Abstract

In this paper, we study the solution of the difference equation , where the initials are positive real numbers.

1. Introduction

Difference equations appear naturally as discrete analogues in many sciences such as biology, ecology, and physics. In recent years, many authors studied the solution form of difference equations. For instance, Cinar [1–3] studiedrespectively.

DeVault et al. [4] examined

Elsayed [5] dealt with

Simsek et al. [6–9] studiedrespectively. For some more results concerning difference equations, we refer the reader to [10–21].

In this work, we deal with the following nonlinear difference equation:where is investigated.

2. Main Results

Let be the unique equilibrium of equation (5); then,so is obtained. For every and , notation means .

Theorem 1. For (5), the following statements are true:(a)The sequences , , …, are decreased, and exists such that(b) or(c)If there exists such that for all , then(d)We can generate the following formulas:(e)If , then as .
If , then as . .

Proof. (a)Firstly, for all , from (5), one gets . So,(b)In view of equation (5), Then, or is obtained.(c)If there exists such that then if , and we obtained the result.(d)Subtracting from both sides in (5), we have and the following formula, for , From (20), we get So, Also, Moreover, On the contrary, Also, Moreover, Now, we get the above formulas: where and hold.(e)Suppose that . By (d), we produce the following formulas:Similarly,Similarly,Similarly,Similarly,Similarly,Similarly,From equations (29) and (30),Thus, .
From equations (30) and (31),Thus, .
From equations (31) and (32),Thus, .
From equations (32) and (33),Thus, .
From equations (33) and (34),Thus, .
From equations (34) and (35),Thus, .

3. Examples

In this section, we consider some numerical examples.

Example 1. Assume that, for , we get .Then, we have the graph in Figure 1.


Figure 1 
Dynamics of equation (5) with initial conditions in Example 1.

Example 2. If we select the initial conditions as follows,then we have the graph in Figure 2.


Figure 2 
Dynamics of equation (5) with initial conditions in Example 2.

Data Availability

All the data utilized in this article have been included, and the sources from where they were adopted are cited accordingly.

Conflicts of Interest

The authors declare that they have no conflicts of interest.