Abstract
In this paper, we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case as a special case of the following system: with initial conditions , are nonzero real numbers. Moreover, we study some behavior of the systems such as the boundedness of solutions for such systems. Finally, we present some numerical examples by giving some numerical values for the initial values of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program MATLAB to confirm the obtained results.
1. Introduction
We believe that difference equations, also referred to as recursive sequence, are a hot topic here as there has been increasing interest in the study of qualitative analysis of difference equations and systems of difference equations. Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economics, physics, computer sciences, and so on. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solution (see [1–9] and the references cited therein). Recently, a great effort has been made in studying the qualitative analysis of rational difference equations and rational difference system (see [10–35]).
The study of the nonlinear rational difference equations is quite challenging and rewarding [2, 8]. The results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations [1]. Recently, many researchers have investigated the behavior of the solution of difference equations. Difference equations arise in the situations in which the discrete values of the independent variable involve. Many practical phenomena are modeled with the help of difference equations [1]. In engineering, difference equations arise in control engineering, digital signal processing, electrical networks, etc. In social sciences, difference equations arise to study the national income of a country and then its variation with time, Cobweb phenomenon in economics, etc.
There are many papers related to the difference equation system, for example, the periodicity of the positive solutions of the rational difference equations system:has been obtained by Cinar in [4].
Khan et al. [6] studied the equilibrium points, local asymptotic stability of an equilibrium point, instability of equilibrium points, periodicity behavior of positive solutions, and global character of an equilibrium point of a fourth-order system of rational difference equations of the form:
Elabbasy et al. [7] has obtained the solution of particular cases of the following general system of difference equations:
In [36], Elsayed et al. dealt with the solutions of the systems of the difference equations:
Kurbanli [13–15] investigated the behavior of the solutions of the difference equation systems:
In [21], Yalcinkaya et al. studied the periodic character of the following two systems of difference equations:where the initial values are nonzero real numbers for .
In [37], Zhang et al. studied the boundedness, the persistence, and global asymptotic stability of the positive solutions of the system of difference equations:
Zkan and Kurbanli [38] have investigated the periodical solutions of the following system of third-order rational difference equations:
Similar to the references above, this paper is devoted to obtain the form of the solutions of the following third-order systems of rational difference equations:where the initial conditions , are nonzero real numbers.
2. The System:
We obtain the form of solutions of the following system of difference equations:
Theorem 1. If be the solution of (10), then
Proof. Obviously, results are true for . Suppose that it is also true for , i.e.,From equation (10), it follows thatNow, one can see thatMoreover, from (10), one hasThen,Finally, from equation (10),Thus,In a similar way, other relations can also be proved.
Lemma 1. If is the solution of (10), then it is bounded as well as converge to zero.
Proof. From (10),we see thatTherefore, subsequences are nonincreasing and so are bounded from above by . Similarly, the subsequences are nonincreasing and hence bounded above by and are nonincreasing and hence bounded from above by .
Example 1. Figure 1 represents the dynamics of (10) with , , and .
3. The System:
Here, we will discuss solutions of the following system:where and nonzero initial conditions such that , , , and .
Theorem 2. If are solutions of (21), then solutions of (21) are represented by the following formulas for
Proof. Obviously, results are true for . Now, one suppose that, for , it is also true, i.e.,Now, from equation (21), it follows thatAlso, from equation (21), one can see thatIn a similar pattern, one can prove the other relation.
Example 2. Figure 2 represents the dynamics of (21) with , , and .
In a similar pattern, one can prove the following cases.
4. The System: , , and
We discuss the solutions of the following system:where and nonzero initial conditions such that , and .
Theorem 3. If are solutions of (26), then
Example 3. Figure 3 represents the behavior of (26) if , .
5. The System: , , and
We discuss the solutions of the following system:where and nonnegative initial conditions such that , , , and .
Theorem 4. If are solutions of (28), then
Example 4. Figure 4 represents the dynamics of (28) with , , and .
6. The System: , , and
We will obtain the solutions of the following system:where and nonzero initial conditions such that , , , and .
Theorem 5. If are solutions of (30), then
Example 5. Figure 5 represents the dynamics of (30) with , .
7. Conclusion
This paper discussed boundedness and the form expressions of the solutions of some special cases of difference equation systems , after the introduction in Section 1, we investigated in Section 2, the boundedness behavior of equation (10), and we have obtained the solutions of this system. In Section 3, we obtained the form of the solutions of the system . Also, in Section 4, we gave the solution form of the system . In Section 5, we dealt with the solution of system (28). Finally, in Section 6, we have the expressions of the form of the solutions of the system , and in each case, we presented some numerical examples of each case and drew it by using MATLAB.
Data Availability
All the data utilized in this article have been included, and the sources were it was adopted were cited accordingly.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
The work of H. S. Alayachi and M. S. M. Noorani was financially supported by the UKM (grant number DIP-2017-011) and Ministry of Education Malaysia (grant number FRGS/1/2017/STG06/UKM/01/1), while A. Q. Khan’s research was partially supported by the Higher Education Commission of Pakistan.