic with period two.

Lemma 1.7. *Equation (1.6) has no right semicycle with an infinite terms for the positive initial conditions and .*

*Proof. *Conversely, assume that (1.6) has a right semicycle with an infinite terms. And, let be periodic sequence of natural numbers with period two such that . Without loss of generality, we denote by the first term of right semicycle with an infinite terms. There is at least . For all , we can write
which implies
But this is a contradiction which completes the proof.

We omit the proof of the following lemma, since it can easily be obtained similarly.

Lemma 1.8. *Equation (1.6) has no right semicycle with an infinite terms for the negative initial conditions and .*

#### 2. Main Results

Since is a two periodic, it has the form . If , then (1.6) becomes , from which it follows that every well-defined solution is periodic with period two. Hence, in the sequel, we will consider the case when at least one of and is not zero.

##### 2.1. The Case

Theorem 2.1. *If and at least one of the initial conditions is arbitrary positive real number, then every well-defined solution of (1.6) is eventually periodic with period two.*

*Proof. *Firstly, assume that . Then, we have . There are two cases to be considered.(a)If , then . Hence,
From Lemma 1.4, the result follows.(b)If , then . We have
There are two subcases to be considered.(b_{1})If , then . Hence,
From Lemma 1.4, the result follows in this case.(b_{2})If , then . We have
There are two subcases to be considered.(b_{21})If , then . We have
From Lemma 1.4, the result follows in this case.(b_{22})If , then . The result follows Lemma 1.7. Secondly, assume that , then we have
From Lemmas 1.5 and 1.6, the result follows (the case is similar, so it will be omitted) which completes the proof.

*Remark 2.2. *If and , then every well-defined solution of (1.6) is not periodic.

##### 2.2. The Case or

Theorem 2.3. *If or , then every well-defined solution of (1.6) is eventually periodic with period two.*

*Proof. *First assume that . Then, we have
From Lemmas 1.5 and 1.6, the result follows. The case is similar, so it will be omitted.

##### 2.3. The Other Cases

If at least one of and greater than one, then we have the well-defined solutions of (1.6), where the positive initial conditions are not periodic. So, there are many cases in which solutions of (1.6) are not periodic. If the solutions of (1.6) are not periodic, then general solution of (1.6) can be obtained for many subcases.

Theorem 2.4. *Assume that is a well-defined solution of (1.6) for and .*(a)*If and or , then
* (b)*If and or , then
*

*Proof. *(a) It can be proved by induction. Let and . For , (2.8) holds. Assume that (2.8) holds for all . We may assume that is even (the case is odd is similar, so it will be omitted). Then, by the inductive hypothesis, we have
which completes the proof.

(b) Also, this case can be proved similarly.

Now, we describe the behavior of solutions of (1.6) for some other cases. We omit the proof of the following theorem, since it can easily be obtained by induction.

Theorem 2.5. *Assume that is a well-defined solution of (1.6).*(a)*If and , then*(b)*If and , then
*(c)*If and ,,, then
*

There are many different cases. The different cases can be obtained similarly.

Theorem 2.6. *If and initial conditions are negative, then every well-defined solution of (1.6) is eventually periodic with period two.*

*Proof. *Assume that . Then,
There are two cases to be considered.(a)If , then ,, . Then, the result follows Lemma 1.4.(b)If , then . There are two subcases.(b_{1})If , then , . Then the result follows Lemma 1.4.(b_{2})If , then there will be subcases and from Lemmas 1.4 and 1.8 which completes the proof.

#### Acknowledgment

I am grateful to the anonymous referees for their valuable suggestions that improved the quality of this study.

#### References

- A. D. Myškis, βSome problems in the theory of differential equations with deviating argument,β
*Uspekhi Matematicheskikh Nauk*, vol. 194, no. 32-2, pp. 173β202, 1977. View at: Google Scholar - A. Gelisken, C. Cinar, and R. Karatas, βA note on the periodicity of the Lyness max equation,β
*Advances in Difference Equations*, vol. 2008, Article ID 651747, 5 pages, 2008. View at: Google Scholar | Zentralblatt MATH - A. Gelisken, C. Cinar, and I. Yalcinkaya, βOn the periodicity of a difference equation with maximum,β
*Discrete Dynamics in Nature and Society*, Article ID 820629, 11 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Gelişken and C. Çinar, βOn the global attractivity of a max-type difference equation,β
*Discrete Dynamics in Nature and Society*, Article ID 812674, 5 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Çinar, S. Stević, and I. Yalçinkaya, 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: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. M. Kent and M. A. Radin, βOn the boundedness nature of positive solutions of the difference equation ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{{A}_{n}/{x}_{n},{B}_{n}/{x}_{n-1}\right\}$ with periodic parameters,β
*Dynamics of Continuous, Discrete & Impulsive Systems B*, supplement, pp. 11β15, 2003. View at: Google Scholar - 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 Site | Google Scholar | Zentralblatt MATH | 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 | Zentralblatt MATH - D. Simsek, C. Cinar, and I. Yalcinkaya, βOn the solutions of the difference equation ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{{x}_{n-1},1/{x}_{n-1}\right\}$,β
*International Journal of Contemporary Mathematical Sciences*, vol. 10, no. 1, pp. 481β487, 2006. View at: Google Scholar | Zentralblatt MATH - D. Simsek, βOn the solutions of the difference equation ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{{x}_{n-2},1/{x}_{n-2}\right\}$,β
*Selcuk University Journal of Education Faculty*, vol. 23, pp. 367β377, 2007. View at: Google Scholar - E. A. Grove and G. Ladas,
*Periodicities in Nonlinear Difference Equations*, vol. 4, Chapman & Hall, Boca Raton, Fla, USA, 2005. - E. C. Pielou,
*An Introduction to Mathematical Ecology*, Wiley-Interscience, New York, NY, USA, 1969. - E. C. Pielou,
*Population and Community Ecology*, Gordon and Breach, 1974. - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, βOn the periodic nature of some max-type difference equations,β
*International Journal of Mathematics and Mathematical Sciences*, no. 14, pp. 2227β2239, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, βGlobal attractivity and periodic character of a fractional difference equation of order three,β
*Yokohama Mathematical Journal*, vol. 53, no. 2, pp. 89β100, 2007. View at: Google Scholar | Zentralblatt MATH - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, βQualitative behavior of higher order difference equation,β
*Soochow Journal of Mathematics*, vol. 33, no. 4, pp. 861β873, 2007. View at: Google Scholar | Zentralblatt MATH - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, βQualitative behavior of some max-type difference equations,β
*Vietnam Journal of Mathematics*, vol. 36, no. 1, pp. 47β61, 2008. View at: Google Scholar | Zentralblatt MATH - E. M. Elabbasy and E. M. Elsayed, βOn the solution of the recursive sequence ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{{x}_{n-2},1/{x}_{n-2}\right\}$,β
*Fasciculi Mathematici*, no. 41, pp. 55β63, 2009. View at: Google Scholar | Zentralblatt MATH - E. M. Elsayed, βOn the solutions of higher order rational system of recursive sequences,β
*Mathematica Balkanica*, vol. 22, no. 3-4, pp. 287β296, 2008. View at: Google Scholar | Zentralblatt MATH - E. M. Elsayed and S. Stević, βOn the max-type equation ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{A/{x}_{n},{x}_{n-2}\right\}$,β
*Nonlinear Analysis*, vol. 71, no. 3-4, pp. 910β922, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. M. Elsayed and B. D. Iričanin, βOn a max-type and a min-type difference equation,β
*Applied Mathematics and Computation*, vol. 215, no. 2, pp. 608β614, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. M. Elsayed, βDynamics of a recursive sequence of higher order,β
*Communications on Applied Nonlinear Analysis*, vol. 16, no. 2, pp. 37β50, 2009. View at: Google Scholar | Zentralblatt MATH - E. J. Janowski, V. L. Kocic, G. Ladas, and S. W. Schultz, βGlobal behavior of solutions of ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{{x}_{n},A\right\}/{x}_{n-1}$,β in
*Proceedings of the 1st International Conference on Difference Equations*, pp. 273β282, Gordon and Breach Science, San Antonio, Tex, USA. View at: Google Scholar | Zentralblatt MATH - E. P. Popov,
*Automatic Regulation and Control*, Nauka, Moscow, Russia, 1966. - F. Sun, βOn the asymptotic behavior of a difference equation with maximum,β
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 243291, 6 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. Szalkai, βOn the periodicity of the sequence ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{{A}_{0}/{x}_{n},{A}_{1}/{x}_{n-1},\dots ,{A}_{k}/{x}_{n-k}\right\}$,β
*Journal of Difference Equations and Applications*, vol. 5, no. 1, pp. 25β29, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. Yalçinkaya, B. D. Iričanin, and C. Çinar, βOn a max-type difference equation,β
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 47264, 10 pages, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. Yalcinkaya, C. Cinar, and A. Gelisken, βOn the recursive sequence ${x}_{n+1}=\mathrm{max}\{{x}_{n},A\}/{x}_{n}^{2}{x}_{n-1}$,β
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 583230, 13 pages, 2010. View at: Publisher Site | Google Scholar - J. Feuer, βOn the eventual periodicity of ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{1/{x}_{n},{A}_{n}/{x}_{n-1}\right\}$ with a period-four parameter,β
*Journal of Difference Equations and Applications*, vol. 12, no. 5, pp. 467β486, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Feuer and K. T. McDonnell, βOn the eventual periodicity of ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{1/{x}_{n},{A}_{n}/{x}_{n-1}\right\}$ with a period-five parameter,β
*Computers and Mathematics with Applications*, vol. 56, no. 4, pp. 883β890, 2008. View at: Publisher Site | Google Scholar - J. Feuer, βPeriodic solutions of the Lyness max equation,β
*Journal of Mathematical Analysis and Applications*, vol. 288, no. 1, pp. 147β160, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. S. Berenhaut, J. D. Foley, and S. Stević, 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Abu-Saris and F. Allan, βPeriodic and nonperiodic solutions of the difference equation ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{{x}_{n}^{2},A\right\}/{x}_{n}{x}_{n-1}$,β in
*Advances in Difference Equations*, pp. 9β17, Gordon and Breach, Amsterdam, The Netherlands, 1997. View at: Google Scholar | Zentralblatt MATH - R. M. Abu-Sarris and F. M. Allan, βRational recursive sequences involving the maximum function,β
*Far East Journal of Mathematical Sciences*, vol. 1, no. 3, pp. 335β342, 1999. View at: Google Scholar | Zentralblatt MATH - S. Stević, βOn the recursive sequence ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{{c,x}_{n}^{p}/{x}_{n-1}^{p}\right\}$,β
*Applied Mathematics Letters*, vol. 21, no. 8, pp. 791β796, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Sun, B. Qin, H. Xi, and C. Han, βGlobal behavior of the max-type difference equation ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\left\{1/{x}_{n},{A}_{n}/{x}_{n-1}\right\}$,β
*Abstract and Applied Analysis*, vol. 2009, Article ID 152964, 10 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2012 İbrahim Yalçınkaya. 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.