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On the system of high order rational difference equations.

1. Introduction

In this paper, we study the global behavior of solutions of the following system:

[x.sub.n+1] = A + [x.sub.n]/[[SIGMA].sup.k.sub.i=l] [y.sub.n-i], [y.sub.n+1] = B + [y.sub.n]/[[SIGMA].sup.k.sub.i=l] [x.sub.n-i] (1)

n = 0,1, ...,

where A, B are positive constants and initial conditions [x.sub.-i], [y.sub.-i] [member of] (0, [infinity]), i = 0,1,2, ... ,k.

A pair of sequences of positive real numbers {([x.sub.n], [y.sub.n])} that satisfies (1) is a positive solution of (1). If a positive solution of (1) is a pair of positive constants (x, y), then the solution is the equilibrium solution.

A positive solution {([x.sub.n], [y.sub.n])} of (1) is bounded and persists, if there exist positive constants M, N such that

M [less than or equal to] [x.sub.n], [y.sub.n] [less than or equal to] N, n = -2,-1, .... (2)

In 1998, DeVault et al. [1] proved that every positive solution of the difference equation

[x.sub.n+1] = A + [x.sub.n]/ [x.sub.n-1], n = 0,l, ..., (3)

where A [member of] (0, [infinity]), oscillates about the positive equilibrium c = 1 + A of (3). Moreover, every positive solution of (3) is bounded away from zero and infinity. Also the positive equilibrium of (3) is globally asymptotically stable.

In 2003, Abu-Saris and DeVault [2] studied the following recursive difference equation:

[x.sub.n+1] =A + [x.sub.n]/[x.sub.n-k], n = 0,1, ..., (4)

where A [member of] (1, + [infinity]), [x.sub.-k], [x.sub.-k+1], ..., [x.sub.0] are positive real numbers.

Papaschinopoulos and Schinas [3] investigated the global behavior for a system of the following two nonlinear difference equations:

[x.sub.n+1] = A + [y.sub.n]/[x.sub.n-p], [y.sub.n+1] = A + [x.sub.n]/[y.sub.n-q], (5)

n = 0,1, ...,

where A is a positive real number, p, q are positive integers, and [x.sub.-p], ...,[x.sub.0], [y.sub.-q] , ..., [y.sub.0] are positive real numbers.

In 2012, Zhang et al. [4] investigated the global behavior for a system of the following third-order nonlinear difference equations:

[x.sub.n+1] = A + [x.sub.n]/[y.sub.n-1] + [y.sub.n-2], [y.sub.n+1] = B [y.sub.n]/[x.sub.n-1] + [x.sub.n-2], (6)

where A,B [member of] (0,[infinity]), and the initial values [x.sub.-i],[y.sub.-i] [member of] (0, [infinity]), i = 0,1. For other related results, the reader can refer to [5-18].

Motivated by the discussion above, we study the global asymptotic behavior of solutions for system (1). More precisely, we prove the following: if A > 1/k, B > 1/k then every positive solution {([x.sub.n], [y.sub.n])} of (1) is persistent and bounded. Moreover, we prove that every positive solution {([x.sub.n], [y.sub.n])} of (1) converges the unique positive equilibrium (x, y) as n [right arrow] [infinity].

2. Main Results

In the following lemma, we show boundedness and persistence of the positive solutions of (1).

Lemma 1. Consider (1). Suppose that

A > 1/k, B > 1/k (7)

are satisfied. Then, every positive solution ([x.sub.n], [y.sub.n]) of (1) is satisfied, for n = k + 1, k + 2, ...

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Proof. Let {([x.sub.n], [y.sub.n])} be a positive solution of (1). Since [x.sub.n] > 0 and [y.sub.n] > 0 for all n [greater than or equal to] 1, (1) implies that

[x.sub.n] [greater than or equal to] A, [y.sub.n] [greater than or equal to] B, n = 1,2,3, ... (9)

Moreover, using (1) and 9), we have

[x.sub.n] [less than or equal to] A + 1/kB [x.sub.n-i], [y.sub.n] [less than or equal to] B + 1/kA [y.sub.n-i], (10)

n = k + 1,k + 2, ....

Let [v.sub.n], [w.sub.n] be the solution of the system, respectively,

[v.sub.n] = A + 1/kB [V.sub.n-1], [W.sub.n] = B + 1/kA [W.sub.n-1], n [greater than or equal to] k + 1, (11)

such that

[v.sub.i] = [x.sub.i], [w.sub.i] = [y.sub.i], i = 1,2, ...,k. (12)

We prove by induction that

[x.sub.n] [less than or equal to] [v.sub.n], [y.sub.n] [less than or equal to] [w.sub.n], n [greater than or equal to] k + 1. (13)

Suppose that (13) is true for n = m [greater than or equal to] k + 1. Then, from (10), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Therefore, (13) is true. From (11) and (12), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Then, from (9), (13), and (15), the proof of the relation (8) follows immediately.

Theorem 2. Consider the system of difference equation (1). If relation (7) is satisfied, then the following statements are true.

(i) Equation (1) has a unique positive equilibrium (x, y) given by

x = [k.sup.2] AB - 1/k(kB - 1), y = [k.sup.2] AB - 1/k(kA - 1). (16)

(ii) Every positive solution ([x.sub.n], [y.sub.n]) of system (1) tends to the positive equilibrium (x, y) of (1) as n [right arrow] [infinity].

Proof. (i) Let x and y be positive numbers such that

x = A + x/ky, y = B + y/kx, (17)

Then, from (7) and (17), we have that the positive solution (x, y) is given by (16). This completes the proof of Part (i). (ii) From (1) and 8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where [l.sub.i], [L.sub.i] [member of] (0, [infinity]), i = 1,2. Then, from (1) and (18), we get

[L.sub.1] [less than or equal to] A + [L.sub.1]/[kl.sub.2], [l.sub.1] [greater than or equal to] A + [l.sub.1]/[kL.sub.2] (19)

[L.sub.2] [less than or equal to] B + [L.sub.2]/[kl.sub.1], [l.sub.2] [greater than or equal to] B + [l.sub.2]/[kL.sub.1]

from which we have

[L.sub.1] (kB - 1) [less than or equal to] [l.sub.2] (kA - 1), [L.sub.2] (kA - 1) [less than or equal to] [l.sub.1] (kB - 1). (20)

Then, relations (7) and (20) imply that [L.sub.1][L.sub.2] [less than or equal to] [l.sub.1][l.sub.2], from which it follows that

[L.sub.1][L.sub.2] = [l.sub.1][l.sub.2]. (21)

We claim that

[L.sub.1] = [l.sub.1], [L.sub.2] = [l.sub.2]. (22)

Suppose on the contrary that [l.sub.1] < [L.sub.1]. Then, from (21), we have [L.sub.1][L.sub.2] = [l.sub.1][l.sub.2] < [L.sub.1][l.sub.2] and so [L.sub.2] < [l.sub.2] which is a contradiction. So [L.sub.1] = [l.sub.1]. Similarly, we can prove that [L.sub.2] = [l.sub.2]. Therefore, (22) is true. Hence, from (1) and (22), there exist the lim [x.sub.n] and lim [y.sub.n], as n [right arrow] [infinity] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

where (x, y) is the unique positive equilibrium of (1). This completes the proof of Part (ii). The proof of Theorem 2 is completed.

Theorem 3. Consider the system of difference equation (1). If relation (7) is satisfied and assuming that

[k.sup.2] AB - 1/kA - 1 [k.sup.2] AB - 1/kB - 1 < 1, (24)

then the unique positive equilibrium (x, y) is locally asymptotically stable.

Proof. From Theorem 2, the system of difference equation (1) has a unique equilibrium (x, y). The linearized equation of system (1) about the equilibrium point (x, y) is

[[PSI].sub.n+i] = B[[PSI].sub.n], (25)

where [[PSI].sub.n] = [([x.sub.n], ..., [x.sub.n-k], [y.sub.n], ..., [y.sub.n-k]).sup.T], and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Let [[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.2k+2] denote the eigenvalues of matrix B and let D = diag([d.sub.1],[d.sub.2], ..., [d.sub.2k+2]) be a diagonal matrix, where [d.sub.1] = [d.sub.k+2] = 1, [d.sub.i] = [d.sub.k+1+i] = 1 - i[epsilon] (i = 2, ... ,k + 1), and

0 < [epsilon] < min {1/k + 1(1 x + y/k[y.sup.2]), 1/k + 1 (1 x + y/k[x.sup.2])}. (27)

Clearly, D is invertible: Computing matrix DBD 1,we obtain that

[DBD.sup.-1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

From [d.sub.1] > [d.sub.2] > ... > [d.sub.k+1] > 0 and [d.sub.k+2] > [d.sub.k+3] > ... > [d.sub.2k+2] > 0, imply that

[d.sub.2][d.sup.-1.sub.1] < 1, [d.sub.3] [d.sup.-1.sub.2] < l, ...,[d.sub.k+1][d.sup.-1.sub.k] < 1, (29)

[d.sub.k+3] [d.sup.-1.sub.k+2] < 1, ..., [d.sub.2k+2] [d.sup.-1.sub.2k+1] < 1.

Furthermore, noting (7), (24), and (27), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

It is well known that B has the same eigenvalues as [DBD.sup.-1]; we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

This implies that the equilibrium (x, y) of (1) is locally asymptotically stable.

Combining Theorem 2 with Theorem 3, we obtain the following theorem.

Theorem 4. Consider the system of difference equation (1). If relations (7) and (24) are satisfied, then the unique positive equilibrium (x, y) is globally asymptotically stable.

3. Some Numerical Examples

In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to nonlinear difference equations and system of nonlinear difference equations.

[FIGURE 1 OMITTED]

Example 1. Consider the following difference equations:

[x.sub.n+1] = 0.8 + [x.sub.n]/[y.sub.n-1] + [y.sub.n-2] + [y.sub.n-3] (32)

[y.sub.n+1] = 0.6 + [y.sub.n]/[X.sub.n-1] + [X.sub.n-2] + [X.sub.n-3]

with the initial values [x.sub.-i] = [y.sub.-i] = 0.5 (i = 1,2,3). Then, the solution ([x.sub.n], [y.sub.n]) of system (32) is bounded and persists and the system has a unique equilibrium (x,y) = (1.3833, 0.7905) which is globally asymptotically stable (see Figure 1).

Example 2. Consider the following difference equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

with the initial values [x.sub.-i] = [y.sub.-i] = 1.5 (i = 1,2, 3,4). Then, the solution ([x.sub.n], [y.sub.n]) of system (33) is bounded and persists and the system has a unique equilibrium (x,y) = (1.1929, 0.7591) which is globally asymptotically stable (see Figure 2).

4. Conclusion

In this paper, we study the dynamics of a system of high order difference equation

[x.sub.n+1] = A + [x.sub.n]/[[SIGMA].sup.k.sub.i=1] [y.sub.n-1], [y.sub.n+1] = B + [y.sub.n]/[[SIGMA].sup.k.sub.i=1] [x.sub.n-1] (34)

N = 0,1, ..., k [member of] {1,2, ...}.

[FIGURE 2 OMITTED]

It concluded that, under condition A > 1/k,B > 1/k, the positive solution ([x.sub.n], [y.sub.n]) of this system is bounded and persists; moreover, if (([k.sup.2]AB - 1)/(kA - 1)) + (([k.sup.2] AB - 1)/(kB - 1)) < 1, it converges asymptotically the unique equilibrium (x,y).

We conclude the paper by presenting the following open problem.

Open Problem. Consider the system of difference equation (1) with A [less than or equal to] 1/k and B [less than or equal to] 1/k. Find the set of all initial conditions that generate bounded solutions. In addition, investigate global behavior of these solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

http://dx.doi.org/10.1155/2014/760502

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grant no. 11361012), the Scientific Research Foundation of Guizhou Provincial Science and Technology Department ([2013]J2083, [2009]J2061), and the Natural Science Foundation of Guizhou Provincial Educational Department (no. 2008040).

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Qianhong Zhang, (1) Wenzhuan Zhang, (1) Yuanfu Shao, (2) and Jingzhong Liu (3)

(1) Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, Guizhou 550004, China

(2) School of Science, Guilin University of Technology, Guilin, Guangxi 541000, China

(3) Department of Mathematics and Physics, Hunan Institute of Technology, Hengyang, Hunan 421002, China

Correspondence should be addressed to Qianhong Zhang; zqianhong68@163.com

Received 15 March 2014; Revised 2 July 2014; Accepted 22 July 2014; Published 29 October 2014

Academic Editor: Chunrui Zhang
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Title Annotation:Research Article
Author:Zhang, Qianhong; Zhang, Wenzhuan; Shao, Yuanfu; Liu, Jingzhong
Publication:International Scholarly Research Notices
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Date:Jan 1, 2014
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