On the Behavior of the Positive Solutions of the System of Two Higher-Order Rational Difference Equations

Abstract

We study the convergence of the positive solutions of the system of the following two difference equations:

where K is a positive integer, the parameters A,B, α, β  and the initial conditions are positive real numbers. Our results generalize well known results in [1,2].

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Q. Wang, G. Zhang and L. Fu, "On the Behavior of the Positive Solutions of the System of Two Higher-Order Rational Difference Equations," Applied Mathematics, Vol. 4 No. 8, 2013, pp. 1220-1225. doi: 10.4236/am.2013.48164.

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

Conflicts of Interest

The authors declare no conflicts of interest.

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