Research Article | Open Access
On the System of High Order Rational Difference Equations
This paper is concerned with the boundedness, persistence, and global asymptotic behavior of positive solution for a system of two rational difference equations , where and
In this paper, we study the global behavior of solutions of the following system: where are positive constants and initial conditions , .
A pair of sequences of positive real numbers that satisfies (1) is a positive solution of (1). If a positive solution of (1) is a pair of positive constants , then the solution is the equilibrium solution.
A positive solution of (1) is bounded and persists, if there exist positive constants such that
In 1998, DeVault et al.  proved that every positive solution of the difference equation where , oscillates about the positive equilibrium 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  studied the following recursive difference equation: where , are positive real numbers.
Papaschinopoulos and Schinas  investigated the global behavior for a system of the following two nonlinear difference equations: where is a positive real number, are positive integers, and , are positive real numbers.
In 2012, Zhang et al.  investigated the global behavior for a system of the following third-order nonlinear difference equations: where , and the initial values , . 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 then every positive solution of (1) is persistent and bounded. Moreover, we prove that every positive solution of (1) converges the unique positive equilibrium as .
2. Main Results
In the following lemma, we show boundedness and persistence of the positive solutions of (1).
Proof. Let be a positive solution of (1). Since and for all , (1) implies that
Moreover, using (1) and (9), we have Let be the solution of the system, respectively, such that We prove by induction that Suppose that (13) is true for . Then, from (10), we get Therefore, (13) is true. From (11) and (12), we have 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 given by (ii)Every positive solution of system (1) tends to the positive equilibrium of (1) as .
Proof. (i) Let and be positive numbers such that
Then, from (7) and (17), we have that the positive solution is given by (16). This completes the proof of Part (i).
(ii) From (1) and (8), we have where , . Then, from (1) and (18), we get from which we have Then, relations (7) and (20) imply that , from which it follows that We claim that Suppose on the contrary that . Then, from (21), we have and so which is a contradiction. So . Similarly, we can prove that . Therefore, (22) is true. Hence, from (1) and (22), there exist the and , as and where is the unique positive equilibrium of (1). This completes the proof of Part (ii). The proof of Theorem 2 is completed.
Proof. From Theorem 2, the system of difference equation (1) has a unique equilibrium . The linearized equation of system (1) about the equilibrium point is where , and Let denote the eigenvalues of matrix and let be a diagonal matrix, where , , and Clearly, is invertible. Computing matrix , we obtain thatFrom and , imply that Furthermore, noting (7), (24), and (27), we have It is well known that has the same eigenvalues as ; we have that This implies that the equilibrium of (1) is locally 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.
Example 1. Consider the following difference equations: with the initial values . Then, the solution of system (32) is bounded and persists and the system has a unique equilibrium which is globally asymptotically stable (see Figure 1).
Example 2. Consider the following difference equations: with the initial values . Then, the solution of system (33) is bounded and persists and the system has a unique equilibrium which is globally asymptotically stable (see Figure 2).
In this paper, we study the dynamics of a system of high order difference equation It concluded that, under condition , the positive solution of this system is bounded and persists; moreover, if , it converges asymptotically the unique equilibrium .
We conclude the paper by presenting the following open problem.
Open Problem. Consider the system of difference equation (1) with and . 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.
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 (J2083, J2061), and the Natural Science Foundation of Guizhou Provincial Educational Department (no. 2008040).
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