Abstract
Consider the following system of difference equations: , , , , , ; where is a positive integer, , , and the initial conditions , are positive real numbers. We obtain the expressions of the positive solutions of the system and then give a precise description of the convergence of the positive solutions. Finally, we give some numerical results.
1. Introduction
Difference equation or system of difference equations is a diverse field which impacts almost every branch of pure and applied mathematics. Not only does it provide us with some simple and useful mathematic models to help elucidate interesting phenomena in applications, but also it can kind of display some surprising complicated dynamics comparing with its analogue differential equations. Hence, the systems of difference equations and difference equations have attracted a lot of attention (see, e.g., the systems of difference equations [1–16] and difference equations [17–29] and the references therein). Among them, symmetric and close to symmetric systems of difference equations have attracted a considerable interest.
Papaschinnopoulos and Schinas [1] studied the oscillatory behavior, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the system of the nonlinear difference equations:In [2], they also investigated the boundedness, persistence, the oscillatory behavior, and the asymptotic behavior of the positive solutions of the system of difference equations:
Clark et al. [3, 4] investigated the global asymptotic stability of the system of difference equations:
Camouzis and Papaschinopoulos [5] studied the global asymptotic behavior of positive solutions of the system of rational difference equations:
Yang [6] studied the behavior of positive solutions of the system of difference equations:
Zhang et al. [7] studied the boundedness, the persistence, and global asymptotic stability of the positive solutions of the system of difference equations:
Yalçinkaya and Çinar [8] studied the global asymptotic stability of the system of difference equations:
Kurbanlı et al. [9] studied the behavior of the positive solutions of the following system of difference equations:
Motivated by the above studies, in this note, we consider the following system of difference equations:where is a positive integer, , , and the initial conditions , , are positive real numbers. We perfect and generalize the results in related literature.
2. Main Results
Throughout this paper, let and stand for the set of natural numbers and the set of real numbers, respectively.
Let be a positive solution of (9). If we set then (9) translates intowhere , .
For convenience, in the following we will investigate (11). Set where we appeal to the convention
Combing (12) with (11), we get
By (11), (12), and (13), we get
Lemma 1. Let be a positive solution of (11); then
Proof. From (12) we know from (11) we obtain and combining (19) with (20) we get the conclusion.
This completes the proof.
Lemma 2. Let be a positive solution of (11); then
Proof. For ; , by (18), (13), and (14), we haveHence, (21) holds.
Lemma 3. Let be a positive solution of (11); then where we appeal to the convention
Proof. We will prove the conclusion by induction. For , , it is obvious that (23) holds. For , , from Lemma 2, we know that (23) holds.
Suppose that (23) holds for , then for , by Lemma 2 we haveHence, (23) holds for , from which we get the conclusion.
In the following, setwhere is floor function.
Lemma 4. Let be a positive solution of (11); then
Proof. In fact, for ; by (11) and Lemma 3 we haveHence, (26) holds.
In the following, set It is obvious that
Lemma 5. For , the following statements are true.(1)Suppose that , or , or , ; then .(2)Suppose that , or , ; then .(3)Suppose that , or , ; then .
Proof. (1) Case 1. , . Note that It follows that
Case 2. , . Note that Hence,
Case 3. , . By (16) and (17), we have That is, When , , combining (31) with (33) we get Hence,
(2) Case 1. , . In this case, by (33) we know andHence, Note that the series is convergent, and we have
Case 2. , . Since The series is convergent, and we know that
(3) Case 1. , . Note that Hence,
Case 2. , . Combining (31) with (33) we get Hence, .
Theorem 6. Let be a positive solution of (11). The following statements are true.(1)Suppose that , or , or , ; then , .(2)Suppose that , or , ; then , .(3)Suppose that , or , ; then , .
Proof. By Lemma 4 and (28) we know The conclusion follows by Lemma 5.
Theorem 7. Let be a positive solution of (9). The following statements are true.(1)Suppose that , or , or , ; then , .(2)Suppose that , or , ; then , .(3)Suppose that , or , ; then , .
Proof. The proof follows by Theorem 6 and (10).
3. Numerical Results
In this section, we give some numerical simulations to illustrate our results. Consider the following system of difference equations:
For convenience, set , and , .
Example 1. In (41), we take , , From Table 1 and Figure 1(a) we see that
(a)
(b)
(c)
(d)
Example 2. In (41), we take , , From Table 2 and Figure 1(b) we see that
Example 3. In (41), we take , , From Table 3 and Figure 1(c) we see that
Example 4. In (41), we take , , From Table 4 and Figure 1(d) we see that
Example 5. In (41), we take , , From Table 5 and Figure 2(a) we see that
(a)
(b)
Example 6. In (41), we take , , From Table 6 and Figure 2(b) we see that
Example 7. In (41), we take , , From Table 7 and Figure 3(a) we see that
(a)
(b)
Example 8. In (41), we take , , From Table 8 and Figure 3(b) we see that
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This research is supported by National Key Scientific Research Project (no. 11631005), the National Natural Science Foundation of China (nos. 11461002, 11471085, 91230104, and 11301103), Program for Changjiang Scholars and Innovative Research Team in University (Grant no. IRT1226), Program for Yangcheng Scholars in Guangzhou (Grant no. 12A003S), and Science and Technology Research Project of Colleges and Universities of Guangxi (no. LX2014194).