Abstract
We investigate the solutions of the system of difference equations , , , where .
1. Introduction
Recently, there has been great interest in studying difference equation systems. One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real life situations in population biology, economic, probability theory, genetics, psychology, and so forth. There are many papers related to the difference equations system, for example, the following papers.
In [1], รinar studied the solutions of the systems of the difference equations
In [2] Papaschinopoulos and Schinas studied the oscillatory behavior, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the system of nonlinear difference equations
In [3] Papaschinopoulos and Schinas proved the boundedness, persistence, the oscillatory behavior, and the asymptotic behavior of the positive solutions of the system of difference equations
In [4, 5] รzban studied the positive solutions of the system of rational difference equations
In [6, 7] Clark and Kulenoviฤ investigate the global asymptotic stability
In [8] Camouzis and Papaschinopoulos studied the global asymptotic behavior of positive solutions of the system of rational difference equations
In [9] Yang et al. considered the behavior of the positive solutions of the system of the difference equations
In [10] Kulenoviฤ and Nurkanoviฤ studied the global asymptotic behavior of solutions of the system of difference equations
In [11] Zhang et al. investigated the behavior of the positive solutions of the system of difference equations
In [12] Zhang et al. studied the boundedness, the persistence, and global asymptotic stability of the positive solutions of the system of difference equations
In [13] Yalcinkaya studied the global asymptotic behavior of a system of two nonlinear difference equations.
In [14] Yalcinkaya et al. investigated the solutions of the system of difference equations
In [15] Yalcinkaya studied the global asymptotic stability of the system of difference equations
In [16] Iriฤanin and Steviฤ studied the positive solutions of the system of difference equations
In [17] Kurbanli et al. studied the behavaior of positive solutions of the system of rational difference equations
Also see references.
In this paper, we investigate the behavior of the solutions of the difference equations system where the initial conditions are arbitrary real numbers.
Theorem 1.1. Let be arbitrary real numbers and and let be a solution of the system (1.15). Also, assume that and then all solutions of (1.15) are
Proof. For , we have
for assume that
are true. Then for we will show that (1.16) is true. From (1.15), we have
Also, similarly from (1.15), we have
from properties of Binomial coefficients,
written and accurate.
Also, we have
wheredenotes.
Corollary 1.2. Let be only positive real numbers or negative real numbers and let be arbitrary nonnegative real numbers, and let be a solution of the system (1.15). If and then one has
Proof. From and we have and . Hence, we have
Then,
Also,
Example 1.3. If , then the solutions of (1.15) can be represented by Table 1.
Corollary 1.4. Let be only positive real numbers or negative real numbers and let be arbitrary nonnegative real numbers, and let be a solution of the system (1.15) If and then one has
Proof.
From and we have and . Hence, we have
Then,
Also,
Example 1.5. If then the solutions of (1.15) can be represented by Table 2.
Corollary 1.6. Let be only positive real numbers or negative real numbers and let be arbitrary nonnegative real numbers, and let be a solution of the system (1.15) If and then one has
Proof. From and we have and . Hence, we have
Then,
Also,
Example 1.7. If then the solutions of (1.15) can be represented by Table 3.
Corollary 1.8. Let be only positive real numbers or negative real numbers and let be arbitrary nonnegative real numbers, and let be a solution of the system (1.15). If and then one has
Proof. From and we have and . Hence, we have
Then,
Also,
Example 1.9. If then the solutions of (1.15) can be represented by Table 4.
Acknowledgment
The authors are grateful to the anonymous referees for their valuable suggestions that improved the quality of this study.