Mathematical Problems in Engineering

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Discrete Fractional-Order Systems with Applications in Engineering and Natural Sciences 2021

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Research Article | Open Access

Volume 2021 |Article ID 1902473 | https://doi.org/10.1155/2021/1902473

Abeer Alshareef, Faris Alzahrani, Abdul Qadeer Khan, "Dynamics and Solutions’ Expressions of a Higher-Order Nonlinear Fractional Recursive Sequence", Mathematical Problems in Engineering, vol. 2021, Article ID 1902473, 12 pages, 2021. https://doi.org/10.1155/2021/1902473

Dynamics and Solutions’ Expressions of a Higher-Order Nonlinear Fractional Recursive Sequence

Academic Editor: Abdelalim Elsadany
Received19 Apr 2021
Accepted20 May 2021
Published09 Jun 2021

Abstract

The principle purpose of this article is to examine some stability properties for the fixed point of the below rational difference equation where , and are arbitrary real numbers. Moreover, solutions for some special cases of the proposed difference equation are introduced.

1. Introduction

In recent years, many researchers have tended to use difference equations in mathematical models to explain the problems in different sciences since they have a lot of features such as they enable the scientists to introduce the predictions of their study and it gives more accurate results. In addition, there are various types of nonlinear difference equations that can be studied; one of the most commonly used is rational nonlinear difference equations. However, the research studies in the area of difference equations have two directions: first one is the analysis of the behavior of solutions. Therefore, there are a huge number of articles published to investigate the stability of the equilibrium points and the existence of the periodic solutions for the nonlinear difference equations (see, for example, [15]). The second direction is to obtain the expressions of the solution if it is possible since there is no explicit and enough methods to find the solution of nonlinear difference equations (see, for example, [611]).

Saleh and Farhat [12] investigated the stability properties and the period two solutions of all nonnegative solutions of the difference equation:

In [13], Jia studied the solutions’ behavior of the high-order fuzzy difference equation:

Kerker et al. [14] investigated the global behavior of the rational difference equation:

Khaliq and Elsayed [15] examined the dynamics behavior and existence of the periodic solution of the difference equation:

In [16], Saleh et al. studied the properties’ stability for a nonlinear rational difference equation of a higher order:

Sadiq and Kalim [17] obtained the solution behavior of the difference equation:

To see more related work on the nonlinear difference equation, refer to [1843]. Our aim of this article is to investigate the dynamics of the solution for the below difference equation:where , and are arbitrary real numbers with initial conditions for .

This paper is collected as follows: in Section 2, the boundedness of the solution is presented, and we prove that the periodic solution of period two does not exist in the next section. Following that, we state the conditions of the local and global stability of the equilibrium point in Sections 4 and 5, respectively. Then, we introduce the solutions’ forms for some special cases in Section 6. Finally, we give some numerical examples in order to illustrate the behavior of the solutions.

2. Boundedness of Solution

Theorem 1. If the following conditionis true, then every solution of (7) is bounded.

Proof. Assume that is a solution of (7). Then, from (7), we haveHence,Implies that the subsequences , , and are nonincreasing. Thus, they are bounded from above by , where .

3. Periodicity of the Solution

Theorem 2. For nonlinear difference equation (7), there is no periodic solution of period two.

Proof. To prove Theorem 2, suppose that (7) has a positive prime period two solutions presented as . Then,Similarly,Subtracting (11) from (12), we getSince , thus , and this contradicts the fact that .

4. The Equilibrium Point and Local Stability

The fixed points of (7) are given by

If , then (7) has only one equilibrium point which is .

Assume is a continuously differentiable function defined by

Therefore,

Then,

Hence,

Theorem 3. The fixed point is said to be a locally asymptotically stable if the relationis satisfied.

Proof. From Theorem 5.10 in [44], it follows that is asymptotically stable ifwhere and . Then,Hence,Finally, the proof is done.

5. Global Attractivity of the Fixed Point

Theorem 4. The fixed point of (7) has to be a global attracting when

Proof. From (16), we see that the function , which defined in (15), is increasing in and decreasing in . Let be a solution of the system:Therefore,Subtracting (25) from (26), we getand then, if . Thus, from Theorem 5.20 in [44], we observe that there exists only one solution for (7) and it is a global attractor if .

6. Special Cases

Now, we present the solutions’ expressions for special cases of (7):where the initial conditions areand are arbitrary real numbers.

6.1. First Equation

We solve the equation

Theorem 5. Assume is a solution of (30); thus, for ,where is the Fibonacci sequence.

Proof. We show that the expressions in (31) are solutions of (30) by applying mathematical induction. First, the results hold for . Second, we suppose that the forms are satisfied for and . Now, we prove that the results are satisfied for :From (30), it follows thatTherefore,Similarly, one can investigate other expressions. The proof is done.

6.2. Second Equation

In this section, we introduce the solution of the following equation:

Theorem 6. Let be a solution of (35); then, for ,where is the Fibonacci sequence.

Proof. The proof will be the same as proof of Theorem 5, so it is therefore omitted.

6.3. Third Equation

In this section, we present the solution of the following equation:

Theorem 7. Let be a solution of (37); then, for ,where .

Proof. By using mathematical induction, we prove that (38) are solutions of (37). First, the results for are true. Second, assume that the assumption holds for and .Now, from (37), we haveThus,Similarly, one can see that the other forms are true. The proof is complete.

6.4. Fourth Equation

We study the following equation:

Theorem 8. Suppose that is a solution of (42), then there exists a periodic solution with period 54. Moreover, takes the form

Proof. The proof of this case will be the same as the proof presented for Theorem 7 and will be omitted therefore.

7. Numerical Examples

To illustrate the solution behavior of (7) for various cases, we present some numerical examples.

Example 1. To show the stability of (7), we set two groups for the values of the coefficients: (i) and (ii) , and the initial conditions areand . The result is obtained in Figure 1. It is clear that (i) condition (23) is satisfied, which implies that the solution tends to the fixed point , while the solution moves away from the fixed point for (ii) since condition (23) failed.
The following examples have explained the solutions of special case equations (30)–(42).

Example 2. We choose the initial conditions asand . The solution is given in Figure 2.

Example 3. In Figure 3, we set the initial conditions:

Example 4. For (37), we choose the initial conditions asand then, the result is shown in Figure 4.

Example 5. We set the valuesThe solution is given in Figure 5. Clearly, the solution is periodic that means the result conforms with Theorem 8.

Data Availability

The data used to support the findings of the study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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