Discrete Dynamics in Nature and Society
VolumeÂ 2012Â (2012), Article IDÂ 327437, 9 pages
http://dx.doi.org/10.1155/2012/327437
Research Article

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

Department of Mathematics, Ahmet Kelesoglu Education Faculty, Konya University, 42090 Meram Campus, Meram Yeni Yol, Konya, Turkey

Received 30 September 2011; Accepted 12 November 2011

Copyright Â© 2012 Ä°brahim YalĂ§Ä±nkaya. 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 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 [1â€“39] 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 SteviÄ‡ showed that every well-defined solution of the difference equation where , is eventually periodic with period three.

In [21], Elsayed and IriÄŤanin 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], YalĂ§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.(b1)If , then . Hence, From Lemma 1.4, the result follows in this case.(b2)If , then . We have There are two subcases to be considered.(b21)If , then . We have From Lemma 1.4, the result follows in this case.(b22)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.(b1)If , then , . Then the result follows Lemma 1.4.(b2)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

1. A. D. Myškis, â€śSome problems in the theory of differential equations with deviating argument,â€ť Uspekhi Matematicheskikh Nauk, vol. 194, no. 32-2, pp. 173â€“202, 1977.
2. A. Gelisken, C. Cinar, and R. Karatas, â€śA note on the periodicity of the Lyness max equation,â€ť Advances in Difference Equations, vol. 2008, Article ID 651747, 5 pages, 2008.
3. A. Gelisken, C. Cinar, and I. Yalcinkaya, â€śOn the periodicity of a difference equation with maximum,â€ť Discrete Dynamics in Nature and Society, Article ID 820629, 11 pages, 2008.
4. A. Gelişken and C. Çinar, â€śOn the global attractivity of a max-type difference equation,â€ť Discrete Dynamics in Nature and Society, Article ID 812674, 5 pages, 2009.
5. C. Çinar, S. Stević, and I. Yalçinkaya, â€śOn positive solutions of a reciprocal difference equation with minimum,â€ť Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 307â€“314, 2005.
6. C. M. Kent and M. A. Radin, â€śOn the boundedness nature of positive solutions of the difference equation ${x}_{n+1}=\mathrm{max} \left\{{A}_{n}/{x}_{n},{B}_{n}/{x}_{n-1}\right\}$ with periodic parameters,â€ť Dynamics of Continuous, Discrete & Impulsive Systems B, supplement, pp. 11â€“15, 2003.
7. D. P. Mishev, W. T. Patula, and H. D. Voulov, â€śA reciprocal difference equation with maximum,â€ť Computers & Mathematics with Applications, vol. 43, no. 8-9, pp. 1021â€“1026, 2002.
8. D. P. Mishev, W. T. Patula, and H. D. Voulov, â€śPeriodic coefficients in a reciprocal difference equation with maximum,â€ť Panamerican Mathematical Journal, vol. 13, no. 3, pp. 43â€“57, 2003.
9. D. Simsek, C. Cinar, and I. Yalcinkaya, â€śOn the solutions of the difference equation ${x}_{n+1}=\mathrm{max} \left\{{x}_{n-1},1/{x}_{n-1}\right\}$,â€ť International Journal of Contemporary Mathematical Sciences, vol. 10, no. 1, pp. 481â€“487, 2006.
10. D. Simsek, â€śOn the solutions of the difference equation ${x}_{n+1}=\mathrm{max} \left\{{x}_{n-2},1/{x}_{n-2}\right\}$,â€ť Selcuk University Journal of Education Faculty, vol. 23, pp. 367â€“377, 2007.
11. E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, vol. 4, Chapman & Hall, Boca Raton, Fla, USA, 2005.
12. E. C. Pielou, An Introduction to Mathematical Ecology, Wiley-Interscience, New York, NY, USA, 1969.
13. E. C. Pielou, Population and Community Ecology, Gordon and Breach, 1974.
14. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, â€śOn the periodic nature of some max-type difference equations,â€ť International Journal of Mathematics and Mathematical Sciences, no. 14, pp. 2227â€“2239, 2005.
15. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, â€śGlobal attractivity and periodic character of a fractional difference equation of order three,â€ť Yokohama Mathematical Journal, vol. 53, no. 2, pp. 89â€“100, 2007.
16. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, â€śQualitative behavior of higher order difference equation,â€ť Soochow Journal of Mathematics, vol. 33, no. 4, pp. 861â€“873, 2007.
17. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, â€śQualitative behavior of some max-type difference equations,â€ť Vietnam Journal of Mathematics, vol. 36, no. 1, pp. 47â€“61, 2008.
18. E. M. Elabbasy and E. M. Elsayed, â€śOn the solution of the recursive sequence ${x}_{n+1}=\mathrm{max} \left\{{x}_{n-2},1/{x}_{n-2}\right\}$,â€ť Fasciculi Mathematici, no. 41, pp. 55â€“63, 2009.
19. E. M. Elsayed, â€śOn the solutions of higher order rational system of recursive sequences,â€ť Mathematica Balkanica, vol. 22, no. 3-4, pp. 287â€“296, 2008.
20. E. M. Elsayed and S. Stević, â€śOn the max-type equation ${x}_{n+1}=\mathrm{max} \left\{A/{x}_{n},{x}_{n-2}\right\}$,â€ť Nonlinear Analysis, vol. 71, no. 3-4, pp. 910â€“922, 2009.
21. E. M. Elsayed and B. D. Iričanin, â€śOn a max-type and a min-type difference equation,â€ť Applied Mathematics and Computation, vol. 215, no. 2, pp. 608â€“614, 2009.
22. E. M. Elsayed, â€śDynamics of a recursive sequence of higher order,â€ť Communications on Applied Nonlinear Analysis, vol. 16, no. 2, pp. 37â€“50, 2009.
23. E. J. Janowski, V. L. Kocic, G. Ladas, and S. W. Schultz, â€śGlobal behavior of solutions of ${x}_{n+1}=\mathrm{max} \left\{{x}_{n},A\right\}/{x}_{n-1}$,â€ť in Proceedings of the 1st International Conference on Difference Equations, pp. 273â€“282, Gordon and Breach Science, San Antonio, Tex, USA.
24. E. P. Popov, Automatic Regulation and Control, Nauka, Moscow, Russia, 1966.
25. F. Sun, â€śOn the asymptotic behavior of a difference equation with maximum,â€ť Discrete Dynamics in Nature and Society, vol. 2008, Article ID 243291, 6 pages, 2008.
26. H. D. Voulov, â€śPeriodic solutions to a difference equation with maximum,â€ť Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2155â€“2160, 2003.
27. H. D. Voulov, â€śOn the periodic nature of the solutions of the reciprocal difference equation with maximum,â€ť Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 32â€“43, 2004.
28. I. Szalkai, â€śOn the periodicity of the sequence ${x}_{n+1}=\mathrm{max} \left\{{A}_{0}/{x}_{n},{A}_{1}/{x}_{n-1},\dots ,{A}_{k}/{x}_{n-k}\right\}$,â€ť Journal of Difference Equations and Applications, vol. 5, no. 1, pp. 25â€“29, 1999.
29. I. Yalçinkaya, B. D. Iričanin, and C. Çinar, â€śOn a max-type difference equation,â€ť Discrete Dynamics in Nature and Society, vol. 2007, Article ID 47264, 10 pages, 2007.
30. I. Yalcinkaya, C. Cinar, and A. Gelisken, â€śOn the recursive sequence ${x}_{n+1}=\mathrm{max}\left\{{x}_{n},A\right\}/{x}_{n}^{2}{x}_{n-1}$,â€ť Discrete Dynamics in Nature and Society, vol. 2010, Article ID 583230, 13 pages, 2010.
31. J. Feuer, â€śOn the eventual periodicity of ${x}_{n+1}=\mathrm{max} \left\{1/{x}_{n},{A}_{n}/{x}_{n-1}\right\}$ with a period-four parameter,â€ť Journal of Difference Equations and Applications, vol. 12, no. 5, pp. 467â€“486, 2006.
32. J. Feuer and K. T. McDonnell, â€śOn the eventual periodicity of ${x}_{n+1}=\mathrm{max} \left\{1/{x}_{n},{A}_{n}/{x}_{n-1}\right\}$ with a period-five parameter,â€ť Computers and Mathematics with Applications, vol. 56, no. 4, pp. 883â€“890, 2008.
33. J. Feuer, â€śPeriodic solutions of the Lyness max equation,â€ť Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 147â€“160, 2003.
34. K. S. Berenhaut, J. D. Foley, and S. Stević, â€śBoundedness character of positive solutions of a max difference equation,â€ť Journal of Difference Equations and Applications, vol. 12, no. 12, pp. 1193â€“1199, 2006.
35. R. Abu-Saris and F. Allan, â€śPeriodic and nonperiodic solutions of the difference equation ${x}_{n+1}=\mathrm{max} \left\{{x}_{n}^{2},A\right\}/{x}_{n}{x}_{n-1}$,â€ť in Advances in Difference Equations, pp. 9â€“17, Gordon and Breach, Amsterdam, The Netherlands, 1997.
36. R. M. Abu-Sarris and F. M. Allan, â€śRational recursive sequences involving the maximum function,â€ť Far East Journal of Mathematical Sciences, vol. 1, no. 3, pp. 335â€“342, 1999.
37. S. Stević, â€śOn the recursive sequence ${x}_{n+1}=\mathrm{max} \left\{{c,x}_{n}^{p}/{x}_{n-1}^{p}\right\}$,â€ť Applied Mathematics Letters, vol. 21, no. 8, pp. 791â€“796, 2008.
38. T. Sun, B. Qin, H. Xi, and C. Han, â€śGlobal behavior of the max-type difference equation ${x}_{n+1}=\mathrm{max} \left\{1/{x}_{n},{A}_{n}/{x}_{n-1}\right\}$,â€ť Abstract and Applied Analysis, vol. 2009, Article ID 152964, 10 pages, 2009.
39. W. T. Patula and H. D. Voulov, â€śOn a max type recurrence relation with periodic coefficients,â€ť Journal of Difference Equations and Applications, vol. 10, no. 3, pp. 329â€“338, 2004.