Abstract

Difference equations are of growing importance in engineering in view of their applications in discrete time-systems used in association with microprocessors. We will check out the global stability and boundedness for a nonlinear generalized high-order difference equation with delay.

1. Introduction

Recently, there is a tremendous rate of interest in examining difference formulas. Among the factors, this is a necessity for techniques that we can use in checking out equations emerging in mathematical models.

Difference formulas have been investigated in different mathematical branches for an extended period.

Camouzis et al. [1] studied

Elabbasy et al. [2] dealt with

Grove et al. [3] presented a summary of

Kulenovic et al. [4] studied

Kulenovic and Ladas [5] studied

Stevic in [6] studied the positive solution of

Agarwal and Elsayed [7] studied

For other works, we refer to [6, 8–27].

Our objective is to check out global stability and boundedness of solutions forwhere and and with the initials and .

2. Local Stability of Equilibrium

Theorem 1. Equation (8) has the following equilibriums:where

Let be defined by

Therefore,

At , we obtainand we havewhere , for , where

Theorem 2. (i) of (8) is locally asymptotically stable if(ii) of (8) is locally asymptotically stable if

Proof. wherei.e.,Proof of (ii) is the same as the proof of (i).

3. Solutions Boundedness for (8)

In Theorem 3, every solution of (8) is bounded.

Proof.

4. Applications

4.1. Case 1: and

We have

Equation (22) has equilibrium .

Theorem 3. Suppose that where and . Then, the equilibrium of (22) is locally asymptotically stable.

Proof.

4.2. Case 2 [7]:

We have

Equation (25) has equilibriums .

Let be defined by

Hence, the equilibrium of (25) is locally asymptotically stable if .

4.3. Case 3:

and

Here, we have

Equation (27) has the following equilibriums:

Let defined by

So, we have

Theorem 4. (i)The equilibrium of equation (27) is locally asymptotically stable if(ii)The equilibrium of equation (27) is locally asymptotically stable if

4.4. Case 4

We will consider the difference equation as a particular case of (8):

Theorem 5. Suppose

Then, the equilibrium of (32) is locally asymptotically stable.

Proof. Let be defined byTherefore,

Theorem 6. Every solution of (32) is bounded.

Proof. There are many cases in which the solution of (32) is bounded:(1)If (2)If (3)If (4)If

Data Availability

The data used to support the findings of the study are available within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.