# On Stability Analysis of Higher-Order Rational Difference Equation

Authors: Abdul Khaliq; H. S. Alayachi; M. S. M. Noorani; A. Q. Khan
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, global behavior of equilibrium points, boundedness and periodicity of the rational recursive sequencewn+1=wn−pα+βwn/γwn+δwn−r, where γwn≠−δwn−r for r∈0,∞,α,β,γ,δ∈0,∞,andr>p≥0. With initial values w−p,w−p+1,…,w−r,w−r+1,…,w−1,andw0 are positive real numbers. Some numerical examples are given to verify our theoretical results.


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## Body

## 1. Introduction

It is very amusing to explore the nature of the solutions of a higher-order rational difference equation and to explain the local asymptotic stability of its equilibrium points. The inspection of some properties associated with these equations is a very enormous activity. Discrete dynamical systems or difference equations are diverse fields because various biological systems and models naturally lead to their study by means of a discrete variable. Applications of discrete dynamical systems and difference equations have appeared recently in many fields of science and technology. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. Rational difference equations are a special form of nonlinear difference equations. Delay difference equations have rich dynamics to study. Due to adequate computational outcomes, discrete dynamical systems are awful lot better than allied structures in differential equations. Specifically, in case of nonoverlapping generations, difference equations are greater apposite to take a look at the behavior of population models [1, 2]. Also, an epidemiological approach to insurgent population modeling is mentioned in [3, 4]. The research of delay difference equations is a divergent field that involves most aspects of mathematics, including both applied and pure. Recently, there has been a symbolic development in the applications of difference equations. It is very interesting to investigate the behavior of solutions of a system of nonlinear difference equations and to discuss the local asymptotic stability of their equilibrium points. There are many papers in which systems and behavior of rational difference equations have been studied see [5–20]. Rational difference equations have been studied by several authors. There has been a great interest exclusively in the study of the attractiveness of the solution of such equations.Here, we recall some basics and preliminaries that will be useful in our results and investigation.


## 2. Preliminaries and Definitions

Definition 1.
LetI be some interval of real numbers, and let(1)F:Ik+1⟶I,be a continuously differentiable function. Then, for every set of initial conditions S−k,S−k+1,…,S0∈I, the difference equation(2)Sn+1=RSn,Sn−1,…,Sn−k,n=0,1,…,has a unique solution Snn=−k∞.Definition 2.
A pointS¯∈I is called an equilibrium point of equation (2) if(3)S¯=RS¯,S¯,…,S¯.That is, Sn=S¯, for all n≥0, is a solution of equation (2), or equivalently, S¯ is a fixed point of R.Definition 3.
(periodicity). A sequenceSnn=−k∞ is said to be periodic with period p if Sn+p=Sn for all n≥−k.Definition 4.
(Fibonacci sequence). The sequenceFmm=1∞=1,2,3,5,8,13,….i.e.Fm=Fm−1+Fm−2≥0,F−2=0,F−1=1 is called Fibonacci sequence.Definition 5.
Equation (2) is called permanent and bounded if there exist numbers t and T with 0<t<T<∞ such that for any initial values S−k,S−k+1,…,S−1,S0∈I, there exists a positive integer N which depends on these initial values such that t≤Sn≤T, for all n≥N.Definition 6.
(stability)(i)
The equilibrium pointS¯∈I of equation (2) is locally stable if for every ε>0, there exists δ>0 such that for all S−k,S−k+1,…,S−1,S0∈I with(4)S−k−S¯+S−k+1−S¯+⋯+S0−S¯<δ,we have(5)Sn−S¯<εfor alln≥−k.(ii)
The equilibrium pointS¯ of equation (2) is locally asymptotically stable if S¯ is a locally stable solution of equation (2) and there exists γ>0, such that for all S−k,S−k+1,…,S−1, S0∈K with(6)S−k−S¯+S−k+1−S¯+⋯+S0−S¯<γ,we have(7)limn⟶∞Sn=S¯.(iii)
The equilibrium pointS¯ of equation (2) is global attractor if for all S−k,S−k+1,…,S−1, S0∈I, we have(8)limn⟶∞Sn=S¯.(iv)
The equilibrium pointS¯ of equation (2) is globally asymptotically stable if S¯ is locally stable and S¯ is also a global attractor of equation (2).(v)
The equilibrium pointS¯ of equation (2) is unstable if S¯ is not locally stable.Definition 7.
The linearized equation of equation (2) about the equilibrium S¯ is defined by the following equation:(9)yn+1=∑i=03ρiyn−i,where(10)ρ0=∂FS¯,S¯,…,S¯∂Sn,ρ1=∂FS¯,S¯,…,S¯∂Sn−1,ρ2=∂FS¯,S¯,…,S¯∂Sn−2,ρ3=∂FS¯,S¯,…,S¯∂Sn−3.Lemma 1 (see[21]).
Assume thatpi∈R and i∈1,2,…,k. Then,(11)∑i=1kpi<1,is a sufficient condition for the asymptotic stability of the following difference equation:(12)Sn+k+p1Sn+k−1+⋯+pkSn=0,n=0,1,2,….The following theorem will be useful for the proof of our results in this paper.Theorem 1.
Letα,β be an interval of real numbers and assume that g:α,β2⟶α,β is a continuous function, and consider the following equation:(13)xn+1=gxn,xn−1,n=0,1,…,satisfying the following conditions:(a)
gx,y is nondecreasing in x∈α,β for each fixed y∈α,β, and gx,y is nonincreasing in y∈α,β for each fixed x∈α,β(b)
Ifm,M∈α,β×α,β is a solution of the systems(14)M=gM,m,m=gm,M,thenm=M and equation (2) has a unique equilibrium S¯∈α,β and every solution of equation (2) converges to S¯.
Some published works include the following:
Ibrahim and El-Moneam [22] investigate the local and global stability of the following recursive sequence(15)Sn+1=Sn−pα+bSn−qcSn−q+dSn−r,where cS−q+k≠−dS−r+k for k=0,1,…,minq,randα,b,c, and d are real numbers.
Aboutaleb et al. [23] studied the global attractiveness of the positive equilibrium of the following rational recursive equation:(16)ym+1=α−βymγ+ym−1,m=0,1,2,…,where the coefficients α,β,andγ are the nonnegative real numbers.
Several other researchers have studied the behavior of the solution of difference equations; for example, Devault et al. [24] investigated the global behavior of all positive solutions of the following equation:(17)Sn+1=p+Sn−kqSn−k.
Elabbasy et al. [25] studied the boundedness, global stability, and periodicity character and gave the solution of some special cases of the following difference equation:(18)xn+1=αxn−l+βxn−kAxn−l+Bxn−k.
Elabbasy [26] gave the solution of the following difference equation:(19)Sn+1=αSn−bSncSn−dSn−1.
El-Moneam and Zayed [27] investigated the global stability and periodicity character and gave the solution of some special cases of the following difference equation:(20)xn+1=axn−k+bxn−kcxn+dxn−k.
Zayed and El-Moneam [28] studied the global behavior of the following rational recursive sequence:(21)xn+1=α0xn+α1xn−2+α2xn−m+α3xn−kβ0xn+β1xn−2+β2xn−m+β3xn−k.
Motivated by the above study, the main focus of this article is to discuss some qualitative behaviors of the solutions of the following rational recursive sequence:(22)wn+1=wn−pα+βwnγwn+δwn−r,where γwn≠−δwn−r for r∈0,∞,α,β,γ,δ∈0,∞,andr>p≥0. With initial values w−p,w−p+1,….w−r,w−r+1,…w−1,andw0 are the positive real numbers. Some numerical examples are given to verify our theoretical results.


## 3. Local Stability of Equation (22)

This section deals with the local stability character of the solutions of equation (22). Equation (22) has a unique equilibrium point, and it is given from(23)w¯=w¯α+βw¯γw¯+δw¯,orw¯21−αγ+δ=βw¯2,if(24)1−αγ+δ≠β.Then, the unique equilibrium point isw¯=0.Letf:0,∞3⟶0,∞ be a function defined by(25)fs,u,v=αs+βsuγu+δv.Therefore, it follows that(26)fs=α+βuγu+δv,fu=δβvsγu+δv2,fv=−βsuγu+δv2.We see that(27)fsw¯,w¯,w¯=α+βγ+δ=−a2,fuw¯,w¯,w¯=δβγ+δ2=−a1,fvw¯,w¯,w¯=−βδγ+δ2=−a0.The linearized equation of equation (22) about w¯ is(28)Tn+1=a0Tn+a1Tn−p+a2Tn−r.The characteristic equation of equation (28) is(29)λn+1=a0λn+a1λn−p+a2λn−r.Theorem 2.
Assume that(30)γ+δ21−α>βγ+3δ.
Then, the equilibrium point of equation (22) is locally asymptotically stable.Proof.
It follows by Lemma 7 that equation (28) is asymptotically stable if(31)α+βγ+δ+δβγ+δ2+−βδγ+δ2<1,orα+βγ+δ+2δβγ+δ2<1,and so(32)βγ+3βδγ+δ2<1−α.Hence,(33)γ+δ21−α>βγ+3δ.


## 4. Boundedness of the Solutions

In this section, we show the boundedness of the positive solutions of equation (22).Theorem 3.
Every solution of equation (22) is bounded from above by(34)M=maxw−p+1,w−p,…,w−2,w−1,w0,if(35)α+βγ<1.Proof.
LetFnn=−p∞ be a solution of equation (22). It follows from equation (22) that(36)wn+1=αwn−p+βwnwn−pγwn+δwn−r≤αwn−p+βwnwn−pγwn=α+βγwn−p.
Then, for alln≥N,(37)wn+1≤wn−p,∀n≥0.
Then, all the subsequenceswp+1n+jn=0∞,i=0,1,2,…,p, are decreasing and so are bounded from above by(38)M=maxw−p+1,w−p,…,w−2,w−1,w0.Theorem 4.
Ifα>1, then each solution of equation (22) will be unbounded.Proof.
Letwnn=−p∞ be a solution of equation (22). Then, from equation (22), we see that(39)wn+1=αwn−p+βwnwn−pγwn+δwn−r>αwn−p,∀n≥1.
From above, the right-hand side can be written as follows:(40)Tn+1=αTn−p⟹Tn=αnT0,and this equation is unstable because α>1 and limn⟶∞Tn=∞.
Then, by using the ratio test,wnn=−p∞ is unbounded from above.


## 5. Periodic Solutions of Equation (22)

In this section, we satisfy the periodic solutions of equation (22).Theorem 5.
Equation (22) has no positive solutions of prime period two solution ∀α,β,γ,δ∈0,∞ in the following case, if p is even and r is odd.Proof.
Assume that there exists two distinct positive real numbersϕ,φ∈0,∞ such that ϕ,φ, ϕ,φ, ϕ,φ,… is prime period two solution of equation (22). We have(41)ϕ=φα+βφγφ+δϕ,(42)φ=ϕα+βϕγϕ+δφ.By simplifying equations (41) and (42), we obtain(43)ϕ−φαγφ−δϕ=βφ2φ−ϕαγϕ−δφ=βϕ2.By subtracting, we deduce that(44)ϕ−φαγφ−δϕ−φ−ϕαγϕ−δφ=βφ2−βϕ2δ+β+αγϕ2=β+αγ+δφ2⟹ϕ2−φ2=0.This implies that equation (22) has no positive solutions of prime period two.Theorem 6.
Equation (22) has positive period two solution ∀α,β,γ,δ∈0,∞ in the following cases:(i)
γ+δ1+α+β=0pandrare even(ii)
α=1pandrare odd(iii)
γ+δ1−α−β=0pis odd;ris evenProof.
We will prove case (i) and the other cases (ii) and (iii) are obtained by the same way. Assume that there exist two distinct positive real numbersϕ,φ∈0,∞ such that ϕ,φ, ϕ,φ, ϕ,φ,… is prime period two solution of equation (22):(45)ϕ=φα+βφγφ+δφ,(46)φ=ϕα+βϕγϕ+δϕ.
From equation (45) and equation (46), we obtain(47)ϕ−φαγ+δ=βφφ−ϕαγ+δ=βϕ,
By solving these equations, we obtain(48)ϕ−φγ+δ1+α+β=0.
Sinceα,β,γ,andδ are nonzero positive real numbers, and ϕ≠φ.
This implies that(49)γ+δ1+α+β=0.


## 6. Global Stability

We will study the global asymptotic stability of the positive solutions of equation (22) in this section.Theorem 7.
The equilibrium pointw¯ of equation (22) is a global attractor if(50)β≠1−αγ,Proof.
Leta and b are real numbers and assume that f:a,b3⟶a,b be function defined by fs,u,v=αs+βsu/γu+δv. Thus, we see that the function fs,u,v is increasing in sandu and is decreasing in v. Let g,G is a solution of the system G=fG,G,g and g=fg,g,G. Then, from (22), we see that(51)G=αG+βG2γG+δg,g=αg+βg2γg+δG,⟹(52)G1−α=βG2γG+δg,g1−α=βg2γg+δG.
Then,(53)γ1−αG2+δ1−αGg=G2β,γ1−αg2+δ1−αGg=g2β.By subtracting these equations, we obtain(54)γ1−αG2−g2=βG2−g2,γ1−α≠β.
Thus,(55)G=g.
It follows that the equilibrium pointw¯ of equation (22) is a global attractor.


## 7. Applications

In this section, we will discuss the solution of some special cases of equation (22).Case 1.
Whenα=0,p=r=0. In this case, we have the following special type of difference equation:(56)wn+1=βwn2γwn+δwn=wnβγ+δ.Theorem 8.
The general solution of equation (56) is(57)wn=aβnγ+δn,where(58)w0=a.Proof.
Proof is obtained by induction, and it is easy to do.Case 2.
Whenα=1andβ=γ=δ=1,p=3andr=4. In this case, we have the following special case of difference equation:(59)wn+1=wn−3+wnwn−3wn+wn−4,where w−4,w−3,w−2,w−1, and w0 are the arbitrary real numbers.Theorem 9.
Letwnn=−4∞ be a solution of equation (59). Then, for n=0,1,2,…,(60)w4n=e∏i=0n−1F8i+8a+F8i+9eF8i+7a+F8i+8e,w4n−1=d∏i=0n−1F8i+6a+F8i+7eF8i+5a+F8i+6e,w4n−2=c∏i=0n−1F8i+4a+F8i+5eF8i+3a+F8i+4e,w4n−3=b∏i=0n−1F8i+2a+F8i+3eF8i+1a+F8i+2e,where w−4=a,w−3=b,w−2=c,w−1=d, w0=e, and Fmm=0∞=0,1,1,2,3,5,8,13,21,…,.Proof.
Forn=0, the result holds. Now, suppose that n>0 and that our assumption holds for n−1andn−2:(61)w4n−4=e∏i=0n−2F8i+8a+F8i+9eF8i+7a+F8i+8e,w4n−5=d∏i=0n−2F8i+6a+F8i+7eF8i+5a+F8i+6e,w4n−6=c∏i=0n−2F8i+4a+F8i+5eF8i+3a+F8i+4e,w4n−7=b∏i=0n−2F8i+2a+F8i+3eF8i+1a+F8i+2e,w4n−8=e∏i=0n−3F8i+8a+F8i+9eF8i+7a+F8i+8e.
Now, it follows from (59) that(62)w4n=wn−4−w4n−1w4n−4w4n−1−w4n−5=e∏i=0n−2F8i+8a+F8i+9eF8i+7a+F8i+8e−d∏i=0n−1F8i+6a+F8i+7e/F8i+5a+F8i+6ee∏i=0n−2F8i+8a+F8i+9e/F8i+7a+F8i+8ed∏i=0n−1F8i+6a+F8i+7e/F8i+5a+F8i+6e−d∏i=0n−2F8i+6a+F8i+7e/F8i+5a+F8i+6e=e∏i=0n−2F8i+8a+F8i+9eF8i+7a+F8i+8e1−d∏i=0n−1F8i+6a+F8i+7e/F8i+5a+F8i+6ed∏i=0n−1F8i+6a+F8i+7e/F8i+5a+F8i+6e−d∏i=0n−2F8i+6a+F8i+7e/F8i+5a+F8i+6e=e∏i=0n−2F8i+8a+F8i+9eF8i+7a+F8i+8e1−d∏i=0n−1F8i+6a+F8i+7e/F8i+5a+F8i+6ed∏i=0n−1F8i+6a+F8i+7e/F8i+5a+F8i+6eF8n−2a+F8n−1e/F8n−3a+F8n−2e−1=e∏i=0n−2F8i+8a+F8i+9eF8i+7a+F8i+8e1−F8n−2a+F8n−1e/F8n−3a+F8n−2eF8n−2a+F8n−1e−F8n−3a−F8n−2e/F8n−3a+F8n−2e=e∏i=0n−2F8i+8a+F8i+9eF8i+7a+F8i+8e1−F8n−2a+F8n−1eF8n−1a+F8ne=e∏i=0n−2F8i+8a+F8i+9eF8i+7a+F8i+8eF8na+F8n+1eF8n−1a+F8ne.
Thus, we have(63)w4n=e∏i=0n−1F8i+8a+F8i+9eF8i+7a+F8i+8e.
Hence, the proof is completed.
Similarly, we can prove other relations.Case 3.
Whenα=0andβ=γ=δ=1,p=r=3. In this case, we have the following special case of difference equation:(64)wn+1=wnwn−3wn+wn−3.
The following theorem gives the solution of equation (64).Theorem 10.
Letwnn=−p∞ be a solution of equation (64). Then, the solution for equation (64) is given by(65)wn=w0w−1w−2w−3Fn−2w0w−1w−3+Fn−1w0w−2w−3+Fnw−1w−3w−2+Fn+1w0w−1w−2,where Fn=Fn−1+Fn−3,n≥4,F0=0,andF1=F2=F3=1 and initial values w−3=d,w−2=c,w−1=b,andw0=a are the arbitrary positive real numbers.Proof.
Forn=1,(66)w1=w0w−3w0+w−3=w0w−1w−2w−3w0w−1w−2+w−1w−2w−3=w0w−1w−2w−3F0w0w−1w−3+F1w0w−2w−3+F2w−1w−3w−2+F3w0w−1w−2.Hence, the relation holds true. Now, suppose that relation (64) holds for n=k.
That is,(67)wk=w0w−1w−2w−3Fk−2w0w−1w−3+Fk−1w0w−2w−3+Fkw−1w−3w−2+Fk+1w0w−1w−2.We now want to show that relation (64) holds true for n=k+1:(68)wk+1=wkwk−3wk+wk−3=w0w−1w−2w−3/Fk−2w0w−1w−3+Fk−1w0w−2w−3+Fkw−1w−3w−2+Fk+1w0w−1w−2w0w−1w−2w−3/Fk−5w0w−1w−3+Fk−4w0w−2w−3+Fk−3w−1w−3w−2+Fk−2w0w−1w−2w0w−1w−2w−3/Fk−2w0w−1w−3+Fk−1w0w−2w−3+Fkw−1w−3w−2+Fk+1w0w−1w−2+w0w−1w−2w−3/Fk−5w0w−1w−3+Fk−4w0w−2w−3+Fk−3w−1w−3w−2+Fk−2w0w−1w−2=w0w−1w−2w−3Fk−5w0w−1w−3+Fk−4w0w−2w−3+Fk−3w−1w−3w−2+Fk−2w0w−1w−2+Fk−2w0w−1w−3+Fk−1w0w−2w−3+Fkw−1w−3w−2+Fk+1w0w−1w−2=w0w−1w−2w−3Fk−5+Fk−2w0w−1w−3+Fk−4+Fk−1w0w−2w−3+Fk−3+Fkw−1w−3w−2+Fk−2+Fk+1w0w−1w−2=w0w−1w−2w−3Fk−2w0w−1w−3+Fk−1w0w−2w−3+Fkw−1w−3w−2+Fk+1w0w−1w−2.Case 4.
Whenα=1andβ=γ=δ=1,p=2andr=3. In this case, we have the following special case of difference equation:(69)wn+1=wn−2+wnwn−2wn+wn−3,where w−3,w−2,w−1, and w0 are the arbitrary real numbers.Theorem 11.
Letwnn=−3∞ be a solution of equation (69). Then, for n=0,1,2,…,(70)w3n=d∏i=0n−1F6i+6a+F6i+7dF6i+5a+F6i+6d,w3n+2=c∏i=0n−1F6i+4a+F6i+5dF6i+3a+F6i+4d,w3n+1=b∏i=0n−1F6i+2a+F6i+3dF6i+1a+F6i+2d,where w−3=d,w−2=c,w−1=b, w0=a, and Fmm=0∞=0,1,1,2,3,5,8,13,21,…,.Proof.
Forn=0, the result holds. Now, suppose that n>0 and that our assumption holds for n−1andn−2:(71)w3n−3=d∏i=0n−2F6i+6a+F6i+7dF6i+5a+F6i+6d,w3n−1=c∏i=0n−2F6i+4a+F6i+5dF6i+3a+F6i+4d,w3n−2=b∏i=0n−2F6i+2a+F6i+3dF6i+1a+F6i+2d.
Now, it follows from (69) that(72)w3n=w3n−3−w3n+1w3n−3w3n+1−w3n−2=d∏i=0n−2F6i+6a+F6i+7dF6i+5a+F6i+6d−b∏i=0n−1F6i+2a+F6i+3d/F6i+1a+F6i+2dd∏i=0n−2F6i+6a+F6i+7d/F6i+5a+F6i+6db∏i=0n−1F6i+2a+F6i+3d/F6i+1a+F6i+2d−b∏i=0n−2F6i+2a+F6i+3d/F6i+1a+F6i+2d=d∏i=0n−2F6i+6a+F6i+7dF6i+5a+F6i+6d1−b∏i=0n−1F6i+2a+F6i+3d/F6i+1a+F6i+2db∏i=0n−1F6i+2a+F6i+3d/F6i+1a+F6i+2d−b∏i=0n−2F6i+2a+F6i+3d/F6i+1a+F6i+2d=d∏i=0n−2F6i+6a+F6i+7dF6i+5a+F6i+6d1−b∏i=0n−1F6i+2a+F6i+3d/F6i+1a+F6i+2db∏i=0n−2F6i+2a+F6i+3d/F6i+1a+F6i+2dF6in−4a+F6n−3d/F6n−5a+F6in−4d−1=d∏i=0n−2F6i+6a+F6i+7dF6i+5a+F6i+6d1−F6in−4a+F6n−3d/F6n−5a+F6in−4dF6in−4a+F6n−3d−F6n−5a−F6in−4d/F6n−5a+F6in−4d=d∏i=0n−2F6i+6a+F6i+7dF6i+5a+F6i+6d1−F6n−4a+F6n−3dF6n−3a+F6n−2d=d∏i=0n−2F6i+6a+F6i+7dF6i+5a+F6i+6dF6n−3a−F6n−2d−F6n−4a+F6n−3dF6n−3a+F6n−2d=d∏i=0n−2F6i+6a+F6i+7dF6i+5a+F6i+6dF6n−2a+F6n−1dF6n−3a+F6n−2d,w3n=d∏i=0n−1F6i+6a+F6i+7dF6i+5a+F6i+6d.
Hence, the proof is complete.By using the same way, we can prove other relations.


## 8. Numerical Examples

This section discusses some numerical results of our previous results.Example 1.
Figure1 shows the behavior of the solution bounded of equation (56) when we take(73)w0=0.1,α=0,β=γ=δ=1,α=0,p=r=0.Figure 1
Bounded solution of equation (56).Example 2.
Figure2 shows the behavior of solution of equation (59) when we take(74)w−4=1,w−3=2,w−2=3,w−1=4,w0=5,α=1,β=γ=δ=1,p=3,r=4.Figure 2
Unbounded solution of equation (59).Example 3.
Figure3 shows that the solution of equation (64) is globally asymptotically stable when we take(75)w−3=1,w−2=2,w−1=3,w0=4,α=0,β=γ=δ=1,p=r=3.Figure 3
Global asymptotic stability of equation (64).Example 4.
Figure4 shows the solution of equation (69) has no prime period two solution if we take(76)w−3=0.75,w−2=20,w−1=30,w0=40,α=1,β=γ=δ=1,p=2,r=3.Figure 4
Equation (69) has no positive prime period two solution.


## 9. Concluding Remarks

In the literature, several articles are related to qualitative behavior of rational difference equations. It is a very interesting mathematical problem to study the dynamics of such equations because these are closely related to models in population dynamics and biological sciences. We have investigated the existence and uniqueness of positive equilibrium points, and boundedness and persistence of positive solutions are proved. Moreover, we have shown that they are locally as well as globally asymptotically stable. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. We have taken some special cases as applications of equation (22) in Section 7 and found the closed form of the solutions. Finally, some illustrative examples are provided to support our theoretical discussion.



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