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

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.'erefore, 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, [1–5]). 'e second direction is to obtain the expressions of the solution if it is possible since there is no explicit and enoughmethods to find the solution of nonlinear difference equations (see, for example, [6–11]). Saleh and Farhat [12] investigated the stability properties and the period two solutions of all nonnegative solutions of the difference equation:


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. erefore, 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, [1][2][3][4][5]). e 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, [6][7][8][9][10][11]).
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 highorder fuzzy difference equation: Kerker et al. [14] investigated the global behavior of the rational difference equation: V n+1 � a n + V n a n + V n− k .
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: To see more related work on the nonlinear difference equation, refer to . Our aim of this article is to investigate the dynamics of the solution for the below difference equation: where ξ, ε, μ, and κ are arbitrary real numbers with initial conditions U j for j � − 17, − 16, . . . , 0. is 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. en, 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.

Boundedness of Solution
is true, then every solution of (7) is bounded.
Proof. Assume that U n ∞ n�− 17 is a solution of (7). en, from (7), we have Hence, Implies that the subsequences U 9n− 8 ∞ n�− 17 are nonincreasing. us, they are bounded from above by U max , where

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

The Equilibrium Point and Local Stability
e fixed points of (7) are given by If (1 − ξ)(μ + κ) ≠ ε, then (7) has only one equilibrium point which is U � 0.
Assume g: (0, ∞) 2 ⟶ (0, ∞) is a continuously differentiable function defined by erefore, en, Hence, Theorem 3. e fixed point U � 0 is said to be a locally asymptotically stable if the relation is satisfied.
Proof. From eorem 5.10 in [44], it follows that U is asymptotically stable if where Hence, Finally, the proof is done.

Global Attractivity of the Fixed Point
Theorem 4. e fixed point U of (7) has to be a global attracting when Proof. From (16), we see that the function g(v, w), which defined in (15), is increasing in v and decreasing in w. Let (ρ, τ) be a solution of the system: erefore, Subtracting (25) from (26), we get and then, ρ � τ if μ(1 − ξ) ≠ ε. us, from eorem 5.20 in [44], we observe that there exists only one solution for (7) and it is a global attractor if μ(1 − ξ) ≠ ε.

Special Cases
Now, we present the solutions' expressions for special cases of (7): where the initial conditions are and U 0 are arbitrary real numbers.

First Equation.
We solve the equation Mathematical Problems in Engineering (30); thus, for n � 0, 1, . . ., Proof. We show that the expressions in (31) are solutions of (30) by applying mathematical induction. First, the results hold for n � 0. Second, we suppose that the forms are satisfied for n − 1 and n − 2. Now, we prove that the results are satisfied for n: From ( erefore, Similarly, one can investigate other expressions. e proof is done.

Second Equation.
In this section, we introduce the solution of the following equation: Theorem 6. Let U n ∞ n�− 17 be a solution of (35); then, for n � 0, 1, . . .,

Mathematical Problems in Engineering 5
Proof. e proof will be the same as proof of eorem 5, so it is therefore omitted. □

ird Equation.
In this section, we present the solution of the following equation: Theorem 7. Let U n ∞ n�− 17 be a solution of (37); then, for n � 0, 1, . . ., where Proof. By using mathematical induction, we prove that (38) are solutions of (37). First, the results for n � 0 are true. Second, assume that the assumption holds for n − 2 and n − 1.
Mathematical Problems in Engineering Now, from (37), we have Mathematical Problems in Engineering us, Similarly, one can see that the other forms are true. e proof is complete.
We study the following equation: Theorem 8. Suppose that U n Example 1. To show the stability of (7), we set two groups for the values of the coefficients: (i) ξ � 0.5, ε � 0.1, μ � 1.6, and κ � 0.2 and (ii) ξ � 0.5, ε � 5, μ � 10, and κ � 0.001, and the initial conditions are and U 0 � 5.2. e 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 U � 0, while the solution moves away from the fixed point for (ii) since condition (23) failed.
e following examples have explained the solutions of special case equations (30)- (42).

Example 2.
We choose the initial conditions as and U 0 � 5.02. e solution is given in Figure 2.
Example 3. In Figure 3, we set the initial conditions: and then, the result is shown in Figure 4.  e solution is given in Figure 5. Clearly, the solution is periodic that means the result conforms with eorem 8.

Data Availability
e data used to support the findings of the study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.