# Qualitative Behavior of a Nonlinear Generalized Recursive Sequence with Delay

Authors: Tarek F. Ibrahim; Abdul Qadeer Khan; Abdelhameed Ibrahim
Journal: Mathematical Problems in Engineering
 (2021)

---

## 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.


---

## Body

## 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(1)ϒn+1=βϒn21+ϒn−12.Elabbasy et al. [2] dealt with(2)ϒn+1=αϒn−kβ+γ∏i=0kϒn−i.Grove et al. [3] presented a summary of(3)ϒn+1=A1+B1ϒn+C1ϒn−1+D1ϒn−2A2+B2ϒn+C2ϒn−1+D2ϒn−2.Kulenovic et al. [4] studied(4)ϒn+1=A1ϒn+B1ϒn−1C1ϒn+D1ϒn−1.Kulenovic and Ladas [5] studied(5)ϒn+1=A1+B1ϒn+C1ϒn−1A2+B2ϒn+C2ϒn−1.Stevic in [6] studied the positive solution of(6)ϒn+1=ϒn−kϒn+⋯+ϒn−k+1+1.Agarwal and Elsayed [7] studied(7)ϒn+1=aϒn+bϒnϒn−3cϒn−2+dϒn−3.For other works, we refer to [6, 8–27].Our objective is to check out global stability and boundedness of solutions for(8)ϒn+1=α+∑i=0kaiϒn−i+ϒnϒn−kβ+∑j=0kbjϒn−j,where ai and bi∈0,∞ and α,β≥1 with the initials ϒ−1,ϒ−p+1,…,ϒ−p and ϒ0∈0,∞.


## 2. Local Stability of Equilibrium

Theorem 1. Equation (8) has the following equilibriums:(9)ϒ¯1=−12D1,ϒ¯2=−12D2,D1=−β+αB+Aβ−C−B+AB+1,D2=−β+αB+Aβ+C−B+AB+1,where(10)C=β2+2βαB−2Aβ2+α2β2−2αβAB+A2β2−4αβ,A=∑i=0kai,B=∑j=0kbj.Letf be defined by(11)fv0,v1,v2,…,vk=α+∑i=0kaivi+v0vkβ+∑i=0kbivi.Therefore,(12)∂fv0,v1,v2,…,vk∂v0=a0+β+∑i=0kbivivk−v0vkb0β+∑i=0kbivi2,∂fv0,v1,v2,…,vp∂v1=a1+β+∑i=0kbivi0−v0vkb1β+∑i=0kbivi2=a1−v0vkb1β+∑i=0kbivi2,∂fv0,v1,v2,…,vp∂v2=a2+β+∑i=0kbivi0−v0vkb2β+∑i=0kbivi2=a2−v0vkb2β+∑i=0kbivi2,⋮∂fv0,v1,v2,…,vp∂vk−1=ak−1−v0vkbk−1β+∑i=0kbivi2,∂fv0,v1,v2,…,vp∂vk=ak+β+∑i=0kbiviv0−v0vkbkβ+∑i=0kbivi2.Atϒ¯1=−1/2D1, we obtain(13)∂fϒ¯,ϒ¯,ϒ¯,…,ϒ¯∂v0=a0+β+Bϒ¯ϒ¯−ϒ¯2b0β+x¯B2=a0+−1/2βD1+1/4BD12−1/4D12b0β−1/2BD12,∂fϒ¯,ϒ¯,ϒ¯,…,ϒ¯∂v1=a1−1/4D12b1β−1/2BD12,∂fϒ¯,ϒ¯,ϒ¯,…,ϒ¯∂v2=a2−1/4D12b2β−1/2BD12,⋮∂fϒ¯,ϒ¯,ϒ¯,…,ϒ¯∂vk−1=ak−1−1/4D12bk−1β−1/2BD12,∂fϒ¯,ϒ¯,ϒ¯,…,ϒ¯∂vk=ak+−1/2βD1+1/4BD12−1/4D12bkβ−1/2BD12,and we have(14)yn+1+∑i=0kdiyn−i=0,where di=−fviϒ¯,ϒ¯,…,ϒ¯, for i=0,1,…,k, where(15)λk+1+∑i=0kdiλi=0.Theorem 2.
(i)
ϒ¯1=−1/2D1 of (8) is locally asymptotically stable if(16)A+D1−β+1/4BD1β−1/2BD12<1;(ii)
ϒ¯1=−1/2D2 of (8) is locally asymptotically stable if(17)A+D2−β+1/4BD2β−1/2BD22<1.Proof.
(18)a0+−1/2βD1+1/4BD12−1/4D12b0β−1/2BD12+a1−14D12b1β−1/2BD12+a2−14D12b2β−1/2BD12+⋯+ak−1−14D12bk−1β−1/2BD12+ak+−1/2βD1+1/4BD12−1/4D12bkβ−1/2BD12<1,∑i=0kai+1β−1/2BD12−12βD1+14BD12−14D12b0+−12βD1+14BD12−14D12bk−D124β−1/2BD12∑i=1k−1bi<1,∑i=0kai+1β−1/2BD12−βD1+12BD12−14D12b0+bk−D124β−1/2BD12∑i=1k−1bi<1,A+1β−1/2BD12−βD1+12BD12−14D12b0+bk−D124β−1/2BD12B−b0−bk<1,A+1β−1/2BD12−βD1+12BD12−D124b0−D124bk−D124B+D124b0+D124bk<1,A+D1−β+1/4BD1β−1/2BD12<1,where(19)D1=−β+αB+Aβ−C−B+AB+1,i.e.,(20)A+−β+αB+Aβ−C/−B+AB+1−β+1/4B−β+αB+Aβ−C/−B+AB+1β−1/2B−β+αB+Aβ−C/−B+AB+12<1.
Proof of (ii) is the same as the proof of (i).


## 3. Solutions Boundedness for (8)

In Theorem3, every solution of (8) is bounded.Proof.
(21)ϒn+1=α+∑i=0kaiϒn−i+ϒnϒn−kβ+∑j=0kbjϒn−j≤α+∑i=0kaiϒn−i+ϒnϒn−kbkϒn−k≤α+∑i=0kaiϒn−i+ϒnbk.


## 4. Applications

### 4.1. Case 1:α=0 and β=0

We have(22)ϒn+1=∑i=0kaiϒn−i+ϒnϒn−k∑j=0kbjϒn−j.Equation (22) has equilibrium ϒ¯=0.Theorem 3.
Suppose that(23)A+1B<1, where A=∑i=0kai and B=∑j=0kbj. Then, the equilibrium of (22) is locally asymptotically stable.Proof.
(24)a0+1/4B−1/4b0−1/2B2+a1−14b1−1/2B2+a2−14b2−1/2B2+⋯+ak−1−14bk−1−1/2B2+ak+1/4B−1/4bk−1/2B2<1,∑i=0kai+1−1/2B214B−14b0+14B−14bk−14−1/2B2∑i=1kbi<1,∑i=0kai+1−1/2B212B−14b0−14bk−14−1/2B2∑i=1k−1bi<1,∑i=0ka1+1−1/2B212B−14b0−14bk−14−1/2B2B−b0−bk<1,∑i=0kai+1−1/2B212B−14b0−14bk−14B−b0−bk<1,∑i=0kai+1−1/2B212B−14B<1,A+1B<1.


### 4.2. Case 2 [7]:

α=0andβ=0withk=3,aii=0k&bii=0k=0excepta0,b2,b3≠0We have(25)ϒn+1=a0ϒn+ϒnϒn−3b2ϒn−2+b3ϒn−3.Equation (25) has equilibriums ϒ¯=0.Letf be defined by(26)fv0,v2,v3=a0v0+v0v3b2v2+b3v3,a0+1/4b2+b3−1/2b2+b32+−14b2−1/2b2+b32+1/4b2+b3−1/4b3−1/2b2+b32<1,a0+b2+b3b2+b32+−b2b2+b32+b2+b3−b3b2+b32<1,a0+b2+b3b2+b32+−b2b2+b32+b2+b3−b3b2+b32<1,a0+1b2+b3+−b2b2+b32+b2b2+b32<1,a0+1b2+b3+2b2b2+b32<1,3b2+b3b2+b32<1−a0.Hence, the equilibriumϒ¯=0 of (25) is locally asymptotically stable if 3b2+b3/b2+b32<1−a0.


### 4.3. Case 3:

α=0 and β=1withk=2,aii=0k&bii=0k=0excepta0,a1,b1,b2≠0Here, we have(27)ϒn+1=a0ϒn+a1ϒn−1+ϒnϒn−21+b1ϒn−1+b2ϒn−2.Equation (27) has the following equilibriums:(28)ϒ¯1=0,ϒ¯2=−a0+a1−11+a0+a1−1b1+b2.Letf defined by(29)fv0,v1,v2=a0v0+a1v1+v0v21+b1v1+b2v2,∂fv0,v1,v2∂v0=a0+b1v1+1+b2v2v2b1v1+1+b2v22=a0+v2b1v1+1+b2v2,∂fv0,v1,v2∂v1=a1+−v0v2b1b1v1+1+b2v22,∂fv0,v1,v2∂v2=1+b1v1+b2v2v0−v0v2b21+b1v1+b2v22=b1v1+1v0b1v1+1+b2v22,a0+v2b1v1+1+b2v2+a1+−v0v2b1b1v1+1+b2v22+b1v1+1v0b1v1+1+b2v22<1,a0+v2b1v1+1+b2v2+a1−v0v2b1b1v1+1+b2v22+b1v1+1v0b1v1+1+b2v22<1,a0+b1v1+1+b2v2v2b1v1+1+b2v22+a1−v0v2b1b1v1+1+b2v22+b1v1+1v0b1v1+1+b2v22<1,b1v1+1+b2v2v2−v0v2b1+1+b1v1v0b1v1+1+b2v22<1−a0−a1,v2+b1v1v2+b2v22−v0v2b1+v0+b1v0v1b1v1+1+b2v22<1−a0−a1.So, we haveTheorem 4.
(i)
The equilibriumϒ¯1=0 of equation (27) is locally asymptotically stable if(30)0<1−a0−a1;(ii)
The equilibriumϒ¯1=−a0+a1−1/1+a0+a1−1b1+b2 of equation (27) is locally asymptotically stable if(31)−1+a0+a12a0b1+b2+2a1b1+b2−b1+b1a0+b1a1−b2+b2a0+b2a1a0b1+b2+a1b1+b2−b1+b1a0+b1a1−b2+b2a0+b2a12<1−a0−a1.


### 4.4. Case 4

We will consider the difference equation as a particular case of (8):(32)ϒn+1=a0ϒn+a2ϒn−2+a4ϒn−4+ϒnϒn−4b0ϒn+b2ϒn−2+b4ϒn−4.Theorem 5.
Suppose(33)1<1−a0−a2−a4b0+b4+b2.Then, the equilibriumϒ¯=0 of (32) is locally asymptotically stable.Proof.
Letf be defined by(34)fv0,v2,v4=a0v0+a2v2+a4v4+v0v4b0v0+b2v2+b4v4.
Therefore,(35)∂fv0,v2,v4∂v0=a0+b0v0+b2v2+b4v4v4−v0v4b0b0v0+b2v2+b4v42=a0+b2v2v4+b4v42b0v0+b2v2+b4v42,∂fv0,v2,v4∂v2=a2+b0v0+b2v2+b4v40−v0v4b2b0v0+b2v2+b4v42=a2+−v0v4b2b0v0+b2v2+b4v42,∂fv0,v2,v4∂v4=a4+b0v0+b2v2+b4v4v0−v0v4b4b0v0+b2v2+b4v42=a4+b0v02+b2v2v0b0v0+b2v2+b4v42,∂fϒ¯,ϒ¯,ϒ¯∂v0=a0+b2+b4b0+b4+b22,∂fϒ¯,ϒ¯,ϒ¯∂v2=a2+−b2b0+b4+b22,∂fϒ¯,ϒ¯,ϒ¯∂v4=a4+b0+b2b0+b4+b22,a0+b2+b4b0+b4+b22+a2+−b2b0+b4+b22+a4+b0+b2b0+b4+b22<1,b4+b0+b2b0+b4+b22<1−a0−a2−a4,1<1−a0−a2−a4b0+b4+b2.Theorem 6.
Every solution of (32) is bounded.Proof.
(36)ϒn+1=a0ϒn+a2ϒn−2+a4ϒn−4+ϒnϒn−4b0ϒn+b2ϒn−2+b4ϒn−4<a0+1b4ϒn+a2ϒn−2+a4ϒn−4.
There are many cases in which the solution of (32) is bounded:(1)
Ifa0+1/b4<1,a2=0anda4=0.(2)
Ifa0+a2+1/b4<1,ϒn−2<ϒnanda4=0.(3)
Ifa0+a4+1/b4<1,ϒn−4<ϒnanda2=0.(4)
Ifa0+a2+a4+1/b4<1,ϒn−4<Υnandϒn−2<ϒn.


## 4.1. Case 1:α=0 and β=0

We have(22)ϒn+1=∑i=0kaiϒn−i+ϒnϒn−k∑j=0kbjϒn−j.Equation (22) has equilibrium ϒ¯=0.Theorem 3.
Suppose that(23)A+1B<1, where A=∑i=0kai and B=∑j=0kbj. Then, the equilibrium of (22) is locally asymptotically stable.Proof.
(24)a0+1/4B−1/4b0−1/2B2+a1−14b1−1/2B2+a2−14b2−1/2B2+⋯+ak−1−14bk−1−1/2B2+ak+1/4B−1/4bk−1/2B2<1,∑i=0kai+1−1/2B214B−14b0+14B−14bk−14−1/2B2∑i=1kbi<1,∑i=0kai+1−1/2B212B−14b0−14bk−14−1/2B2∑i=1k−1bi<1,∑i=0ka1+1−1/2B212B−14b0−14bk−14−1/2B2B−b0−bk<1,∑i=0kai+1−1/2B212B−14b0−14bk−14B−b0−bk<1,∑i=0kai+1−1/2B212B−14B<1,A+1B<1.


## 4.2. Case 2 [7]:

α=0andβ=0withk=3,aii=0k&bii=0k=0excepta0,b2,b3≠0We have(25)ϒn+1=a0ϒn+ϒnϒn−3b2ϒn−2+b3ϒn−3.Equation (25) has equilibriums ϒ¯=0.Letf be defined by(26)fv0,v2,v3=a0v0+v0v3b2v2+b3v3,a0+1/4b2+b3−1/2b2+b32+−14b2−1/2b2+b32+1/4b2+b3−1/4b3−1/2b2+b32<1,a0+b2+b3b2+b32+−b2b2+b32+b2+b3−b3b2+b32<1,a0+b2+b3b2+b32+−b2b2+b32+b2+b3−b3b2+b32<1,a0+1b2+b3+−b2b2+b32+b2b2+b32<1,a0+1b2+b3+2b2b2+b32<1,3b2+b3b2+b32<1−a0.Hence, the equilibriumϒ¯=0 of (25) is locally asymptotically stable if 3b2+b3/b2+b32<1−a0.


## 4.3. Case 3:

α=0 and β=1withk=2,aii=0k&bii=0k=0excepta0,a1,b1,b2≠0Here, we have(27)ϒn+1=a0ϒn+a1ϒn−1+ϒnϒn−21+b1ϒn−1+b2ϒn−2.Equation (27) has the following equilibriums:(28)ϒ¯1=0,ϒ¯2=−a0+a1−11+a0+a1−1b1+b2.Letf defined by(29)fv0,v1,v2=a0v0+a1v1+v0v21+b1v1+b2v2,∂fv0,v1,v2∂v0=a0+b1v1+1+b2v2v2b1v1+1+b2v22=a0+v2b1v1+1+b2v2,∂fv0,v1,v2∂v1=a1+−v0v2b1b1v1+1+b2v22,∂fv0,v1,v2∂v2=1+b1v1+b2v2v0−v0v2b21+b1v1+b2v22=b1v1+1v0b1v1+1+b2v22,a0+v2b1v1+1+b2v2+a1+−v0v2b1b1v1+1+b2v22+b1v1+1v0b1v1+1+b2v22<1,a0+v2b1v1+1+b2v2+a1−v0v2b1b1v1+1+b2v22+b1v1+1v0b1v1+1+b2v22<1,a0+b1v1+1+b2v2v2b1v1+1+b2v22+a1−v0v2b1b1v1+1+b2v22+b1v1+1v0b1v1+1+b2v22<1,b1v1+1+b2v2v2−v0v2b1+1+b1v1v0b1v1+1+b2v22<1−a0−a1,v2+b1v1v2+b2v22−v0v2b1+v0+b1v0v1b1v1+1+b2v22<1−a0−a1.So, we haveTheorem 4.
(i)
The equilibriumϒ¯1=0 of equation (27) is locally asymptotically stable if(30)0<1−a0−a1;(ii)
The equilibriumϒ¯1=−a0+a1−1/1+a0+a1−1b1+b2 of equation (27) is locally asymptotically stable if(31)−1+a0+a12a0b1+b2+2a1b1+b2−b1+b1a0+b1a1−b2+b2a0+b2a1a0b1+b2+a1b1+b2−b1+b1a0+b1a1−b2+b2a0+b2a12<1−a0−a1.


## 4.4. Case 4

We will consider the difference equation as a particular case of (8):(32)ϒn+1=a0ϒn+a2ϒn−2+a4ϒn−4+ϒnϒn−4b0ϒn+b2ϒn−2+b4ϒn−4.Theorem 5.
Suppose(33)1<1−a0−a2−a4b0+b4+b2.Then, the equilibriumϒ¯=0 of (32) is locally asymptotically stable.Proof.
Letf be defined by(34)fv0,v2,v4=a0v0+a2v2+a4v4+v0v4b0v0+b2v2+b4v4.
Therefore,(35)∂fv0,v2,v4∂v0=a0+b0v0+b2v2+b4v4v4−v0v4b0b0v0+b2v2+b4v42=a0+b2v2v4+b4v42b0v0+b2v2+b4v42,∂fv0,v2,v4∂v2=a2+b0v0+b2v2+b4v40−v0v4b2b0v0+b2v2+b4v42=a2+−v0v4b2b0v0+b2v2+b4v42,∂fv0,v2,v4∂v4=a4+b0v0+b2v2+b4v4v0−v0v4b4b0v0+b2v2+b4v42=a4+b0v02+b2v2v0b0v0+b2v2+b4v42,∂fϒ¯,ϒ¯,ϒ¯∂v0=a0+b2+b4b0+b4+b22,∂fϒ¯,ϒ¯,ϒ¯∂v2=a2+−b2b0+b4+b22,∂fϒ¯,ϒ¯,ϒ¯∂v4=a4+b0+b2b0+b4+b22,a0+b2+b4b0+b4+b22+a2+−b2b0+b4+b22+a4+b0+b2b0+b4+b22<1,b4+b0+b2b0+b4+b22<1−a0−a2−a4,1<1−a0−a2−a4b0+b4+b2.Theorem 6.
Every solution of (32) is bounded.Proof.
(36)ϒn+1=a0ϒn+a2ϒn−2+a4ϒn−4+ϒnϒn−4b0ϒn+b2ϒn−2+b4ϒn−4<a0+1b4ϒn+a2ϒn−2+a4ϒn−4.
There are many cases in which the solution of (32) is bounded:(1)
Ifa0+1/b4<1,a2=0anda4=0.(2)
Ifa0+a2+1/b4<1,ϒn−2<ϒnanda4=0.(3)
Ifa0+a4+1/b4<1,ϒn−4<ϒnanda2=0.(4)
Ifa0+a2+a4+1/b4<1,ϒn−4<Υnandϒn−2<ϒn.



---

*Source: 6162320-2021-08-06.xml*