Abstract
In this paper, we obtain the solution forms of fifth order systems of rational difference equations and . Where the initial values are nonzero real numbers. Numerical examples are also provided.
1. Introduction
The applications of discrete dynamical systems and difference equations have grown rapidly, especially in economics and modern sciences. In [1], El-Metwally et al. investigated the asymptotic behavior of the population model:
Elsayed et al. [2] studied the discrete population model
In economy, Askar [3] studied the discrete dynamical systems of duopoly games
Many of authors investigated the behavior of nonlinear systems of difference equations when the solutions are difficult to obtain. For example, Khaliq and Zubair in [4] studied properties of solutions such as persistence, boundedness, global stability, and locally asymptotically stable of equilibrium point
It is very interesting to get solutions of systems of nonlinear difference equations if possible. Other authors were able to obtain the solutions of difference equations. For instance, Alayachi et al. in [5] studied the formula of solutions of the following difference equations system of order three
Elsayed et al. [6] obtained the solution expressions and studied the behavior of three-dimensional systems of difference equations
See other related results on difference equations [7–17].
Through such a paper, we will get the solution form for six cases of the fifth order systems given by following form:where the initial conditions are any real number. Also, some numerical examples and figures will be presented to confirm the results of the obtained solutions.
2. The First Case
Through this section, we obtain the solution formula of system of the three difference equations
Theorem 1. Assume that are solutions of difference equations system. Then, for , we see that all solutions of system (8) are given by the following formulas:
Proof. For the result holds. Now suppose that and that our assumption holds for , that is,Now, we find from system (8) that
Example 1. Figure 1 shows numerical solution of system (8) with the initial conditions .
Example 2. This illustrates the numerical solution of system (8) with the initial conditions (See Figure 2).
3. Second Case
Through this section, we obtain the solution form of the system of three difference equations given by the following:
Theorem 2. Assume that are solutions of difference equation system. Then for , we see that all solutions of system (12) are given by the following formulas:
Proof. The proof is derived from the proof of the previous theorem.
Example 3. Figure 3 shows numerical solution of system (12) with the initial conditions .
Example 4. This illustrates the numerical solution of system (12) with the initial conditions (See Figure 4).
4. Third Case
In this section, we investigate the solutions of the difference equation system
Theorem 3. Assume that are solutions of difference equation system. Then for , we see that all solutions of system (14) are given by the following formulas:
Proof. This proof is obtained similar to the First Theory.
Example 5. Figure 5 shows numerical solution of system (14) with the initial conditions .
Example 6. This illustrates the numerical solution of system (14) with the initial conditions (See Figure 6).
5. Fourth Case
In this section, we investigate the solutions of the difference equation system:
Theorem 4. Assume that are solutions of difference equation system. Then for , we see that all solutions of system (16) are given by the following formulas:
Proof. This proof is obtained similar to the First Theory.
Example 7. Figure 7 shows numerical solution of system (16) with the initial conditions .
Example 8. For illustrates the numerical solution of system (16) with the initial conditions (See Figure 8).
6. Fifth Case
In this section, we investigate the solutions of the difference equation system:
Theorem 5. Assume that are solutions of difference equation system. Then for , we see that all solutions of system (18) are given by the following formulas:
Proof. This proof is derived from the proof of First theorem.
Example 9. Figure 9 shows the numerical solution of system (18) with the initial conditions .
Example 10. This illustrates the numerical solution of system (18) with the initial conditions (See Figure 10).
7. Sixth Case
In this section, we investigate the solutions of the difference equation system
Theorem 6. Assume that are solutions of difference equation system. Then for , we see that all solutions of system (20) are given by the following formulas:
Proof. This proof is obtained similar to the First theorem.
Example 11. Figure 11 shows numerical solution of system (20) with the initial conditions .
Example 12. This illustrates the numerical solution of system (20) with the initial conditions (See Figure 12).
Data Availability
All the data utilized in this article have been included, and the sources from where they were adopted were cited accordingly.
Conflicts of Interest
The authors declare that they have no conflicts of interest.