# On the Solutions of a System of Third-Order Rational Difference Equations.

1. Introduction

The prime objective of this paper is to analyze the forms of the solution of the following difference equation systems:

[mathematical expression not reproducible], (1)

in which the initial conditions [x.sub.-2], [x.sub.-1], [x.sub.0], [y.sub.-2], [y.sub.-1], [y.sub.0] are arbitrary positive real numbers.

Difference equations appear as a natural model of evolution phenomena. Furthermore, considerable findings in difference equation theory have been retrieved as discrete analogues and as numerical solutions of differential equations. This is notably true in the case of Lyapunov theory of stability. What is more, it has applications in biology, ecology, economy, physics, and so on. Due to this matter, there has been a rise in the interest in the study of qualitative analysis of scalar rational difference equations and rational system of difference equations. Although difference equations look simple in form, it is quite difficult to understand thoroughly the behaviors of their solutions because some prototypes for the development of the basic theory of the global behavior of nonlinear difference equation come from the results of rational difference equations (see [1-6]) and the references cited therein.

El-Dessoky and Elsayed [7] have analyzed the form of the solutions and the periodicity character of the following systems of rational difference equations:

[mathematical expression not reproducible]. (2)

Elsayed and Alghamdi [8] have examined the form of the solutions of the following nonlinear difference equations systems:

[mathematical expression not reproducible]. (3)

Elsayed et al. [9] have investigated the form of the solutions of the following difference equations systems:

[mathematical expression not reproducible]. (4)

Kurbanli [10] has explored the following system of difference equations:

[mathematical expression not reproducible]. (5)

Mansour et al. [11] have got the form of the solutions of some systems of the following rational difference equations:

[mathematical expression not reproducible]. (6)

Touafek and Elsayed [12] have studied the periodicity and gave the form of the solutions of the following systems of difference equations of order two:

[mathematical expression not reproducible]j. (7)

Equations similar to difference equations and nonlinear systems of rational difference equations were investigated; see [13-20].

2. Main Results

11. The First System: [mathematical expression not reproducible]. The solutions of the system of two difference equations are analyzed in this subsection.

[mathematical expression not reproducible], (8)

where the initial conditions [x.sub.-2], [x.sub.-1], [x.sub.0], [y.sub.-2], [y.sub.-1], [y.sub.0] are arbitrary positive real numbers.

Theorem 1. Suppose that [[x.sub.n], [y.sub.n]} are solutions of system (8). Then for n = 0,1,2,..., one has

[mathematical expression not reproducible], (9)

where [x.sub.0] = a, [x.sub.-1] = b, [x.sub.-2] = c, [y.sub.0] = d, [y.sub.-1] = e, [y.sub.-2] = f and where [f.sub.n] is the Fibonacci Sequence with [f.sub.-2] = 1, [f.sub.-1] = 0. Proof. For n = 0, the result holds. By using mathematical induction now suppose that n > 0 and that our assumption holds for -1,n-2. That is,

[mathematical expression not reproducible]. (10)

Now, it follows from system (8) substitution of the above equations that

[mathematical expression not reproducible]. (11)

Also, we obtain

[mathematical expression not reproducible]. (12)

Hence the proof is complete.

Lemma 2. Let {[x.sub.n], [y.sub.n]} be a positive solution of system (8); then every solution of system (8) is bounded and converges to zero.

Proof. It follows from system (8) that

[mathematical expression not reproducible]; (13)

thus

[mathematical expression not reproducible]. (14)

Then the subsequences [{[x.sub.2n-1]}.sup.[infinity].sub.n=0], [{[x.sub.2n]}.sup.[infinity].sub.n=0] are decreasing and so are bounded from above by

M = max {[x.sub.-2],[x.sub.-1],[x.sub.0]}. (15)

Similarly the subsequences [{[y.sub.2n-1]}.sup.[infinity].sub.n=0], [{[y.sub.2n]}.sup.[infinity].sub.n=0] are decreasing and so are bounded from above by

N = max {[y.sub.-2], [y.sub.-1],[y.sub.0]}. (16)

22 The Second System: [mathematical expression not reproducible]. The structure of the solutions of the following system of the difference equations is examined in this subsection.

[mathematical expression not reproducible], (17)

with nonzero positive real numbers of initial conditions [mathematical expression not reproducible].

Theorem 3. Suppose that {[x.sub.n], [y.sub.n]} are solutions of the system

[mathematical expression not reproducible]. (18)

Then for n = 0,1,2,..., one has

[mathematical expression not reproducible], (19)

where [mathematical expression not reproducible] is the Fibonacci Sequence with [f.sub.-2] = 1, [f.sub.-1] = 0.

Proof. For n = 0 the result holds. By using mathematical induction now suppose that n > 0 and that our assumption holds for n - 1,n-2. That is,

[mathematical expression not reproducible]. (20)

Now, it follows from system (17) substitution of the above equations that

[mathematical expression not reproducible]. (21)

Also, we obtain

[mathematical expression not reproducible]. (22)

Hence the proof is complete.

2.3. The Third System: [mathematical expression not reproducible]. In this subsection, the framework of the solution of the following system of difference equations is acquired.

[mathematical expression not reproducible], (23)

where the initial conditions [x.sub.-2], [x.sub.-1], [x.sub.0], [y.sub.-2], [y.sub.-1], [y.sub.0] are arbitrary positive real numbers with [x.sub.-2] [not equal to] [y.sub.-1].

Theorem 4. Suppose that {[x.sub.n], [y.sub.n]} are solutions of system (23). Then for n = 0, 1, 2, ..., one sees that all solutions of system (23) are given by the following formulas:

[mathematical expression not reproducible], (24)

where [mathematical expression not reproducible] is the Fibonacci Sequence with [f.sub.-2] = 1, [f.sub.-1] = 0.

Proof. For n = 0 the result holds. By using mathematical induction now suppose that n > 0 and that our assumption holds for n - 1. That is,

[mathematical expression not reproducible]. (25)

Now, it follows from system (23) substitution of the above equations that

[mathematical expression not reproducible]. (26)

Also, we obtain

[mathematical expression not reproducible]. (27)

Thus, the other relations can be proven in a similar way. Hence the proof is complete.

2.4. The Fourth System: [mathematical expression not reproducible]. In this subsection, we analyze the solutions of the system of the following two difference equations.

[mathematical expression not reproducible], (28)

with the initial conditions [x.sub.-2], [x.sub.-1], [x.sub.0] and [y.sub.-2], [y.sub.-1], [y.sub.0] being nonzero positive real numbers.

Theorem 5. Let [{[x.sub.n], [y.sub.n]}.sup.+[infinity].sub.n=-1] be solutions of system (28). Then [{[x.sub.n]}.sup.+[infinity].sub.n=-1] and [{[y.sub.n]}.sup.+[infinity].sub.n=-1] are given by the following formulas for n= 0, 1, 2, ...:

[mathematical expression not reproducible], (29)

where [x.sub.0] = a, [x.sub.-1] = b, [x.sub.-2] = c, [y.sub.0] = d, [y.sub.-1] = e, [y.sub.-2] = f and where fn is the Fibonacci Sequence with [f.sub.-2] = 1, [f.sub.-1] = 0.

Proof. For n = 0 the result holds. By using mathematical induction now suppose that n > 0 and that our assumption holds for n - 1. That is,

[mathematical expression not reproducible]. (30)

Now, it follows from system (28) substitution of the above equations that

[mathematical expression not reproducible]. (31)

Also, we obtain

[mathematical expression not reproducible]. (32)

Thus, the other relations can be proven in a similar way. Hence the proof is complete.

3. Numerical Examples

To illustrate the results of foregoing sections and to support our theoretical discussions, we take into account several interesting numerical examples in this section.

Example 1 (first system: [mathematical expression not reproducible]. Figure 1 considers the solution of the difference equations system (8) with the initial conditions [x.sub.-2] = 1, [x.sub.-1] = 2, [x.sub.0] = 3 and [y.sub.-2] = 4, [y.sub.-1] = 5, [y.sub.0] = 6.

Example 2 (second system: [mathematical expression not reproducible]. Figure 2 indicates the solution of the difference equation system (17) with the initial conditions [x.sub.-2] = 4, [x.sub.-1] = 7, [x.sub.0] = 2 and [y.sub.-2] = 2, [y.sub.-1] = 5, [y.sub.0] = 1.

Example 3 (third system: [mathematical expression not reproducible]. Figure 3 considers the solution of the difference equation system (23) with the initial conditions [x.sub.-2] = 4, [x.sub.-1] = 7, [x.sub.0] = 2 and [y.sub.-2] = 2, [y.sub.-1] = 5, [y.sub.0] = 1

Example 4 (fourth system: [mathematical expression not reproducible]. Figure 4 shows the solution of the difference equation system (28) with the initial conditions [x.sub.-2] = 4, [x.sub.-1] =7, [x.sub.0] = 2 and [y.sub.-2] = 2, [y.sub.-1] = 5, [y.sub.0] = 1.

4. Conclusion

In this analysis, we oversee the structure of the solutions of four cases of the difference equations system [mathematical expression not reproducible]. Additionally, some behavior of the solutions such as boundedness is investigated. Subsequently, some numerical examples are displayed by presenting some numerical values for the initial values per case and figures provided to justify the behavior of the obtained solutions in the case of numerical examples.

https://doi.org/10.1155/2018/1743540

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is financially supported by UKM Grant DIP 2017-011 and Ministry of Education Malaysia Grant FRGS/1/ 2017/STG06/UKM/01/1.

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A. M. Alotaibi (iD), (1) M. S. M. Noorani, (1) and M. A. El-Moneam (2)

(1) School of Mathematical Sciences, Universiti Kebangsaan Malaysia, Bangi, Selangor, Malaysia

(2) Mathematics Department, Faculty of Science, Jazan University, Jazan, Saudi Arabia

Correspondence should be addressed to A. M. Alotaibi; ab-alo@hotmail.com

Received 21 January 2018; Accepted 20 March 2018; Published 3 May 2018

Caption: Figure 1: This figure shows the solution of (8).

Caption: Figure 2: This figure shows the solution of (17).

Caption: Figure 3: This figure shows the solution of (23).

Caption: Figure 4: This figure shows the solution of (28).