Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 9129354, 21 pages
https://doi.org/10.1155/2018/9129354

Research Article

Dynamics and Solutions of a Fifth-Order Nonlinear Difference Equation

M. M. El-Dessoky,1,2 E. M. Elabbasy,² and Asim Asiri¹

¹King Abdulaziz University, Faculty of Science, Mathematics Department, P.O. Box 80203, Jeddah 21589, Saudi Arabia
²Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Correspondence should be addressed to M. M. El-Dessoky; dessokym@mans.edu.eg

Received 17 April 2018; Accepted 17 May 2018; Published 19 July 2018

Academic Editor: Chris Goodrich

Copyright © 2018 M. M. El-Dessoky et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main objective of this paper is to study the behavior of the rational difference equation of the fifth-order , where , and are real numbers and the initial conditions and are positive real numbers such that . Also, we obtain the solution of some special cases of this equation.

1. Introduction

In recent years, there has been a great interest in studying the rational difference equations. These equations describe real life situations in stochastic time series, combinatorial analysis, electrical network, number theory, biology, genetics, probability theory, physics, ecology, statistical problems, and economics, for example [1–5]. It is so important to investigate the asymptotic behavior of solutions of a nonlinear difference equations and to discuss the boundedness, periodicity, and stability (local and global) of their equilibrium points; see [6–36] and references therein.

Kalabušić et al. [6] investigated the periodic nature, the boundedness character, and the global asymptotic stability of solutions of the difference equation

In [7], Elabbasy et al. got the solution and the periodicity character of the recursive sequence

Cinar [8] found the solution of the difference equation

In [9] Obaid et al. studied the global stability, boundedness, and the periodicity of solutions of the rational difference equation

Elsayed et al. [10] studied the dynamical analysis of rational difference equation

Elabbasy et al. [11] obtained the global behavior of the solutions of the difference equation

Aloqeili [12] investigated the dynamics of the difference equation

The aim of this paper is to study some qualitative behavior of the positive solutions of the difference equation

where , and are real numbers and the initial conditions , and are positive real numbers such that .

2. Preliminaries

Let be some interval of real numbers and let

be a continuously differentiable function. Then for every set of initial conditions , the difference equation

has a unique solution .

Definition 1 (equilibrium point). A point is called an equilibrium point of the difference equation (10) if

That is, for is a solution of the difference equation (10) or, equivalently, is a fixed point of

Definition 2 (stability). Let be an equilibrium point of the difference equation (10). Then, we have the following:
(i) The equilibrium point of the difference equation (10) is called locally stable if for every ,there exists such that for all with

we have

(ii) The equilibrium point of the difference equation (10) is called locally asymptotically stable if is locally stable solution of (10) and there exists , such that, for all with

we have

(iii) The equilibrium point of the difference equation (10) is called global attractor if for all , we have

(iv) The equilibrium point of the difference equation (10) is called globally asymptotically stable if is locally stable, and is also a global attractor of the difference equation (10).
(v) The equilibrium point of the difference equation (10) is called unstable if is not locally stable.

Definition 3 (periodicity). A sequence is said to be periodic with period if for all A sequence is said to be periodic with prime period if is the smallest positive integer having this property.

Definition 4 (Fibonacci sequence). The sequence , i.e., , is called Fibonacci sequence.

Definition 5. The linearized equation of the difference equation (10) about the equilibrium is the linear difference equation

Now, assume that the characteristic equation associated with (17) is

where

Theorem 6 (see [1]). Assume that , , and is nonnegative integer. Then

is a sufficient condition for the asymptotic stability of the difference equation

Theorem 7 (see [2]). Let be a continuous function and be an interval of real numbers. Consider the difference equation

Assume that satisfies the following conditions:
(i) is nondecreasing in and in for all and nonincreasing in for all and in
(2) If is a solution of the system

then

Then (22) has a unique equilibrium point and every solution of (22) converges to .

3. Dynamics of (8)

In this section, we study the local stability, global stability of the solutions, and the boundedness of

where , and are positive real numbers.

3.1. Local Stability of the Equilibrium Point

In this subsection, we study the local stability of the equilibrium point of (8).

Equation (8) has a unique equilibrium point and is given by

or

If , then the only equilibrium point is .

Theorem 8. Let

Then the equilibrium point of (8) is locally asymptotically stable.

Proof. Let be a continuous function defined by

Therefore, it follows that

Then

and the linearized equation of (8) about is

It follows by Theorem 6 that (8) is asymptotically stable if and only if

and so

and, thus,

The proof is complete.

Example 9. Figure 1 shows that the zero solution of the difference Equation (8) is local stability if , , , and and the initial conditions , , , , and .

Figure 1: Plot the behavior of the zero solution of (8).

3.2. Global Stability of the Equilibrium Point

In this subsection we study the global stability of the positive solutions of (8).

Theorem 10. The equilibrium point of (8) is global stability if

Proof. Let and be nonnegative real numbers and assume that is a function defined by

Then we can see that the function is increasing in and and decreasing .
Assume that is a solution of the system

Then from (8), we see that

and then

Subtracting these two equations, we obtain

and if , then we see that
According to Theorem 7 the equilibrium point is a global attractor of (8). The proof is complete.

Example 11. The zero solution of the difference Equation (8) is global stability if , , , and and the initial conditions , , , , and (See Figure 2).

Figure 2: Plot the behavior of the zero solution of (8) is global stable.

3.3. Existence of Boundedness and Unboundedness Solutions

Here we look at the boundedness and unboundedness solutions of (8).

Theorem 12. Every solution of (8) is bounded if

Proof. Let be a solution of (8). It follows from (8) that

Then

Then the sequence is decreasing and so is bounded from the above by .

Theorem 13. Every solution of (8) is unbounded if

Proof. Let be a solution of (8). It follows from (8) that

We see that the right hand side can be written as follows:

and this equation is unstable because and Then by using ratio test is unbounded from above.

Example 14. Figure 3 shows that (8) is bounded when , , , and and the initial conditions , , , , and .

Figure 3: Sketch the behavior of the solution of (8) is bounded.

Example 15. Figure 4 shows that (8) is unbounded when , , , and and the initial conditions , , , , and .

Figure 4: Sketch the behavior of the solution of (8) is unbounded.

4. Special Cases of (8)

In this section we investigate the following special case:

where , , , and are integrals numbers.

4.1. First Case When

Theorem 16. Suppose that is a solution of rational difference equation

Then, for , we see that

where , , , , , , and

Proof. For the result holds. Now, suppose that and that our assumption holds for . That is,

From (48), we see that

Also, we see from this that

Also,

This completes the proof.

4.2. Second Case When and

Theorem 17. Let be a solution of rational difference equation

Then, for , we see that

where are arbitrary nonzero real numbers and , .

Proof. For the result holds. Now, suppose that and that our assumption holds for . That is,

Now, it follows from (55) that

Also, we see that

Also,

and

This completes the proof.

Example 18. Figure 5 illustrates the solutions of (55) when the initial conditions , , , , and .

Figure 5: Sketch the behavior of the solution of (55).

4.3. Third Case When ,

Theorem 19. Assume that is solutions of rational difference equation

and then for , we see that

where , , , , and are arbitrary nonzero real numbers with , , , and

Proof. The proof of Theorem 16 will be omitted.

4.4. Fourth Case When and

Theorem 20. Suppose that is solutions of rational difference equation

and then for , we see that

where , and are arbitrary nonzero real numbers with .

Proof. For the result holds. Now, suppose that and that our assumption holds for . That is,

Now, it follows from (64) that

This completes the proof.

Theorem 21. Every solution of rational difference equation (64) is periodic with period 12.

Proof. The proof follows from the previous theorem and will be omitted.

Example 22. Figure 6 shows that the solution of (64) has a periodic solution with period 12 when the initial conditions , , , and .

Figure 6: Plot the periodicity of the solution of (64).

4.5. Fifth Case When and

Theorem 23. Suppose that is solutions of rational difference equation

Then every solution of (68) is periodic with period 12. Moreover takes the form

where , , , , and are arbitrary nonzero real numbers with , , , and .

Proof. The proof is left to the reader.

Example 24. Figure 7 shows that the solution of (68) has a periodic solution with period 12 when the initial conditions , , , , and .

Figure 7: Plot the periodicity of the solution of (68).

4.6. Sixth Case When and

Theorem 25. Suppose that is solutions of rational difference equation

and then for , we see that

where , , , , , , , , and

4.7. Seventh Case When and

Theorem 26. Suppose that is solutions of rational difference equation

and then for , we see that

where , and are arbitrary nonzero real numbers and ,

4.8. Eighth Case When and

Theorem 27. Suppose that is solutions of rational difference equation

and then, for , we see that

where , , , and .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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