# Dynamics and Solutions’ Expressions of a Higher-Order Nonlinear Fractional Recursive Sequence

Authors: Abeer Alshareef; Faris Alzahrani; Abdul Qadeer Khan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The principle purpose of this article is to examine some stability properties for the fixed point of the below rational difference equationUn+1=ξUn−8+εUn−82/μUn−8+κUn−17 where ξ,ε,μ, and κ are arbitrary real numbers. Moreover, solutions for some special cases of the proposed difference equation are introduced.


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## Body

## 1. Introduction

In recent years, many researchers have tended to use difference equations in mathematical models to explain the problems in different sciences since they have a lot of features such as they enable the scientists to introduce the predictions of their study and it gives more accurate results. In addition, there are various types of nonlinear difference equations that can be studied; one of the most commonly used is rational nonlinear difference equations. However, the research studies in the area of difference equations have two directions: first one is the analysis of the behavior of solutions. Therefore, there are a huge number of articles published to investigate the stability of the equilibrium points and the existence of the periodic solutions for the nonlinear difference equations (see, for example, [1–5]). The second direction is to obtain the expressions of the solution if it is possible since there is no explicit and enough methods to find the solution of nonlinear difference equations (see, for example, [6–11]).Saleh and Farhat [12] investigated the stability properties and the period two solutions of all nonnegative solutions of the difference equation:(1)Vn+1=a1Vn+a2Vn−kA+BVn−k.In [13], Jia studied the solutions’ behavior of the high-order fuzzy difference equation:(2)Vn+1=A1Vn−1Vn−2B2+∑i=3kDiVn−i.Kerker et al. [14] investigated the global behavior of the rational difference equation:(3)Vn+1=an+Vnan+Vn−k.Khaliq and Elsayed [15] examined the dynamics behavior and existence of the periodic solution of the difference equation:(4)Vn+1=α1Vn−2+α2Vn−22γ1Vn−2+γ2Vn−5.In [16], Saleh et al. studied the properties’ stability for a nonlinear rational difference equation of a higher order:(5)Vn+1=β1+β2Vn+β3Vn−kB1Vn+B2Vn−k.Sadiq and Kalim [17] obtained the solution behavior of the difference equation:(6)Vn+1=α1Vn−9+α2Vn−192α3Vn−9+α4Vn−19.To see more related work on the nonlinear difference equation, refer to [18–43]. Our aim of this article is to investigate the dynamics of the solution for the below difference equation:(7)Un+1=ξUn−8+εUn−82μUn−8+κUn−17,where ξ,ε,μ, and κ are arbitrary real numbers with initial conditions Uj for j=−17,−16,…,0.This paper is collected as follows: in Section2, the boundedness of the solution is presented, and we prove that the periodic solution of period two does not exist in the next section. Following that, we state the conditions of the local and global stability of the equilibrium point in Sections 4 and 5, respectively. Then, we introduce the solutions’ forms for some special cases in Section 6. Finally, we give some numerical examples in order to illustrate the behavior of the solutions.


## 2. Boundedness of Solution

Theorem 1.
If the following condition(8)ξ+εμ<1,is true, then every solution of (7) is bounded.Proof.
Assume thatUnn=−17∞ is a solution of (7). Then, from (7), we have(9)Un+1=ξUn−8+εUn−82μUn−8+κUn−17≤ξUn−8+εUn−82μUn−8=ξ+εμUn−8.
Hence,(10)Un+1≤Un−8,∀n≥0.
Implies that the subsequencesU9n−8n=−17∞,U9n−7n=−17∞,U9n−6n=−17∞,U9n−5n=−17∞, U9n−4n=−17∞,U9n−3n=−17∞,U9n−2n=−17∞,U9n−1n=−17∞, and U9nn=−17∞ are nonincreasing. Thus, they are bounded from above by Umax, where Umax=maxU−17,U−16,U−15,U−14,U−13,U−12,U−11,U−10,U−9,U−8,U−7,U−6,U−5,U−4,U−3,U−2,U−1,U0.


## 3. Periodicity of the Solution

Theorem 2.
For nonlinear difference equation (7), there is no periodic solution of period two.Proof.
To prove Theorem2, suppose that (7) has a positive prime period two solutions presented as …,e,f,e,f,…. Then,(11)e=ξf+εf2μf+κe,e=ξμf2+ξκef+εf2μf+κe,ξμ+εf2=μ−ξκef+κe2.
Similarly,(12)f=ξe+εe2μe+κf,f=ξμe2+ξκef+εe2μe+κf,ξμ+εe2=μ−ξκfe+κf2.
Subtracting (11) from (12), we get(13)κe2−f2+ξμ+εe2−f2=0,κ+ξμ+εe2−f2=0.
Sinceκ+ξμ+ε≠0, thus e=f, and this contradicts the fact that e≠f.


## 4. The Equilibrium Point and Local Stability

The fixed points of (7) are given by(14)U¯=ξU¯+εU¯2μU¯+κU¯,1−ξU¯=εU¯2μ+κU¯,1−ξμ+κ−εU¯2=0.If1−ξμ+κ≠ε, then (7) has only one equilibrium point which is U¯=0.Assumeg:0,∞2⟶0,∞ is a continuously differentiable function defined by(15)gv,w=ξv+εv2μv+κw.Therefore,(16)∂g∂v=ξ+εμv2+2εκvwμv+κw2,∂g∂w=−κεv2μv+κw2.Then,(17)∂g∂v|v=w=U¯=ξ+εμ+2εκμ+κ2,∂g∂w|v=w=U¯=−εκμ+κ2.Hence,(18)xn+1−ξ+εμ+2εκμ+κ2xn+εκμ+κ2xn−1=0.Theorem 3.
The fixed pointU¯=0 is said to be a locally asymptotically stable if the relation(19)εμ+3k<1−ξμ+κ2,is satisfied.Proof.
From Theorem 5.10 in [44], it follows that U¯ is asymptotically stable if(20)P0+P1<1,where P0=ξ+εμ+2εκ/μ+κ2 and P1=−εκ/μ+κ2. Then,(21)ξ+εμ+2εκμ+κ2+−εκμ+κ2<1,ξ+εμ+3εκμ+κ2<1.
Hence,(22)εμ+3k<1−ξμ+κ2.
Finally, the proof is done.


## 5. Global Attractivity of the Fixed Point

Theorem 4.
The fixed pointU¯ of (7) has to be a global attracting when(23)μ1−ξ≠ε.Proof.
From (16), we see that the function gv,w, which defined in (15), is increasing in v and decreasing in w. Let ρ,τ be a solution of the system:(24)τ=gτ,ρ,ρ=gρ,τ,τ=ξτ+ετ2μτ+κρ,ρ=ξρ+ερ2μρ+κτ.
Therefore,(25)μ1−ξτ2+κ1−ξτρ=ετ2,(26)μ1−ξρ2+κ1−ξτρ=ερ2.
Subtracting (25) from (26), we get(27)μ1−ξ−ετ2−ρ2=0,and then, ρ=τ if μ1−ξ≠ε. Thus, from Theorem 5.20 in [44], we observe that there exists only one solution for (7) and it is a global attractor if μ1−ξ≠ε.


## 6. Special Cases

Now, we present the solutions’ expressions for special cases of (7):(28)Un+1=Un−8±Un−82Un−8±Un−17,where the initial conditions are(29)U−17,U−16,U−15,U−14,U−13,U−12,U−11,U−10,U−9,U−8,U−7,U−6,U−5,U−4,U−3,U−2,U−1,and U0 are arbitrary real numbers.


### 6.1. First Equation

We solve the equation(30)Un+1=Un−8+Un−82Un−8+Un−17.Theorem 5.
AssumeUnn=−17∞ is a solution of (30); thus, for n=0,1,…,(31)U9n−8=U−8∏i=1nF2i+1U−8+F2iU−17F2iU−8+F2i−1U−17,U9n−4=U−4∏i=1nF2i+1U−4+F2iU−13F2iU−4+F2i−1U−13,U9n−7=U−7∏i=1nF2i+1U−7+F2iU−16F2iU−7+F2i−1U−16,U9n−3=U−3∏i=1nF2i+1U−3+F2iU−12F2iU−3+F2i−1U−12,U9n−6=U−6∏i=1nF2i+1U−6+F2iU−15F2iU−6+F2i−1U−15,U9n−2=U−2∏i=1nF2i+1U−2+F2iU−11F2iU−2+F2i−1U−11,U9n−5=U−5∏i=1nF2i+1U−5+F2iU−14F2iU−5+F2i−1U−14,U9n−1=U−1∏i=1nF2i+1U−1+F2iU−10F2iU−1+F2i−1U−10,U9n=U0∏i=1nF2i+1U0+F2iU−9F2iU0+F2i−1U−9,where Fii=1∞=1,1,2,3,5,…, is the Fibonacci sequence.Proof.
We show that the expressions in (31) are solutions of (30) by applying mathematical induction. First, the results hold for n=0. Second, we suppose that the forms are satisfied for n−1 and n−2. Now, we prove that the results are satisfied for n:(32)U9n−17=U−8∏i=1n−1F2i+1U−8+F2iU−17F2iU−8+F2i−1U−17,U9n−13=U−4∏i=1n−1F2i+1U−4+F2iU−13F2iU−4+F2i−1U−13,U9n−16=U−7∏i=1n−1F2i+1U−7+F2iU−16F2iU−7+F2i−1U−16,U9n−12=U−3∏i=1n−1F2i+1U−3+F2iU−12F2iU−3+F2i−1U−12,U9n−15=U−6∏i=1n−1F2i+1U−6+F2iU−15F2iU−6+F2i−1U−15,U9n−11=U−2∏i=1n−1F2i+1U−2+F2iU−11F2iU−2+F2i−1U−11,U9n−14=U−5∏i=1n−1F2i+1U−5+F2iU−14F2iU−5+F2i−1U−14,U9n−10=U−1∏i=1n−1F2i+1U−1+F2iU−10F2iU−1+F2i−1U−10,U9n−9=U0∏i=1n−1F2i+1U0+F2iU−9F2iU0+F2i−1U−9,U9n−18=U0∏i=1n−2F2i+1U0+F2iU−9F2iU0+F2i−1U−9,U9n−26=U−8∏i=1n−2F2i+1U−8+F2iU−17F2iU−8+F2i−1U−17,U9n−22=U−4∏i=1n−2F2i+1U−4+F2iU−13F2iU−4+F2i−1U−13,U9n−25=U−7∏i=1n−2F2i+1U−7+F2iU−16F2iU−7+F2i−1U−16,U9n−21=U−3∏i=1n−2F2i+1U−3+F2iU−12F2iU−3+F2i−1U−12,U9n−24=U−6∏i=1n−2F2i+1U−6+F2iU−15F2iU−6+F2i−1U−15,U9n−20=U−2∏i=1n−2F2i+1U−2+F2iU−11F2iU−2+F2i−1U−11,U9n−23=U−5∏i=1n−2F2i+1U−5+F2iU−14F2iU−5+F2i−1U−14,U9n−19=U−1∏i=1n−2F2i+1U−1+F2iU−10F2iU−1+F2i−1U−10.
From (30), it follows that(33)U9n−1=U9n−10+U9n−102U9n−10+U9n−19,=U9n−101+U9n−10U9n−10+U9n−19,=U9n−101+U−1∏i=1n−1F2i+1U−1+F2iU−10/F2iU−1+F2i−1U−10U−1∏i=1n−1F2i+1U−1+F2iU−10/F2iU−1+F2i−1U−10+U−1∏i=1n−2F2i+1U−1+F2iU−10/F2iU−1+F2i−1U−10,=U9n−101+U−1∏i=1n−2F2i+1U−1+F2iU−10/F2iU−1+F2i−1U−10F2n−1U−1+F2n−2U−10/F2n−2U−1+F2n−3U−10U−1∏i=1n−2F2i+1U−1+F2iU−10/F2iU−1+F2i−1U−10F2n−1U−1+F2n−2U−10/F2n−2U−1+F2n−3U−10+1,=U9n−101+F2n−1U−1+F2n−2U−10/F2n−2U−1+F2n−3U−10F2n−1U−1+F2n−2U−10/F2n−2U−1+F2n−3U−10+1,=U9n−101+F2n−1U−1+F2n−2U−10F2n−1U−1+F2n−2U−10+F2n−2U−1+F2n−3U−10,=U9n−101+F2n−1U−1+F2n−2U−10F2n−1+F2n−2U−1+F2n−2+F2n−3U−10,=U9n−101+F2n−1U−1+F2n−2U−10F2nU−1+F2n−1U−10,=U9n−10F2n−1+F2nU−1+F2n−2+F2n−1U−10F2nU−1+F2n−1U−10,=U9n−10F2n+1U−1+F2nU−10F2nU−1+F2n−1U−10,=U−1∏i=1n−1F2i+1U−1+F2iU−10F2iU−1+F2i−1U−10F2n+1U−1+F2nU−10F2nU−1+F2n−1U−10,=U−1∏i=1nF2i+1U−1+F2iU−10F2iU−1+F2i−1U−10.
Therefore,(34)U9n−1=U−1∏i=1nF2i+1U−1+F2iU−10F2iU−1+F2i−1U−10.
Similarly, one can investigate other expressions. The proof is done.


### 6.2. Second Equation

In this section, we introduce the solution of the following equation:(35)Un+1=Un−8+Un−82Un−8−Un−17.Theorem 6.
LetUnn=−17∞ be a solution of (35); then, for n=0,1,…,(36)U9n−8=U−8∏i=1nF2i+1U−8−F2iU−17F2iU−8−F2i−1U−17,U9n−4=U−4∏i=1nF2i+1U−4−F2iU−13F2iU−4−F2i−1U−13,U9n−7=U−7∏i=1nF2i+1U−7−F2iU−16F2iU−7−F2i−1U−16,U9n−3=U−3∏i=1nF2i+1U−3−F2iU−12F2iU−3−F2i−1U−12,U9n−6=U−6∏i=1nF2i+1U−6−F2iU−15F2iU−6−F2i−1U−15,U9n−2=U−2∏i=1nF2i+1U−2−F2iU−11F2iU−2−F2i−1U−11,U9n−5=U−5∏i=1nF2i+1U−5−F2iU−14F2iU−5−F2i−1U−14,U9n−1=U−1∏i=1nF2i+1U−1−F2iU−10F2iU−1−F2i−1U−10,U9n=U0∏i=1nF2i+1U0−F2iU−9F2iU0−F2i−1U−9,where Fii=−1∞=1,0,1,1,2,3,5,… is the Fibonacci sequence.Proof.
The proof will be the same as proof of Theorem5, so it is therefore omitted.


### 6.3. Third Equation

In this section, we present the solution of the following equation:(37)Un+1=Un−8−Un−82Un−8+Un−17.Theorem 7.
LetUnn=−17∞ be a solution of (37); then, for n=0,1,…,(38)U9n−8=U−8U−17FnU−8+Fn+1U−17,U9n−4=U−4U−13FnU−4+Fn+1U−13,U9n−7=U−7U−16FnU−7+Fn+1U−16,U9n−3=U−3U−12FnU−3+Fn+1U−12,U9n−6=U−6U−15FnU−6+Fn+1U−15,U9n−2=U−2U−11FnU−2+Fn+1U−11,U9n−5=U−5U−14FnU−5+Fn+1U−14,U9n−1=U−1U−10FnU−1+Fn+1U−10,U9n=U0U−9FnU0+Fn+1U−9,where Fii=−1∞=1,0,1,1,2,3,5,…,.Proof.
By using mathematical induction, we prove that (38) are solutions of (37). First, the results for n=0 are true. Second, assume that the assumption holds for n−2 and n−1.(39)U9n−17=U−8U−17Fn−1U−8+FnU−17,U9n−13=U−4U−13Fn−1U−4+FnU−13,U9n−16=U−7U−16Fn−1U−7+FnU−16,U9n−12=U−3U−12Fn−1U−3+FnU−12,U9n−15=U−6U−15Fn−1U−6+FnU−15,U9n−11=U−2U−11Fn−1U−2+FnU−11,U9n−14=U−5U−14Fn−1U−5+FnU−14,U9n−10=U−1U−10Fn−1U−1+FnU−10,U9n−9=U0U−9Fn−1U0+FnU−9,U9n−18=U0U−9Fn−2U0+Fn−1U−9,U9n−26=U−8U−17Fn−2U−8+Fn−1U−17,U9n−22=U−4U−13Fn−2U−4+Fn−1U−13,U9n−25=U−7U−16Fn−2U−7+Fn−1U−16,U9n−21=U−3U−12Fn−2U−3+Fn−1U−12,U9n−24=U−6U−15Fn−2U−6+Fn−1U−15,U9n−20=U−2U−11Fn−2U−2+Fn−1U−11,U9n−23=U−5U−14Fn−2U−5+Fn−1U−14,U9n−19=U−1U−10Fn−2U−1+Fn−1U−10.
Now, from (37), we have(40)U9n−1=U9n−10−U9n−102U9n−10+U9n−19,=U9n−101−U9n−10U9n−10+U9n−19,=U9n−101−U−1U−10/Fn−1U−1+FnU−10U−1U−10/Fn−1U−1+FnU−10+U−1U−10/Fn−2U−1+Fn−1U−10,=U9n−101−U−1U−10/Fn−1U−1+FnU−10U−1U−10/Fn−1U−1+FnU−101+Fn−1U−1+FnU−10/Fn−2U−1+Fn−1U−10,=U9n−101−11+Fn−1U−1+FnU−10/Fn−2U−1+Fn−1U−10,=U9n−101−Fn−2U−1+Fn−1U−10Fn−2U−1+Fn−1U−10+Fn−1U−1+FnU−10U9n−101−Fn−2U−1+Fn−1U−10FnU−1+Fn+1U−10,=U9n−10Fn−Fn−2U−1+Fn+1−Fn−1U−10FnU−1+Fn+1U−10,=U9n−10Fn−1U−1+FnU−10FnU−1+Fn+1U−10,=U−1U−10Fn−1U−1+FnU−10Fn−1U−1+FnU−10FnU−1+Fn+1U−10,=U−1U−10FnU−1+Fn+1U−10.
Thus,(41)U9n−1=U−1U−10FnU−1+Fn+1U−10.
Similarly, one can see that the other forms are true. The proof is complete.


### 6.4. Fourth Equation

We study the following equation:(42)Un+1=Un−8−Un−82Un−8−Un−17.Theorem 8.
Suppose thatUnn=−17∞ is a solution of (42), then there exists a periodic solution with period 54. Moreover, Unn=−17∞ takes the form(43)U−17,U−16,U−15,U−14,U−13,U−12,U−11,U−10,U−9,U−8,U−7,U−6,U−5,U−4,U−3,U−2,U−1,U0,−U−8U−17U−8−U−17,−U−7U−16U−7−U−16,−U−6U−15U−6−U−15,−U−5U−14U−5−U−14,−U−4U−13U−4−U−13,−U−3U−12U−3−U−12,−U−2U−11U−2−U−11,−U−1U−10U−1−U−10,−U0U−9U0−U−19,−U−17,−U−16,−U−15,−U−14,−U−13,−U−12,−U−11,−U−10,−U−9,−U−8,−U−7,−U−6,−U−5,−U−4,−U−3,−U−2,−U−1,−U0,U−8U−17U−8−U−17,U−7U−16U−7−U−16,U−6U−15U−6−U−15,U−5U−14U−5−U−14,U−4U−13U−4−U−13,U−3U−12U−3−U−12,U−2U−11U−2−U−11,U−1U−10U−1−U−10,U0U−9U0−U−19,U−17,U−16,U−15,U−14,U−13,U−12,U−11,U−10,U−9,U−8,U−7,U−6,U−5,U−4,U−3,U−2,U−1,U0,….Proof.
The proof of this case will be the same as the proof presented for Theorem7 and will be omitted therefore.


## 6.1. First Equation

We solve the equation(30)Un+1=Un−8+Un−82Un−8+Un−17.Theorem 5.
AssumeUnn=−17∞ is a solution of (30); thus, for n=0,1,…,(31)U9n−8=U−8∏i=1nF2i+1U−8+F2iU−17F2iU−8+F2i−1U−17,U9n−4=U−4∏i=1nF2i+1U−4+F2iU−13F2iU−4+F2i−1U−13,U9n−7=U−7∏i=1nF2i+1U−7+F2iU−16F2iU−7+F2i−1U−16,U9n−3=U−3∏i=1nF2i+1U−3+F2iU−12F2iU−3+F2i−1U−12,U9n−6=U−6∏i=1nF2i+1U−6+F2iU−15F2iU−6+F2i−1U−15,U9n−2=U−2∏i=1nF2i+1U−2+F2iU−11F2iU−2+F2i−1U−11,U9n−5=U−5∏i=1nF2i+1U−5+F2iU−14F2iU−5+F2i−1U−14,U9n−1=U−1∏i=1nF2i+1U−1+F2iU−10F2iU−1+F2i−1U−10,U9n=U0∏i=1nF2i+1U0+F2iU−9F2iU0+F2i−1U−9,where Fii=1∞=1,1,2,3,5,…, is the Fibonacci sequence.Proof.
We show that the expressions in (31) are solutions of (30) by applying mathematical induction. First, the results hold for n=0. Second, we suppose that the forms are satisfied for n−1 and n−2. Now, we prove that the results are satisfied for n:(32)U9n−17=U−8∏i=1n−1F2i+1U−8+F2iU−17F2iU−8+F2i−1U−17,U9n−13=U−4∏i=1n−1F2i+1U−4+F2iU−13F2iU−4+F2i−1U−13,U9n−16=U−7∏i=1n−1F2i+1U−7+F2iU−16F2iU−7+F2i−1U−16,U9n−12=U−3∏i=1n−1F2i+1U−3+F2iU−12F2iU−3+F2i−1U−12,U9n−15=U−6∏i=1n−1F2i+1U−6+F2iU−15F2iU−6+F2i−1U−15,U9n−11=U−2∏i=1n−1F2i+1U−2+F2iU−11F2iU−2+F2i−1U−11,U9n−14=U−5∏i=1n−1F2i+1U−5+F2iU−14F2iU−5+F2i−1U−14,U9n−10=U−1∏i=1n−1F2i+1U−1+F2iU−10F2iU−1+F2i−1U−10,U9n−9=U0∏i=1n−1F2i+1U0+F2iU−9F2iU0+F2i−1U−9,U9n−18=U0∏i=1n−2F2i+1U0+F2iU−9F2iU0+F2i−1U−9,U9n−26=U−8∏i=1n−2F2i+1U−8+F2iU−17F2iU−8+F2i−1U−17,U9n−22=U−4∏i=1n−2F2i+1U−4+F2iU−13F2iU−4+F2i−1U−13,U9n−25=U−7∏i=1n−2F2i+1U−7+F2iU−16F2iU−7+F2i−1U−16,U9n−21=U−3∏i=1n−2F2i+1U−3+F2iU−12F2iU−3+F2i−1U−12,U9n−24=U−6∏i=1n−2F2i+1U−6+F2iU−15F2iU−6+F2i−1U−15,U9n−20=U−2∏i=1n−2F2i+1U−2+F2iU−11F2iU−2+F2i−1U−11,U9n−23=U−5∏i=1n−2F2i+1U−5+F2iU−14F2iU−5+F2i−1U−14,U9n−19=U−1∏i=1n−2F2i+1U−1+F2iU−10F2iU−1+F2i−1U−10.
From (30), it follows that(33)U9n−1=U9n−10+U9n−102U9n−10+U9n−19,=U9n−101+U9n−10U9n−10+U9n−19,=U9n−101+U−1∏i=1n−1F2i+1U−1+F2iU−10/F2iU−1+F2i−1U−10U−1∏i=1n−1F2i+1U−1+F2iU−10/F2iU−1+F2i−1U−10+U−1∏i=1n−2F2i+1U−1+F2iU−10/F2iU−1+F2i−1U−10,=U9n−101+U−1∏i=1n−2F2i+1U−1+F2iU−10/F2iU−1+F2i−1U−10F2n−1U−1+F2n−2U−10/F2n−2U−1+F2n−3U−10U−1∏i=1n−2F2i+1U−1+F2iU−10/F2iU−1+F2i−1U−10F2n−1U−1+F2n−2U−10/F2n−2U−1+F2n−3U−10+1,=U9n−101+F2n−1U−1+F2n−2U−10/F2n−2U−1+F2n−3U−10F2n−1U−1+F2n−2U−10/F2n−2U−1+F2n−3U−10+1,=U9n−101+F2n−1U−1+F2n−2U−10F2n−1U−1+F2n−2U−10+F2n−2U−1+F2n−3U−10,=U9n−101+F2n−1U−1+F2n−2U−10F2n−1+F2n−2U−1+F2n−2+F2n−3U−10,=U9n−101+F2n−1U−1+F2n−2U−10F2nU−1+F2n−1U−10,=U9n−10F2n−1+F2nU−1+F2n−2+F2n−1U−10F2nU−1+F2n−1U−10,=U9n−10F2n+1U−1+F2nU−10F2nU−1+F2n−1U−10,=U−1∏i=1n−1F2i+1U−1+F2iU−10F2iU−1+F2i−1U−10F2n+1U−1+F2nU−10F2nU−1+F2n−1U−10,=U−1∏i=1nF2i+1U−1+F2iU−10F2iU−1+F2i−1U−10.
Therefore,(34)U9n−1=U−1∏i=1nF2i+1U−1+F2iU−10F2iU−1+F2i−1U−10.
Similarly, one can investigate other expressions. The proof is done.


## 6.2. Second Equation

In this section, we introduce the solution of the following equation:(35)Un+1=Un−8+Un−82Un−8−Un−17.Theorem 6.
LetUnn=−17∞ be a solution of (35); then, for n=0,1,…,(36)U9n−8=U−8∏i=1nF2i+1U−8−F2iU−17F2iU−8−F2i−1U−17,U9n−4=U−4∏i=1nF2i+1U−4−F2iU−13F2iU−4−F2i−1U−13,U9n−7=U−7∏i=1nF2i+1U−7−F2iU−16F2iU−7−F2i−1U−16,U9n−3=U−3∏i=1nF2i+1U−3−F2iU−12F2iU−3−F2i−1U−12,U9n−6=U−6∏i=1nF2i+1U−6−F2iU−15F2iU−6−F2i−1U−15,U9n−2=U−2∏i=1nF2i+1U−2−F2iU−11F2iU−2−F2i−1U−11,U9n−5=U−5∏i=1nF2i+1U−5−F2iU−14F2iU−5−F2i−1U−14,U9n−1=U−1∏i=1nF2i+1U−1−F2iU−10F2iU−1−F2i−1U−10,U9n=U0∏i=1nF2i+1U0−F2iU−9F2iU0−F2i−1U−9,where Fii=−1∞=1,0,1,1,2,3,5,… is the Fibonacci sequence.Proof.
The proof will be the same as proof of Theorem5, so it is therefore omitted.


## 6.3. Third Equation

In this section, we present the solution of the following equation:(37)Un+1=Un−8−Un−82Un−8+Un−17.Theorem 7.
LetUnn=−17∞ be a solution of (37); then, for n=0,1,…,(38)U9n−8=U−8U−17FnU−8+Fn+1U−17,U9n−4=U−4U−13FnU−4+Fn+1U−13,U9n−7=U−7U−16FnU−7+Fn+1U−16,U9n−3=U−3U−12FnU−3+Fn+1U−12,U9n−6=U−6U−15FnU−6+Fn+1U−15,U9n−2=U−2U−11FnU−2+Fn+1U−11,U9n−5=U−5U−14FnU−5+Fn+1U−14,U9n−1=U−1U−10FnU−1+Fn+1U−10,U9n=U0U−9FnU0+Fn+1U−9,where Fii=−1∞=1,0,1,1,2,3,5,…,.Proof.
By using mathematical induction, we prove that (38) are solutions of (37). First, the results for n=0 are true. Second, assume that the assumption holds for n−2 and n−1.(39)U9n−17=U−8U−17Fn−1U−8+FnU−17,U9n−13=U−4U−13Fn−1U−4+FnU−13,U9n−16=U−7U−16Fn−1U−7+FnU−16,U9n−12=U−3U−12Fn−1U−3+FnU−12,U9n−15=U−6U−15Fn−1U−6+FnU−15,U9n−11=U−2U−11Fn−1U−2+FnU−11,U9n−14=U−5U−14Fn−1U−5+FnU−14,U9n−10=U−1U−10Fn−1U−1+FnU−10,U9n−9=U0U−9Fn−1U0+FnU−9,U9n−18=U0U−9Fn−2U0+Fn−1U−9,U9n−26=U−8U−17Fn−2U−8+Fn−1U−17,U9n−22=U−4U−13Fn−2U−4+Fn−1U−13,U9n−25=U−7U−16Fn−2U−7+Fn−1U−16,U9n−21=U−3U−12Fn−2U−3+Fn−1U−12,U9n−24=U−6U−15Fn−2U−6+Fn−1U−15,U9n−20=U−2U−11Fn−2U−2+Fn−1U−11,U9n−23=U−5U−14Fn−2U−5+Fn−1U−14,U9n−19=U−1U−10Fn−2U−1+Fn−1U−10.
Now, from (37), we have(40)U9n−1=U9n−10−U9n−102U9n−10+U9n−19,=U9n−101−U9n−10U9n−10+U9n−19,=U9n−101−U−1U−10/Fn−1U−1+FnU−10U−1U−10/Fn−1U−1+FnU−10+U−1U−10/Fn−2U−1+Fn−1U−10,=U9n−101−U−1U−10/Fn−1U−1+FnU−10U−1U−10/Fn−1U−1+FnU−101+Fn−1U−1+FnU−10/Fn−2U−1+Fn−1U−10,=U9n−101−11+Fn−1U−1+FnU−10/Fn−2U−1+Fn−1U−10,=U9n−101−Fn−2U−1+Fn−1U−10Fn−2U−1+Fn−1U−10+Fn−1U−1+FnU−10U9n−101−Fn−2U−1+Fn−1U−10FnU−1+Fn+1U−10,=U9n−10Fn−Fn−2U−1+Fn+1−Fn−1U−10FnU−1+Fn+1U−10,=U9n−10Fn−1U−1+FnU−10FnU−1+Fn+1U−10,=U−1U−10Fn−1U−1+FnU−10Fn−1U−1+FnU−10FnU−1+Fn+1U−10,=U−1U−10FnU−1+Fn+1U−10.
Thus,(41)U9n−1=U−1U−10FnU−1+Fn+1U−10.
Similarly, one can see that the other forms are true. The proof is complete.


## 6.4. Fourth Equation

We study the following equation:(42)Un+1=Un−8−Un−82Un−8−Un−17.Theorem 8.
Suppose thatUnn=−17∞ is a solution of (42), then there exists a periodic solution with period 54. Moreover, Unn=−17∞ takes the form(43)U−17,U−16,U−15,U−14,U−13,U−12,U−11,U−10,U−9,U−8,U−7,U−6,U−5,U−4,U−3,U−2,U−1,U0,−U−8U−17U−8−U−17,−U−7U−16U−7−U−16,−U−6U−15U−6−U−15,−U−5U−14U−5−U−14,−U−4U−13U−4−U−13,−U−3U−12U−3−U−12,−U−2U−11U−2−U−11,−U−1U−10U−1−U−10,−U0U−9U0−U−19,−U−17,−U−16,−U−15,−U−14,−U−13,−U−12,−U−11,−U−10,−U−9,−U−8,−U−7,−U−6,−U−5,−U−4,−U−3,−U−2,−U−1,−U0,U−8U−17U−8−U−17,U−7U−16U−7−U−16,U−6U−15U−6−U−15,U−5U−14U−5−U−14,U−4U−13U−4−U−13,U−3U−12U−3−U−12,U−2U−11U−2−U−11,U−1U−10U−1−U−10,U0U−9U0−U−19,U−17,U−16,U−15,U−14,U−13,U−12,U−11,U−10,U−9,U−8,U−7,U−6,U−5,U−4,U−3,U−2,U−1,U0,….Proof.
The proof of this case will be the same as the proof presented for Theorem7 and will be omitted therefore.


## 7. Numerical Examples

To illustrate the solution behavior of (7) for various cases, we present some numerical examples.Example 1.
To show the stability of (7), we set two groups for the values of the coefficients: (i) ξ=0.5,ε=0.1,μ=1.6,andκ=0.2 and (ii) ξ=0.5,ε=5,μ=10,andκ=0.001, and the initial conditions are(44)U−17=0.1,U−16=0.2,U−15=0.3,U−14=0.4,U−13=0.5,U−12=0.6,U−11=0.7,U−10=0.8,U−9=0.9,U−8=1.2,U−7=1.5,U−6=2.2,U−5=2.3,U−4=2.5,U−3=4.2,U−2=4.6,U−1=4.8,and U0=5.2. The result is obtained in Figure 1. It is clear that (i) condition (23) is satisfied, which implies that the solution tends to the fixed point U¯=0, while the solution moves away from the fixed point for (ii) since condition (23) failed.
The following examples have explained the solutions of special case equations (30)–(42).Figure 1
Plotting the solution ofUn+1=ξUn−8+εUn−82/μUn−8+κUn−17.Example 2.
We choose the initial conditions as(45)U−17=0.01,U−16=0.02,U−15=0.03,U−14=0.04,U−13=0.05,U−12=0.06,U−11=0.07,U−10=0.08,U−9=0.09,U−8=1.02,U−7=1.05,U−6=2.02,U−5=2.03,U−4=2.05,U−3=4.02,U−2=4.06,U−1=4.08,and U0=5.02. The solution is given in Figure 2.Figure 2
Plotting of the solution ofUn+1=Un−8+Un−82/Un−8+Un−17.Example 3.
In Figure3, we set the initial conditions:(46)U−17=0.01,U−16=0.02,U−15=0.03,U−14=0.04,U−13=0.05,U−12=0.06,U−11=0.07,U−10=0.08,U−9=0.09,U−8=1.02,U−7=1.05,U−6=2.02,U−5=2.03,U−4=2.05,U−3=4.02,U−2=4.06,U−1=4.08,U0=5.02.Figure 3
The solution behavior ofUn+1=Un−8+Un−82/Un−8−Un−17.Example 4.
For (37), we choose the initial conditions as(47)U−17=2,U−16=2.1,U−15=2.2,U−14=2.3,U−13=2.4,U−12=2.5,U−11=2.6,U−10=2.7,U−9=2.8,U−8=3,U−7=3.1,U−6=3.2,U−5=3.3,U−4=3.4,U−3=3.5,U−2=3.6,U−1=3.7,U0=3.8,and then, the result is shown in Figure 4.Figure 4
The solution behavior ofUn+1=Un−8−Un−82/Un−8+Un−17.Example 5.
We set the values(48)U−17=0.01,U−16=0.02,U−15=0.03,U−14=0.04,U−13=0.05,U−12=0.06,U−11=0.07,U−10=0.08,U−9=0.09,U−8=1.02,U−7=1.05,U−6=2.02,U−5=2.03,U−4=2.05,U−3=4.02,U−2=4.06,U−1=4.08,U0=5.02.
The solution is given in Figure5. Clearly, the solution is periodic that means the result conforms with Theorem 8.Figure 5
Plotting the solution ofUn+1=Un−8−Un−82/Un−8−Un−17.



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