- Research Article
- Open Access
On a Max-Type Difference Equation
© Ali Gelisken et al. 2010
- Received: 8 December 2009
- Accepted: 23 April 2010
- Published: 30 May 2010
- Differential Equation
- Real Number
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
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, [1–20] and the relevant references cited therein. The max operator arises naturally in certain models in automatic control theory (see [13, 14]). Furthermore, difference equation appear naturally as a discrete analogue and as a numerical solution of differential and delay differential equations having applications and various scientific branches, such as in ecology, economy, physics, technics, sociology, and biology.
In this paper, we investigate the asymptotic behavior of the positive solutions of (1.4). We prove that every positive solution of (1.4) converges to Clearly, we can assume that without loss of generality.
We need the following two lemmas in order to prove the main result of this section.
where initial conditions are real numbers.
We need the following lemma in order to prove the main result of this section.
The authors are grateful to the anonymous referees for their valuable suggestions that improved the quality of this study.
- Abu-Saris RM, Allan FM:Periodic and nonperiodic solutions of the difference equation max . In Advances in Difference Equations (Veszprém, 1995). Gordon and Breach, Amsterdam, The Netherlands; 1997:9-17.Google Scholar
- Amleh AM, Hoag J, Ladas G: A difference equation with eventually periodic solutions. Computers & Mathematics with Applications 1998,36(10–12):401-404. 10.1016/S0898-1221(98)80040-0MathSciNetView ArticleMATHGoogle Scholar
- Berenhaut KS, Foley JD, Stević S: Boundedness character of positive solutions of a max difference equation. Journal of Difference Equations and Applications 2006,12(12):1193-1199. 10.1080/10236190600949766MathSciNetView ArticleMATHGoogle Scholar
- Briden WJ, Grove EA, Ladas G, Kent CM:Eventually periodic solutions of . Communications on Applied Nonlinear Analysis 1999,6(4):31-43.MathSciNetMATHGoogle Scholar
- Briden WJ, Grove EA, Ladas G, McGrath LC:On the nonautonomous equation . In New Developments in Difference Equations and Applications (Taipei, 1997). Gordon and Breach, Amsterdam, The Netherlands; 1999:49-73.Google Scholar
- Çinar C, Stević S, Yalçinkaya I: On positive solutions of a reciprocal difference equation with minimum. Journal of Applied Mathematics & Computing 2005,17(1-2):307-314. 10.1007/BF02936057MathSciNetView ArticleMATHGoogle Scholar
- Gelişken A, Çinar C, Karataş R: A note on the periodicity of the Lyness max equation. Advances in Difference Equations 2008, 2008:-5.Google Scholar
- Gelişken A, Çinar C, Yalçinkaya I: On the periodicity of a difference equation with maximum. Discrete Dynamics in Nature and Society 2008, 2008:-11.Google Scholar
- Gelişken A, Çinar C: On the global attractivity of a max-type difference equation. Discrete Dynamics in Nature and Society 2009, 2009:-5.Google Scholar
- Grove EA, Kent C, Ladas G, Radin MA:On the with a period 3 parameter. In Fields Institute Communications. Volume 29. American Mathematical Society, Providence, RI, USA; 2001:161-180.Google Scholar
- Ladas G:On the recursive sequence . Journal of Difference Equations and Applications 1996,2(3):339-341. 10.1080/10236199608808067MathSciNetView ArticleGoogle Scholar
- Mishev DP, Patula WT, Voulov HD: A reciprocal difference equation with maximum. Computers & Mathematics with Applications 2002,43(8-9):1021-1026. 10.1016/S0898-1221(02)80010-4MathSciNetView ArticleMATHGoogle Scholar
- Myškis AD: Some problems in the theory of differential equations with deviating argument. Uspekhi Matematicheskikh Nauk 1977,32(2(194)):173-202.Google Scholar
- Popov EP: Automatic Regulation and Control. Nauka, Moscow, Russia; 1966.Google Scholar
- Szalkai I:On the periodicity of the sequence . Journal of Difference Equations and Applications 1999,5(1):25-29. 10.1080/10236199908808168MathSciNetView ArticleMATHGoogle Scholar
- Stević S:On the recursive sequence . Applied Mathematics Letters 2008,21(8):791-796. 10.1016/j.aml.2007.08.008MathSciNetView ArticleMATHGoogle Scholar
- Sun F: On the asymptotic behavior of a difference equation with maximum. Discrete Dynamics in Nature and Society 2008, 2008:-6.Google Scholar
- Voulov HD: On the periodic character of some difference equations. Journal of Difference Equations and Applications 2002,8(9):799-810. 10.1080/1023619021000000780MathSciNetView ArticleMATHGoogle Scholar
- Yalçinkaya I, Iričanin BD, Çinar C: On a max-type difference equation. Discrete Dynamics in Nature and Society 2007,2007(1):-10.Google Scholar
- Yang X, Liao X, Li C: On a difference equation with maximum. Applied Mathematics and Computation 2006,181(1):1-5. 10.1016/j.amc.2006.01.005MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.