On the Behavior of the Positive Solutions of the System of Two Higher-Order Rational Difference Equations ()
where is a positive integer, the parameters QUOTE and the initial conditions are positive real numbers. Our results generalize well known results in [1,2].
1. Introduction
Difference equations and the system of difference equations play an important role in the analysis of mathematical models of biology, physics and engineering. The study of dynamical properties of nonlinear difference equations and the system of difference equations have been an area of intense interest in recent years (for example, see [1-11]).
In [1], Kurbanlı, Ģinar and Yalçinkaya studied the behavior of the positive solutions of the following system of difference equations
In [2], Stevo Stević investigated the system of the following difference equations
Motivated by the above studies, in this note, we consider the system of the following difference equations
(1)
where is a positive integer, the parameters A, and the initial conditions are positive real numbers.
System (1) is a particular case of the system of the following difference equations
(2)
If , then system (2) is trivial. If , system is reduced to system (1) with
and. Hence from now on, we will consider system (1).
On the other hand, system (1) is a natural generalizetion of the equation
(3)
where is a positive integer, the parameters are positive real numbers. Hence the results which we obtained can also apply to (3).
The essential problem we consider in this paper is the behavior of the positive solutions of system (1). We establish the convergence of the positive solutions of system (1). To some extent, this generalizes the results obtained in [1,2]. It is interesting that a modification of our method enables us to investigate the form of the positive solutions of system (1).
2. Main Results
For convenience, set,
Let be a positive solution of (1). Set
(4)
then (1) translates into
(5)
Set
for
Lemma 2.1 For (5), we have
(6)
Proof. Since
Hence (6) holds for .
For assume that (6) is true.
Note that, for
Then for we have
Hence (6) is true.
Theorem 2.2 Let be a positive solution of (1). For
1) When, we have
,
where
satisfy:
2) When.
a) Suppose, then
b) Suppose, then
c) Suppose
i) If then for
ii) If then for
iii) If then
iv) If then
where
satisfy
Proof. Note that for
Hence
(7)
(8)
(9)
(10)
1) When. In view of (4), (6), (7) and (8), we drive
Thus
Similarly, in view of (4), (6), (9) and (10), we get
When note that the positive series
is convergent, in view of (4) and (6), we drive
Similarly, we get
2) When
a) b) The proof is similar to the proof of 1, we get
,
c) Suppose.
i) If In view of (1), by induction, we drive, for
ii) If, Similarly, we drive for
iii) Note that
Hence, if the positive series
is convergent. If, the negative series is convergent.
In view of (4) and (6), thus if then Similarly, we get
iv) If Similarly, we get
3. Acknowledgements
This work was supported by the NSF of China (No. 11161029), and supported by NSF of Guangxi (No. 2012GXNSFDA276040, 2013GXNSFBA019020), NSF of the Department of Education of Guangxi Province (No. 200103YB157). NSF of Guangxi University of Science and Technology (No. 1166218).
NOTES