On Stability Analysis of Higher-Order Rational Difference Equation

In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, global behavior of equilibrium points, boundedness and periodicity of the rational recursive sequence wn+1 � wn− p(α + (βwn/γwn + δwn− r)), where γwn ≠ − δwn− r for r ∈ (0,∞), α, β, γ, δ ∈ (0,∞), and r>p≥ 0. With initial values w− p, w− p+1, . . . , w− r, w− r+1, . . . , w− 1, andw0 are positive real numbers. Some numerical examples are given to verify our theoretical results.


Introduction
It is very amusing to explore the nature of the solutions of a higher-order rational difference equation and to explain the local asymptotic stability of its equilibrium points. e inspection of some properties associated with these equations is a very enormous activity. Discrete dynamical systems or difference equations are diverse fields because various biological systems and models naturally lead to their study by means of a discrete variable. Applications of discrete dynamical systems and difference equations have appeared recently in many fields of science and technology. ere is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. Rational difference equations are a special form of nonlinear difference equations. Delay difference equations have rich dynamics to study. Due to adequate computational outcomes, discrete dynamical systems are awful lot better than allied structures in differential equations. Specifically, in case of nonoverlapping generations, difference equations are greater apposite to take a look at the behavior of population models [1,2]. Also, an epidemiological approach to insurgent population modeling is mentioned in [3,4]. e research of delay difference equations is a divergent field that involves most aspects of mathematics, including both applied and pure. Recently, there has been a symbolic development in the applications of difference equations. It is very interesting to investigate the behavior of solutions of a system of nonlinear difference equations and to discuss the local asymptotic stability of their equilibrium points. ere are many papers in which systems and behavior of rational difference equations have been studied see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Rational difference equations have been studied by several authors. ere has been a great interest exclusively in the study of the attractiveness of the solution of such equations.
Here, we recall some basics and preliminaries that will be useful in our results and investigation.

Preliminaries and Definitions
Definition 1. Let I be some interval of real numbers, and let be a continuously differentiable function. en, for every set of initial conditions S − k , S − k+1 , . . . , S 0 ∈ I, the difference equation has a unique solution S n ∞ n�− k .

Definition 2.
A point S ∈ I is called an equilibrium point of equation (2) if at is, S n � S, for all n ≥ 0, is a solution of equation (2), or equivalently, S is a fixed point of R. (2) is called permanent and bounded if there exist numbers t and T with 0 < t < T < ∞ such that for any initial values S − k , S − k+1 , . . . , S − 1 , S 0 ∈ I, there exists a positive integer N which depends on these initial values such that t ≤ S n ≤ T, for all n ≥ N.

Definition 5. Equation
we have (ii) e equilibrium point S of equation (2) is locally asymptotically stable if S is a locally stable solution of equation (2) and there exists γ > 0, such that for all we have lim n⟶∞ S n � S.
(iv) e equilibrium point S of equation (2) is globally asymptotically stable if S is locally stable and S is also a global attractor of equation (2). (v) e equilibrium point S of equation (2) (2) about the equilibrium S is defined by the following equation: where ρ 0 � zF(S, S, . . . , S) zS n , Lemma 1 (see [21]). Assume that p i ∈ R and i ∈ 1, 2, . . . , k is a sufficient condition for the asymptotic stability of the following difference equation: S n+k + p 1 S n+k− 1 + · · · + p k S n � 0, n � 0, 1, 2, . . . . (12) e following theorem will be useful for the proof of our results in this paper. Theorem 1. Let [α, β] be an interval of real numbers and assume that g: [α, β] 2 ⟶ [α, β] is a continuous function, and consider the following equation: satisfying the following conditions: 2 Discrete Dynamics in Nature and Society then m � M and equation (2) has a unique equilibrium S ∈ [α, β] and every solution of equation (2) converges to S.
Some published works include the following: Ibrahim and El-Moneam [22] investigate the local and global stability of the following recursive sequence where cS − q+k ≠ − dS − r+k for k � 0, 1, . . . , min(q, r) and α, b, c, and d are real numbers. Aboutaleb et al. [23] studied the global attractiveness of the positive equilibrium of the following rational recursive equation: where the coefficients α, β, and γ are the nonnegative real numbers. Several other researchers have studied the behavior of the solution of difference equations; for example, Devault et al. [24] investigated the global behavior of all positive solutions of the following equation: Elabbasy et al. [25] studied the boundedness, global stability, and periodicity character and gave the solution of some special cases of the following difference equation: Elabbasy [26] gave the solution of the following difference equation: El-Moneam and Zayed [27] investigated the global stability and periodicity character and gave the solution of some special cases of the following difference equation: Zayed and El-Moneam [28] studied the global behavior of the following rational recursive sequence: Motivated by the above study, the main focus of this article is to discuss some qualitative behaviors of the solutions of the following rational recursive sequence: where γw n ≠ − δw n− r for r ∈ (0, ∞), α, β, γ, δ ∈ (0, ∞),

Local Stability of Equation (22)
is section deals with the local stability character of the solutions of equation (22). Equation (22) has a unique equilibrium point, and it is given from if en, the unique equilibrium point is erefore, it follows that We see that e characteristic equation of equation (28) is Theorem 2. Assume that Then, the equilibrium point of equation (22) is locally asymptotically stable.
Proof. It follows by Lemma 7 that equation (28) is asymptotically stable if Discrete Dynamics in Nature and Society and so Hence,

Boundedness of the Solutions
In this section, we show the boundedness of the positive solutions of equation (22). (22) is bounded from above by

Theorem 3. Every solution of equation
Proof. Let F n ∞ n�− p be a solution of equation (22). It follows from equation (22) that en, for all n ≥ N, en, all the subsequences w (p+1)n+j ∞ n�0 , i � 0, 1, 2, . . . , p, are decreasing and so are bounded from above by □ Theorem 4. If α > 1, then each solution of equation (22) will be unbounded.
From above, the right-hand side can be written as follows: and this equation is unstable because α > 1 and lim n⟶∞ T n � ∞. en, by using the ratio test, w n ∞ n�− p is unbounded from above.

Periodic Solutions of Equation (22)
In this section, we satisfy the periodic solutions of equation (22). Theorem 5. Equation (22) has no positive solutions of prime period two solution ∀α, β, γ, δ ∈ (0, ∞) in the following case, if p is even and r is odd.
is implies that

Global Stability
We will study the global asymptotic stability of the positive solutions of equation (22) in this section.

Theorem 7.
e equilibrium point w of equation (22) is a global attractor if Proof. Let a and b are real numbers and assume that f: (a, b) 3 ⟶ (a, b) be function defined by f(s, u, v) � αs + (βsu/γu + δv). us, we see that the function f(s, u, v) is increasing in s and u and is decreasing in v. Let (g, G) is a solution of the system G � f(G, G, g) and g � f(g, g, G). en, from (22), we see that en, By subtracting these equations, we obtain It follows that the equilibrium point w of equation (22) is a global attractor.

Applications
In this section, we will discuss the solution of some special cases of equation (22). Case 1. When α � 0, p � r � 0. In this case, we have the following special type of difference equation: (56) is

Theorem 8. e general solution of equation
where Proof. Proof is obtained by induction, and it is easy to do.
□ Case 2. When α � 1 and β � γ � δ � 1, p � 3 and r � 4. In this case, we have the following special case of difference equation: where w − 4 , w − 3 , w − 2 , w − 1 , and w 0 are the arbitrary real numbers.  Proof. For n � 0, the result holds. Now, suppose that n > 0 and that our assumption holds for n − 1 and n − 2: Discrete Dynamics in Nature and Society 5 Now, it follows from (59) that us, we have Hence, the proof is completed. Similarly, we can prove other relations.
Proof. For n � 1, Hence, the relation holds true. Now, suppose that relation (64) holds for n � k.
at is, We now want to show that relation (64) holds true for n � k + 1: (68) □ Case 4. When α � 1 and β � γ � δ � 1, p � 2 and r � 3. In this case, we have the following special case of difference equation: where w − 3 , w − 2 , w − 1 , and w 0 are the arbitrary real numbers.
Theorem 11. Let w n ∞ n�− 3 be a solution of equation (69). en, for n � 0, 1, 2, . . ., where Proof. For n � 0, the result holds. Now, suppose that n > 0 and that our assumption holds for n − 1 and n − 2: Now, it follows from (69) that Discrete Dynamics in Nature and Society Hence, the proof is complete.
□ By using the same way, we can prove other relations.

Numerical Examples
is section discusses some numerical results of our previous results.
Example 1. Figure 1 shows the behavior of the solution bounded of equation (56) when we take Example 2. Figure 2 shows the behavior of solution of equation (59) when we take Example 3. Figure 3 shows that the solution of equation (64) is globally asymptotically stable when we take 8 Discrete Dynamics in Nature and Society Example 4. Figure 4 shows the solution of equation (69) has no prime period two solution if we take

Concluding Remarks
In the literature, several articles are related to qualitative behavior of rational difference equations. It is a very interesting mathematical problem to study the dynamics of such equations because these are closely related to models in population dynamics and biological sciences. We have investigated the existence and uniqueness of positive equilibrium points, and boundedness and persistence of positive solutions are proved. Moreover, we have shown that they are locally as well as globally asymptotically stable. e main objective of dynamical systems theory is to predict the global  Plot of w n+1 = α.w n-2 + (β w n-2 * w n /(γ w n + δ w n-3 ))) Plot of w n+1 = w n-p (α + (β w n /(γ w n + δ w n-r ))) Plot of w n+1 = w n-3 (α + (β w n /(γ w n + δ w n-4 ))) Discrete Dynamics in Nature and Society behavior of a system based on the knowledge of its present state. We have taken some special cases as applications of equation (22) in Section 7 and found the closed form of the solutions. Finally, some illustrative examples are provided to support our theoretical discussion.

Data Availability
All the data utilized are included in this article and their sources are cited accordingly.

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