On the Solutions of Systems of Difference Equations
© İbrahim Yalçinkaya et al. 2008
Received: 19 March 2008
Accepted: 19 May 2008
Published: 21 May 2008
Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so on . So, recently there has been an increasing interest in the study of qualitative analysis of rational difference equations and systems of difference equations. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solutions. (see [1–11] and the references cited therein).
which can be considered as a natural generalizations of (1.8).
From Lemma 1.1 we see that the rests for of the numbers for obtained by dividing the numbers by , are mutually different, they are contained in the set , make a permutation of the ordered set , and finally is the first number of the form such that
2. The Main Results
In this section, we formulate and prove the main results in this paper.
which yields the result.
which yields the result.
then all solutions of (1.9) are positive.
From (2.16) and (2.17), all solutions of (1.9) are positive.
From (2.16), (2.17), and (2.18), the proof is clear.
From (2.16), (2.17), and (2.19), the proof is clear.
From (2.16) and (2.17), the proof is clear.
then all solutions of (1.10) are positive.
From (2.21) and (2.22), all solutions of (1.10) are positive.
From (2.21), (2.22) and (2.23), the proof is clear.
From (2.21), (2.22) and (2.24), the proof is clear.
From (2.21), (2.22), and (2.24), the proof is clear.
Let . Then the solutions of (1.9), with the initial values and in its invertal of periodicity can be represented by Table 1.
The authors are grateful to the anonymous referees for their valuable suggestions that improved the quality of this study.
- Papaschinopoulos G, Schinas CJ: On a system of two nonlinear difference equations. Journal of Mathematical Analysis and Applications 1998, 219(2):415–426. 10.1006/jmaa.1997.5829MATHMathSciNetView ArticleGoogle Scholar
- Camouzis E, Papaschinopoulos G: Global asymptotic behavior of positive solutions on the system of rational difference equations , . Applied Mathematics Letters 2004, 17(6):733–737. 10.1016/S0893-9659(04)90113-9MATHMathSciNetView ArticleGoogle Scholar
- Çinar C: On the positive solutions of the difference equation system , Applied Mathematics and Computation 2004, 158(2):303–305. 10.1016/j.amc.2003.08.073MATHMathSciNetView ArticleGoogle Scholar
- Çinar C, Yalçinkaya İ: On the positive solutions of difference equation system , , International Mathematical Journal 2004, 5(5):521–524.MATHMathSciNetGoogle Scholar
- Clark D, Kulenović MRS: A coupled system of rational difference equations. Computers & Mathematics with Applications 2002, 43(6–7):849–867. 10.1016/S0898-1221(01)00326-1MATHMathSciNetView ArticleGoogle Scholar
- Grove EA, Ladas G, McGrath LC, Teixeira CT: Existence and behavior of solutions of a rational system. Communications on Applied Nonlinear Analysis 2001, 8(1):1–25.MATHMathSciNetGoogle Scholar
- Özban AY: On the positive solutions of the system of rational difference equations , Journal of Mathematical Analysis and Applications 2006, 323(1):26–32. 10.1016/j.jmaa.2005.10.031MATHMathSciNetView ArticleGoogle Scholar
- Özban AY: On the system of rational difference equations , Applied Mathematics and Computation 2007, 188(1):833–837. 10.1016/j.amc.2006.10.034MATHMathSciNetView ArticleGoogle Scholar
- Papaschinopoulos G, Schinas CJ: On the behavior of the solutions of a system of two nonlinear difference equations. Communications on Applied Nonlinear Analysis 1998, 5(2):47–59.MATHMathSciNetGoogle Scholar
- Papaschinopoulos G, Schinas CJ: Invariants for systems of two nonlinear difference equations. Differential Equations and Dynamical Systems 1999, 7(2):181–196.MATHMathSciNetGoogle Scholar
- Papaschinopoulos G, Schinas CJ, Stefanidou G: On a -order system of lyness-type difference equations. Advances in Difference Equations 2007, 2007:-13.Google Scholar
- Irićanin B, Stević S: Some systems of nonlinear difference equations of higher order with periodic solutions. Dynamics of Continuous, Discrete and Impulsive Systems, Series A Mathematical Analysis 2006, 13: 499–507.MATHMathSciNetGoogle 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.