Research Article | Open Access

Volume 2020 |Article ID 1230979 | https://doi.org/10.1155/2020/1230979

H. S. Alayachi, M. S. M. Noorani, A. Q. Khan, M. B. Almatrafi, "Analytic Solutions and Stability of Sixth Order Difference Equations", Mathematical Problems in Engineering, vol. 2020, Article ID 1230979, 12 pages, 2020. https://doi.org/10.1155/2020/1230979

Analytic Solutions and Stability of Sixth Order Difference Equations

Academic Editor: Ant nio M. Lopes
Accepted12 Sep 2020
Published28 Sep 2020

Abstract

In the present paper, the global attractor, local stability, and boundedness of the solution of sixth order difference equations are investigated analytically and numerically. The exact solutions of three equations are presented by utilizing Fibonacci sequence. We also analyse the periodicity of a sixth order difference equation. The considered difference equations are given by where the initial conditions , and are arbitrary real numbers and the values , and are defined as positive real numbers.

1. Introduction

In a contemporary study, the theory of difference equations has been studied by a huge number of researchers. This can be attributed to the importance of this field in modelling a large number of natural phenomena. Difference equations are used in modelling some real-life problems appeared in biology, physics, economy, engineering, etc. Difference equations become apparent in the study of discretization methods for differential equations. Some results in the theory of difference equations have been obtained in the corresponding results of differential equations as more or less natural discrete analogues. Some recent studies of the dynamic of difference equations are given as follows. Almatrafi [2] obtained the exact solutions of the following systems of the difference equations:

Alotaibi [5] analysed the global stability and examined the periodic solution of the following difference equation:

Also, in [8], the authors dealt with the difference equation and considered some special cases of

Cinar [10] investigated the solution of the difference equation:

El-Dessoky [11] studied the qualitative properties of the fifth order nonlinear difference equation:

BekteÅ¡eviÄ‡ et al. [13] gave a description about the global stability of three special cases of the following recursive equation:

Ibrahim [14] offered some relevant results of the difference equation:

The authors in [18] presented some theoretical investigations for some of the formulae of

One can see the works of [1, 3, 4, 6] and [7, 9, 12, 17, 19, 20, 21, 22, 23, 24] to obtain more information about rational difference equations and systems.

The prime objective of this study is to explore the qualitative behavior of the solutions of the following recursive equation:where the initial conditions , and are arbitrary nonzero real numbers. The constants , and are assumed to be positive.

2. Local Stability of the Equilibrium Point of Equation (9)

This section focuses on discovering the local stability of the solutions around the equilibrium point of equation (9). Equation (9) has a unique equilibrium point given by

If , then the only positive equilibrium point of equation (9) is given by

Let be a continuous function defined by

Hence, we can obtain the following partial derivatives:

Evaluating the previous derivatives at gives

The linearized equation of equation (9) about is now expressed by

Theorem 1. Assume that

Then, the positive fixed point of equation (9) is locally asymptotically stable.

Proof. Theorem A in [15] ensures that equation (9) is asymptotically stable ifthat is,or,which is the required condition.

3. Global Attractivity of the Equilibrium Point of Equation (9)

This section is devoted to investigate the global attractivity of the solutions of equation (9).

Theorem 2. The fixed point of equation (9) is a global attractor if .

Proof. Let and be a real numbers and assume that be a function defined by equation (12). Then, we can easily observe that the function is increasing in and and is decreasing in . Suppose that is a solution of the systemThen, from equation (9), one can notice thatwhich leads toSubtracting equation (23) from (22) givesThus,It follows by Theorem B in [16] that is a global attractor of equation (9). Therefore, the proof is complete.

4. Boundedness of Solutions of Equation (9)

The boundedness of solutions of equation (9) is deeply considered in this section.

Theorem 3. Every solution of (9) is bounded if .

Proof. Let be a solution of equation (9). It follows from equation (9) thatConsequently,

5. Special Cases of Equation (9)

Our main target in this section is to find specific forms for the solutions of some special cases of equation (9) when , and are positive integers. We also aim to give some numerical examples confirming the theoretical work.

5.1. On the Equation

In this section, we present the exact solutions of equation (9):where the initial conditions , and are arbitrary real numbers.

Theorem 4. Let be a solution to equation (28) satisfying , , , , , and . Then, for ,where

Proof. For , the result holds. Now, suppose that and that our assumption holds for , that is,Now, it follows from equation (28) thatThen,Also, we get from equation (28) thatThus, other relations can be similarly proved.

5.2. On the Equation

This section deals with the solutions of the following equation:where the initial conditions , and are nonzero arbitrary real numbers.

Theorem 5. Let be a solution to equation (34) satisfying , , , , , and . Then, for ,

Proof. For , the result holds. Now, suppose that and that our assumption holds for , that is,Now, it can be seen from equation (34) thatIt can also be observed that