- Research Article
- Open Access
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
- Differential Equation
- Qualitative Analysis
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
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
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.
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