# Global Dynamics of Sixth-Order Fuzzy Difference Equation

Authors: Abdul Khaliq; Muhammad Adnan; Abdul Qadeer Khan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Across many fields, such as engineering, ecology, and social science, fuzzy differences are becoming more widely used; there is a wide variety of applications for difference equations in real-life problems. Our study shows that the fuzzy difference equation of sixth order has a nonnegative solution, an equilibrium point and asymptotic behavior.yi+1=Dyi−1yi−2/E+Fyi−3+Gyi−4+Hyi−5,i=0,1,2,…, where yi is the sequence of fuzzy numbers and the parameter D,E,F,G,H and the initial condition y−5,y−4,y−3,y−2,y−1,y0 are nonnegative fuzzy number. The characterization theorem is used to convert each single fuzzy difference equation into a set of two crisp difference equations in a fuzzy environment. So, a pair of crisp difference equations is formed by converting the difference equation. The stability of the equilibrium point of a fuzzy system has been evaluated. By using variational iteration techniques and inequality skills as well as a theory of comparison for fuzzy difference equations, we investigated the governing equation dynamics, such as its boundedness, existence, and local and global stability analysis. In addition, we provide some numerical solutions for the equation describing the system to verify our results.


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## Body

## 1. Introduction

The fuzzy differential equation was first proposed by Zadeh [1]. The fuzzy difference equation was solved in their analysis by Chang and Zadeh [2]; the original value question has been analyzed thoroughly. Their analysis shows that the nonnegative solution is bounded and proceeds.Fuzzy difference equations are difference equations whose initial values, constants, and solutions are all fuzzy numbers (see preliminaries). By using the fuzzy analog of concepts understood from the theory of ordinary difference equations, we extend these solutions to parametric fuzzy difference equations as a means of verifying the behavior of the fuzzy difference equation. According to our findings, the behavior of the parametric fuzzy difference equation does not mirror that of the coinciding parametric ordinary difference equation [3].Since the data on the differential equation model describing many practical issues are incomplete and fluffy setting theory is an effective tool for modifying incertitude and processing vague subjective information in mathematical models, we ought to look into the behavior of the solution to the flouted equation where the parameter is relevant.We cited that Deeba et al. [4], in 1996, investigated the 1st degree difference equation for the historical background for the equation we are studying in this research:(1)Cn+1=aCn+b,n=0,1,2,3,…,where a nonnegative fuzzy number is Cn and C0,b,a is a fuzzy number that occurs in the genetic population.In addition, to calculate the concentration ofCO2 in the blood, the following linearized form of 2nd degree linear fuzzy equation is considered by Deeba and Korvin [3]:(2)Bn+1=Bn−ghBn−1+p,n=0,1,2,…,where g,h,p,B0,andB1 are fuzzy number and Bn is a sequence of a fuzzy nonnegative number.Moreover, Papaschinopoulos and Stefanidou [5] handle the existence, the persistence, the uniqueness, and the boundedness of nonnegative results of the succeeding fuzzy difference equation:(3)Cn+1=∑k=oiBkCn−kqk,where i∈1,2,3,…, the parameters Bi,k∈0,1,2,…,i, are fuzzy nonnegative numbers, the parameters qk,k∈0,1,2,…,i, are real nonnegative constant, and the initial values Ck,k∈−i,−i+1,…,0, are fuzzy nonnegative numbers.Moreover, in 2006, Papaschinopoulos and Stefanidou [6] consider the periodicity of the solution of the following fuzzy difference equation of max-type:(4)Yi+1=maxD0Yi−k,D1Yi−m,where the primary values Yi, i∈−d,−d+1,−1, d=maxk,m are fuzzy nonnegative numbers, k and m are nonnegative integers, and D0 and D1 are the fuzzy nonnegative numbers.More recent, Zhang et al. [7] study the asymptomatic behavior and the existence of nonnegative results of the following nonlinear fuzzy difference equation:(5)yi+1=Ayi+Yi−1B+yi−1,i=0,1,2,…,where yi is the sequence of fuzzy nonnegative number, AandB are nonnegative fuzzy numbers, and the initial conditions y−1 and y0 are nonnegative fuzzy numbers.In 2004, Zhang et al. [8] deal with asymptomatic behavior, the boundedness, and the existence of the nonnegative solution for the 1st degree Ricatti difference equation:(6)zn+1=C+znD+zn,n=0,1,2,3,…,where zn is a sequence of a fuzzy nonnegative number, the initial value z0, and CandD are fuzzy nonnegative number.In recent, in 2006, Zhang et al. [9] inspected the global behavior, persistence, and boundedness of nonnegative result of the 3rd degree rational fuzzy difference equation:(7)yi+1=B+yi−1yi−1yi−2,i=0,1,2,3,…,where initial values y0,y−1,y−2, and B are fuzzy nonnegative numbers.Moreover, in 2007, Khastan et al. [10] investigated global behavior, the uniqueness, and the existence of solution for next two nonequivalent fuzzy difference equation:(8)zi+1−p=azi,i=0,1,2,3,…,where zi is a sequence of fuzzy nonnegative numbers and z0,p,anda are fuzzy nonnegative numbers. It is easy to see that equations (1) and (8) are in-equivalent.In more recent, Changyou Wang, Ping Liu, Xiaolin Su, Rui Li, and Xiaohom Hu [11] investigate the uniqueness and existence of trivial solution and the asymptotical behavior of the equilibrium point of fifth-order nonlinear fuzzy difference equation:(9)βi+1=Aβi−1βi−2B+Cβi−3+Dβi−4,i=0,1,2,…,where βi is the sequence of fuzzy numbers, the initial values β−4,β−3,β−2,β−1,andβ0, and the parameter A,B,C,andD are fuzzy nonnegative number. r theoretical study, we have some numerical simulation.Motivated by the recent conversation, we want to analyze the optimality of constructive results and the asymptotic behavior for the equilibrium points of the nonlinear fuzzy differences equations:(10)yi+1=Dyi−1yi−2E+Fyi−3+Gyi−4+Hyi−5,i=0,1,2,…,where yi is the sequence of fuzzy numbers, the parameter D,E,F,G,H, and the initial condition y−5, y−4, y−3, y−2, y−1, and y0 are nonnegative fuzzy number. When the parameters and the initial values are positive real numbers, Wang et al. [12] considered the global attractivity of the equilibrium point and the asymptotic behavior of the solutions of the difference equation. To demonstrate our theoretical study, we have some numerical simulations.


## 2. Preliminaries and Definitions

We provide the following definitions and preliminaries result for the reader’s convenience. In this section, we will discuss about the fundamental ideas, notations, and definitions of fuzzy difference equation. Some examples will also be given to explain the concepts of result.Definition 1 (membership function, see [13]).
For a setA, we define a membership function μA such as(11)μAx=1,iffx∈A,0,iffx∉A.
We can say that the functionμA maps the elements in the universal set X to the set 0,1. Membership function μA in crisp set maps whole members in universal set X to set 0,1:(12)μA:X⟶0,1,as shown in Figure 1.Figure 1
Membership function.Definition 2 (fuzzy set, see [1]).
Fuzzy set was for the first proposed by Zadeh in 1965 as an extension of classical notion of a set. The word “fuzzy” means “uncertainty or imprecise.” If the information is not clearly defined, then we introduce fuzziness.
A fuzzy set is a collection of elements which correspond to the definition of ‘A ’in the reliability degree equal to 1 or equal to the value belonging to interval0,1. In fuzzy sets, each element is mapped to 0,1 by the membership function:(13)μA:Y⟶0,1,where 0,1 is valid for 0 to 1 (including 0,1). The fuzzy set is thus “the ambiguous boundary set” compared to the crisp set.Definition 3 (fuzzy number, see [14]).
Consider a setY; we denoted the closure of Y as Y¯. We call a function D:R⟶0,1 is a fuzzy number if it fulfills the characteristics:(i)
D is normal which means that there exist y∈R s.t Dy=1.(ii)
D is fuzzy convex which means(14)Dty1+1−ty2≥minDy1,Dy2,∀D∈0,1,y1,y2∈R.(iii)
D is upper semicontinuous on R.(iv)
D is compactly supported which means supy∈R:Dy>0¯ is compact.
Now, consider the set of all fuzzy numbers is represented byRf, with α∈0,1 and D∈Rf, We expressed fuzzy number D with α-cuts as(15)Dα=y∈R:Dy≥α,D0=y∈R:Dy≥0.
We considerD0 the fuzzy number D with support and represent this by supu. Clearly, with Dα limited to R for closed interval, we assumed that D is a nonnegative fuzzy number if Dset0,∞. It is clear that if D is trivial fuzzy number (real nonnegative number), then D is a fuzzy trivial number with Dα=D,D. For p,q∈Rf,pα=pι,α,pr,α,qα=qι,α,qr,α, and ω∈R, the addition p+q, the scalar product ωp, the product pq, and division p/q in the SIA (Standard Interval Arithmetic) setting are defined as(16)p+qα=pα+qα,ωpα=ωpα,∀α∈0,1,pqα=minpι,αqι,α,pι,αqr,α,pr,αqι,α,pr,αqr,α,maxpι,αqι,α,pι,αqr,α,pr,αqi,α,pr,αqr,α,pqα=minpι,αqι,α,pι,αqr,α,pr,αqι,α,pr,αqr,α,maxpι,αqι,α,pι,αqr,α,pr,αqι,α,pr,αqr,α,0∈qα.Definition 4 (LR-fuzzy number, see [15]).
A fuzzy numberB¯ on R is said to be LR-fuzzy number. If there exist a real numbers p,q≥0 such that(17)μBy=Lm−yp,y≤m,Ry−mq,y≥m,in which Ly and Ry are continuous and nondecreasing function on the real line L1=R1=0. L and R are left and right reference functions, respectively, m is the mean value, and p and q are called left and right spreads on the membership function.
“A LR-fuzzy numberB¯ is represented by 3 real number p,q, and m as whose meaning are defined in Figure 2.Figure 2
LR-fuzzy number.Definition 5 (triangular fuzzy number, see [16]).
Consider fuzzy number denoted by 3 points as follows:(18)B=u1,u2,u3.
It is denoted as a membership function as seen in Figure3:(19)μBy=0,y<u1,y−u1u2−u1,u1≤y≥u2,y−u2u3−u2,u2≤y≥u3,0,y>u3.
Now, if we get crisp interval byα−cut operation, interval Bα shall be obtained. ∀α∈0,1 from(20)u1α−u1u2−u1=α,u3α−u3u2−u3=α.
We obtain(21)u1α=u2−u1α+u1,u3α=−u3−u2α+u3.
Thus,(22)Bα=u1α,u3α=u2−u1α+u1,−u3−u2α+u3.Figure 3
Triangular fuzzy numberB=u1,u2,u3.Definition 6 (value of fuzzy number, see [15]).
B¯ is the fuzzy number having α-cut denoted L−1y,R−1y. S is a decreasing function; then, the value of B¯ is determined by(23)ValB¯=∫01fαL−1y+R−1ydα.Definition 7 (boundedness, see [17]).
A series ofΔY=ΔYn of the fuzzy number seems to be bounded if the set ΔYn:n∈N of the fuzzy number is bounded, where ΔY=|Yn−Yn+1. Let mΔ represented the set of all bounded difference series of fuzzy numbers.Definition 8 (metric on fuzzy number, see [11]).
Letpandq be the fuzzy number with(24)pα=pι,α,pr,α,qα=qι,α,qr,α,∀α∈0,1.
Then, we define the metric on fuzzy numbers as follows:(25)Bp,q=supmaxpι,α−qι,α,pr,α−qr,α,where sup is applied, for all α∈0,1. Moreover, Rf,B is the complete metric space. For future analysis, we express 0^∈Rf as(26)0^y=1,ify=0,0,ify≠0.
Thus,0^α=0,0, 0<α≤1.Lemma 1 (see [11]).
LetIuandIv be some intervals of real numbers and f:Iuk+1×Ivk+1⟶Iuandg:Iuk+1×Ivk+1⟶Iv be continuously differentiable function. Thus, for every set of initial conditions un,vj∈Iu×Iv,n=−k,−k+1,…,0,j=−l,−l+1,…,0, the following system of difference equations,(27)ui+1=fin,in−1,…,in−k,vi,vi−1,…,vi−l,vi+1=gui,ui−1,…,ui−k,vi,vi−1,…,vi−l,i=0,1,2,…,has a unique solution un,vjn=−k,j=−l+∞,+∞.Definition 9 (equilibrium points, see [11]).
A pointu,v∈Iu×Iv is called an equilibrium point of system (27) if u=fu,u,…,u,v,v,…,v,v=gu,u,…,u,v,v,…,v. That is, ui,vi=u,v, for i≥0, is the result of system (27), or equivalent, u,v is allotted point f,g of the vector map.
For system (27), we consider the equilibrium point u,v. Then, we have(i)
The equilibrium pointu¯,v¯ said to be locally stable if each ϵ>0, there exist δ>0, such that, for any initial conditions ui,vj∈Iu×Iv, i=−n,−n+1,…,0,j=−k,−k+1,…,0 with ∑i=−n0ui−u¯<ɛ and ∑j=−k0vj−v¯<ɛ; we have un−u¯<ɛ and vn−v¯<ɛ, for any n<0.(ii)
u¯,v¯, the equilibrium point, is said to be attractor if limn⟶∞un=uandlimn⟶∞vn=v, for any initial conditions ui,vj∈Iu×Iv,i=−n,−n+1,…,0,j=−k,−k+1,…,0.(iii)
Ifu¯,v¯ is attractor and stable, then the equilibrium point is said to be asymptotically stable.(iv)
Ifu¯,v¯ is locally unstable, then equilibrium point is said to be unstable.Note 1.
To calculate the stability criteria of the system for a fuzzy difference equation, the equilibrium points are very important.
To calculate the equilibrium point of a fuzzy system, the methods are follows:(i)
Convert the fuzzy system into according crisp system(ii)
From the crisp system calculate the equilibrium pointDefinition 10 (equilibrium points of a vector map, see [11]).
Letu¯,v¯ be equilibrium point of a vector map F=f,ui,ui−1,…,ui−k,g,vi,\\vi−1,…,vi−l, where f and g are continuously differential function at u¯,v¯. The linearized system of (27) about equilibrium point u¯,v¯ is Ui+1=FUi=Fj⋅Ui, where Fj is the Jacobian matrix of system (27) about u¯,v¯ and Ui=ui,ui−1,…,ui−k,vi,vi−1,…,vi−lT.Definition 11 (trivial solution, see [11]).
The trivial solutionj=0^ of equation (10)(i)
The result ofjn∈Dε,n>0 is stable; if given ε>0, there exist δε>0 with Djn,0^<δ,n=−5,−4,…,0, which implies Djn,0^<ε, for n>0, such that jn∈Dδ,n=−5,−4,…,0(ii)
The result ofjn∈Dε,n>0 is attractive if there is a δ>0 such that Djn,0^<δ,n=−5,−4,…,0, one has limi⟶∞Djn,0^=0.(iii)
If (i) and (ii) hold concurrently, then it is asymptotically stableDefinition 12 (monotone, see [11]).
Leta,b,c,andd be the 4 nonnegative whole number such that a+b=n and c+d=m. Splitting u=u1,u2,…,un into u=ua,ub and v=v1,v2,…,vm into v=vc,vd, where uσ denotes the σ-components of u. We say that the function fu1,u2,…,un,v1,v2,…,vm hold a mixed monotone property in subsets Iun×Ivm of Rn×Rm if fua,ub,vcvd is a nondecreasing monotone in every element of (ua, vc) and is nonincreasing monotone in every element of (ub,vd), for u,v∈Iun×Ivm. In specific, if b=0andd=0, then it is called nondecreasing monotone in Iun×Ivm.Lemma 2 (see [11]).
Assume thatUn+1=GUn,n=0,1,2,…, is a system of differential equations and U is the equilibrium point of this system, i.e., GU=U. Then, we have(i)
If all eigenvalues of Jacobian matrixJG about U lies inside the open unit disk ω<1, then U is locally asymptotically stable(ii)
If all eigenvalues of Jacobian matrixJG about U lies outside the open unit disk ω>1, then U is unstableTheorem 1 (characterization theorem, see [16]).
Let us consider the fuzzy difference equation problem:(28)y¯i+1=f¯xn,i,with initial condition y¯i=0=y¯0, where f:E∗×Z≥0⟶E∗ such that(1)
The parametric form of the function is(29)fyi,iα=f¯yn¯α,y¯nα,n,α,f¯yn¯α,y¯nα,n,α.(2)
The functionsf¯yn¯α,y¯nα,n,α and f¯yn¯α,y¯nα,n,α are taken as continuous functions if, for any ϵ1>0, there exist a δ1>0 such that(30)f¯yn¯α,y¯nα,n−f¯yn1¯α,y¯n1α,n1<ɛ1,∀α∈0,1,with(31)yn¯α,y¯nα,n−yn1¯α,y¯n1α,n1<δ1,∀α∈0,1,and ϵ2>0; there exists a δ2>0 such that(32)f¯yn¯α,y¯nα,n,α−f¯yn2¯α,y¯n2α,n2<ɛ2,∀α∈0,1,with(33)yn¯α,y¯nα,n−yn2¯α,y¯n2α,n2<δ2,∀α∈0,1.
Then, the difference equation (28) reduces the system of 2 difference equations as(34)y¯n+1α=f¯yn¯α,y¯nα,n,α,y¯n+1α=f¯yn¯α,y¯nα,n,α,under initial conditions(35)y¯n=0α=y¯0α,y¯n=0α=y¯0α.Note 2.
By using characterization theorem, the single fuzzy difference equation is changed into the system of 2 crisp difference equations. In this paper, in the environment of fuzzy, we take a single fuzzy difference equation. Hence, the difference equation changed into 2 crisp difference equation.Lemma 3 (see [11]).
Assume thatUi+1=GYi,i=z+, is a differential equation’s system and the equilibrium point of the proceeding system is U. Then, about equilibrium point U, characteristics equation of the proceeding system is Qω=b0ωi+b1ωi−1+⋯+bi−1ω+bi=0, with the real coefficient b0>0. Therefore, total answers of the equation Qω lies inside the open disk ω>1 iff Δh>0,h=1,2,…,n, where Δh is the principal minor of order h of the m×m matrix:(36)Δm=b1b3b5…0b0b2b4…00b1b3…0⋮⋮⋮⋱⋮000…bm.


## 3. Main Results

We needed the following lemmas for investigating the uniqueness and existence of a nonnegative solution of (10).Lemma 4 (see [18]).
Consider thatf be the continuous function from R+×R+⟶R+ and D,E,F,G,andH be the fuzzy numbers. Then,(37)fD,E,F,G,Hα=fDα,Eα,Fα,Gα,Hα,α∈0,1.Lemma 5 (see [18]).
Consider thatp∈Rf express pα=pl,α,pr,α,α∈0,1. Therefore, pl,α and pr,α can be regarded as function on 0,1 which holds(i)
pl,α is nondecreasing and continuous on left(ii)
pr,α is nonincreasing and continuous on right(iii)
pl,α≤pr,α
Alternately, for any functionuα and vα belong to 0,1 which hold (i)-(iii) in for the proceeding, there exist a unique p∈Rf such that pα=uα,vα, for any α∈0,1.Theorem 2.
Consider equation (10), where yi is the sequence of fuzzy numbers, the parameter D,E,F,G,andH and the initial condition y−5,y−4,y−3,y−2,y−1,andy0 are nonnegative fuzzy numbers. There exist a unique nonnegative solution yi of equation (10) under initial conditions y−5,y−4,y−3,y−2,y−1,andy0.Proof.
Suppose that there exist a sequence of fuzzy numbersyi satisfying the equation (10) under initial conditions y−5,y−4,y−3,y−2,y−1,andy0.
Consider theα−cuts,α∈0,1:(38)Dα=Dl,α,Dr,α,Eα=El,α,Er,α,Fα=Fl,α,Fr,α,Gα=Gl,α,Gr,α,Hα=Hl,α,Hr,α,Jα=Jl,α,Jr,α,i=−5,−4,….
Then, from (10) and Lemma 4, it follows that(39)yi+1α=Li+1,α,Ri+1,α=Dyi−1yi−2E+Fyi−3+Gyi−4+Hyi−5α=Dyi−1yi−2αEα+Fyi−3α+Gyi−4α+Hyi−5α=Dl,α,Dr,αLi−1,α,Ri−1,αLi−2,α,Ri−2,αEl,α,Er,α+Fl,α,Fr,αLi−3,α,Ri−3,α+Gl,α,Gr,αLi−4,α,Ri−4,α+Hl,α,Hr,αLi−5,α,Ri−5,α=Dl,αLi−1,αLi−2,α,Dr,αRi−1,αRi−2,αEl,α+Fl,αL−3,α+Gl,αLi−4,α+Hl,αLi−5,α,Er,α+Fr,αRi−3,α+Gr,αRi−4,α+Hr,αRi−5,α=Dl,αLi−1,αLi−2,αEr,α+Fr,αRi−3,α+Gr,αRi−4,α+Hr,αRi−5,α,Dr,αRi−1,αRi−2,αEl,α+Fl,αLi−3,α+Gl,αLi−4,α+Hl,αLi−5,α,from the above equation, for i=−5,−4,…, and we have(40)Li+1,α=Dl,αLi−1,αLi−2,αEr,α+Fr,αRi−3,α+Gr,αRi−4,α+Hr,αRi−5,α,Ri+1,α=Dr,αRi−1,αRi−2,αEl,α+Fl,αLi−3,α+Gl,αLi−4,α+Hl,αLi−5,α.
Then, from Lemma1, it is evident that, for any Ln,α,Rn,α,n=−5,−4,−3,−2,−1,0 of the proceeding system (40) under primary conditions, Li,α,Ri,α,i=−5,−4,−3,−2,−1,0,α∈0,1, has a unique solution Li,α,Ri,α.
Alternately, we want to show thatLi+1,α,Ri+1,α,α∈0,1, where Li,α,Ri,α is the solution of system (40) with initial conditions Ln,α,Rn,α,n=−5,−4,−3,−2,−1,0, determines the solution yi of equation (10) with initial conditions y−5,y−4,y−3,y−2,y−1,andy0 such that(41)yiα=Li+1,α,Ri+1,α,α∈0,1,i=−5,−4,….
By Lemma5 and as D,E,F,G,andHandyi,i=−5,−4,−3,−2,−1,0, are fuzzy nonnegative numbers, for any α1,α2∈0,1 and α1>α2, we obtain(42)0<Dl,α1≤Dl,α2≤Dr,α2≤Dr,α1,0<El,α1≤El,α2≤Er,α2≤Er,α1,0<Fl,α1≤Fl,α2≤Fr,α2≤Fr,α1,0<Gl,α1≤Gl,α2≤Gr,α2≤Gr,α1,0<Hl,α1≤Hl,α2≤Hr,α2≤Hr,α1,0<Li,α1≤Li,α2≤Ri,α2≤Ri,α1,i=−5,−4,−3,−2,−1,0.
We solved by mathematical induction that(43)0<Li,α1≤Li,α2≤Ri,α2≤Ri,α1,i=−5,−4,−3,−2,−1,0.
From (42), we find that (43) holds, for i=−5,−4,−3,−2,−1,0.
Consider equation (43) is verifiable, for i≤j,j∈1,2,3,…, then, by using (41)–(43), it pursues that, for i=j+1,(44)Lj+1,α1=Dl,α1Lj−1,α1Lj−2,α1Er,α1+Fr,α1Rj−3,α1+Gr,α1Rj−4,α1+Hr,α1Rj−5,α1≤Dl,α2Lj−1,α2Lj−2,α2Er,α2+Fr,α2Rj−3,α2+Gr,α2Rj−4,α2+Hr,α2Rj−5,α2=Lj+1,α2≤Dr,α2Rj−1,α2Rj−2,α2El,α2+Fl,α2Lj−3,α2+Gl,α2Lj−4,α2+Hl,α2Lj−5,α2=Rj+1,α2≤Dr,α1Rj−1,α1Rj−2,α1El,α1+Fl,α1Lj−3,α1+Gl,α1Lj−4,α1+Hl,α1Rj−5,α1=Rj+1,α1.
Therefore, (42) holds.
Moreover, from (40), we obtain(45)Li+1,α=Dl,αLi−1,αLi−2,αEr,α+Fr,αRi−3,α+Gr,αRi−4,α+Hr,αRi−5,α,Ri+1,α=Dr,αRi−1,αRi−2,αEl,α+Fl,αLi−3,α+Gl,αLi−4,α+Gl,αLi−5,α,α∈0,1.
From Lemma5 and since D,E,F,G,andHandyi,i=−5,−4,−3,−2,−1,0, are the fuzzy nonnegative numbers, we obtain(46)Dl,α,Dr,α,El,α,Er,α,Fl,α,Fr,α,Gl,α,Gr,α,Hl,α,Hr,α,L−1,α,R−1,α,L−2,α,R−2,α,L−3,α,R−3,α,L−4,α,R−4,α,L5,α,R−5,αare left continuous. As a result of (45), we get L1,αandR1,α both are left continuous. Then, we want to show that Li,αandRi,α,i=1,2,3,…, also left continuous by mathematical induction.
Now, we can show that thesupport of yi,supyi=∪α∈0,1Li,α,Ri,α, is compact. It is abundant to show that ∪α∈0,1Li,α,Ri,α is bounded.
Consideri=1; since D,E,F,G,andHandyi,i=−5,−4,−3,−2,−1,0, are the fuzzy nonnegative numbers, there exist constants Pi,Qi>0,i=1,2,3,4,5 such that, for all α∈0,1,(47)Dl,α,Dr,α⊂P1,Q1,El,α,Er,α⊂P2,Q2,Fl,α,Fr,α⊂P3,Q3,Gl,α,Gr,α⊂P4,Q4,Fl,α,Fr,α⊂P5,Q5,Li,α,Ri,α⊂Pi,Qi,i=−5,−4,−3,−2,−1,0.
Therefore, from (45) and (47), we can prove that(48)L1,α,R1,α⊂P1P−1P−2Q5+Q4Q−3+Q3Q−4+Q2Q−5,Q1Q−1Q−2P5+P4P−3+P3P−4+P2P−5,α∈0,1,from which it is obvious that(49)∪α∈0,1L1,α,R1,α⊂P1P−1P−2Q5+Q4Q−3+Q3Q−4+Q2Q−5,Q1Q−1Q−2P5+P4P−3+P3P−4+P2P−5,α∈0,1.
Relation (49) shows that ∪α∈0,1L1,α,R1,α¯ is compact and ∪α∈0,1L1,α,R1,α¯⊂0,∞. Then, from mathematical induction, we want to show that ∪α∈0,1L1,α,R1,α¯ is compact and(50)∪α∈0,1L1,α,R1,α¯⊂0,∞,i=1,2,3,….
So, by Lemma5, relations (43) and (50) and Li,αandRi,α are left continuous, we get Li,α,Ri,α; calculate the sequence of fuzzy nonnegative numbers yi as equation (10) holds.
Now, we show thatyi is the solution of equation (10) under initial conditions y−5,y−4,y−3,y−2,y−1,andy0. Then, for all α∈0,1,(51)yi+1α=Li+1,α,Ri+1,α=Dl,αLi−1,αLi−2,αEr,α+Fr,αRi−3,α+Gr,αRi−4,α+Hr,ℵRi−5,α,Dr,αRi−1,αRi−2,αEl,α+Fl,αLi−3,α+Gl,αLi−4,α+Hl,αLi−5,α,yi+1α=Li+1,α,Ri+1,α=Dji−1ji−2E+Fji−3+Gji−4+Hji−5α,we get that yi is the solution of equation (10) under initial conditions y−5,y−4,y−3,y−2,y−1,andy0.
Consider that there exists one more solutionyi∗ of equation (10) initial values y−5,y−4,y−3,y−2,y−1,andy0; then, we can easily show by explaining as above that(52)yi∗α=Li+1,α,Ri+1,α,α∈0,1,i=1,2,3,….
Then, from (41) and (52), we get that yiα=yi∗α,α∈0,1,i=−5,−4,−3,…, from which it satisfies yi=yi,α∈0,1,i=−5,−4,−3,…. Hence, proved.
With the use of the following theorem, we are investigating the behavior of asymptotic of equation (10) at equilibrium points.
Ifyi is the unique nonnegative solution of equation (10) under the initial conditions y−5,y−4,y−3,y−2,y−1,andy0 such that(53)yiα=Li,α,Ri,α,α∈0,1,i=1,2,3,…,then we see that Li,α,Ri,α is the member of the system of ordinary difference equation family:(54)Li+1,α=Dl,αLi−1,αLi−2,αEr,α+Fr,αRi−3,α+Gr,αRi−4,α+Hr,αRi−5,α,Ri+1,α=Ar,αRi−1,αRi−2,αBl,α+Cl,αLi−3,α+Dl,αLi−4,α+El,αLi−5,α,α∈0,1.
We assume the succeeding system of the ordinary parametric difference equations, in order to investigate the asymptotically behavior of equation (10). Then, from (54),(55)ui+1=dui−1ui−2e+fvi−3+gvi−4+hvi−5,vi+1=pvi−1vi−2q+rui−3+sui−4+tui−5,i=0,1,2,…,where the parameter d,e,f,g,h,p,q,r,s,andt are the real constant of the nonnegative number and u−5,u−4,u−3,u−2,u−1,u0,v−5,v−4,v−3,v−2,v−1,andv0 are initial conditions of nonnegative real constant.
By Lemma1, we came to know that (55) is the system of the ordinary parameter difference equation and has a unique solution ui,vi under any initial conditions.
Moreover, we can easily calculate the equilibrium pointsun,vn of any initial conditions of system (55). There are three equilibrium points of system (55):(56)Y1¯=u1¯,v1¯=0,0,Y2¯=u2¯,v2¯=ed,0,Y3¯=u3¯,v3¯=0,qp.
Ifdp>f+g+hr+s+t, then the fourth nonnegative equilibrium points of system (55) are(57)Y4¯=u4¯,v4¯=ep+qf+g+hdp−f+g+hr+s+t,dq+br+s+tdp−f+g+hr+s+t.Theorem 3.
The system of equation (55) is locally asymptotically stable at equilibrium point Y1¯.Proof.
LetT:R5+⟶R+andU:R5+⟶R+ be the multivariable function defined as(58)Tui−1,ui−2,vi−3,vi−4,vi−5=dui−1ui−2e+fvi−3+gvi−4+hvi−5,Svi−1,vi−2,ui−3,ui−4,ui−5=pvi−1vi−2q+rui−3+sui−4+tui−5.
Moreover, about the equilibrium pointY1¯, we can easily determine the linear equation of system (55) such as(59)φ1=D1φi,where(60)φi=uiui−1ui−2ui−3ui−4ui−5vivi−1vi−2vi−3vi−4vi−5,D1=000000000000100000000000010000000000001000000000000100000000000010000000000001000000000000100000000000010000000000001000000000000100000000000010.
The characteristics polynomial with (59) is(61)ω12=0.
Since we getω<1, from Lemma 2, we see that the equilibrium point Y1¯ of system (55) is locally asymptotically stable, and hence, proved.Theorem 4.
The system of equation (55) is unstable at the equilibrium point Y2¯.Proof.
From (59), about the equilibrium point Y2¯, we can easily calculate the linear equation of system (55) as(62)φi+1=D2φi,where(63)φi=uiui−1ui−2ui−3ui−4ui−5vivi−1vi−2vi−3vi−4vi−5,D2=000000000000100000000000010000000000001000000000000100000000000010000000000−rp−sp−tp011000000000100000000000010000000000001000000000000100000000000010.
The characteristics polynomial with (62) is(64)ω9ω3−ω−1=0.
Since, we getω>1 so that ω9ω3−ω−1=0; it clears that one of the root of characteristic polynomial (59) lies outside the unit disk; therefore, by Lemma 2, we computed the equilibrium point Y2¯ of system of equation (55) is unstable, and hence, proved.Theorem 5.
The system of equation (55) is unstable at the equilibrium point Y3¯.Proof.
From (59), about the equilibrium point Y3¯, we can easily calculate the linear equation of system (55) as(65)φi+1=D3φi,where(66)φi=uiui−1ui−2ui−3ui−4ui−5vivi−1vi−2vi−3vi−4vi−5,D3=011000000−fd−gd−hd100000000000010000000000001000000000000100000000000010000000000000000000000000100000000000010000000000001000000000000100000000000010.
The characteristics polynomial with (65) is(67)ω9ω3−ω−1=0.
Since we haveω>1 so that ω9ω3−ω−1=0, it is the same as equation (64), such that roots of the characteristics polynomial (67) lie outside the unit disk; therefore, by Lemma 2, we get that the equilibrium point Y3¯ of equation (55) is not stable, and hence, proved.Theorem 6.
The system of equation (55) is not stable about the equilibrium point Y4¯.Proof.
From (59), about the equilibrium point Y4¯, we can easily calculate the linear equation of system (55) as(68)φi+1=D4φi,where(69)φi=uiui−1ui−2ui−3ui−4ui−5vivi−1vi−2vi−3vi−4vi−5,D4=011000000−fd−gd−hd100000000000010000000000001000000000000100000000000010000000000−rp−sp−tp011000000000100000000000010000000000001000000000000100000000000010.
The characteristics polynomial with (68) is(70)ω12−2ω10−2ω9+ω8+2ω7+ω6−frdpω4−gr+fsdpω3−hr+gs+ftdpω2−hs+gtdpω−htdp=0.
From (70), we obtained(71)Δ12=0−2−20−gr+fsdp−hs+gtdp0000001−211−frdphr+gs+htdp−htdp0000000−2−20−gr+fsdp−hs+gtdp0000000−211−frdphr+gs+htdp−htdp00000000−20−gr+fsdp−hs+gtdp0000000011−frdphr+gs+htdp−htdp0000000000−gr+fsdp−hs+gtdp0000000001−frdphr+gs+htdp−htdp0000000000−gr+fsdp−hs+gtdp0000000000−frdphr+gs+htdp−htdp0000000000−gr+fsdp−hs+gtdp00000000000hr+gs+htdp−htdp.
We notice that allΔi≯0,i=1,2,3,…,12; by Lemmas 2 and 3, we computed that Y¯4 is not stable, and hence, proved.Theorem 7.
LetIuandIv be some period of real number, and consider that f=Iuk+1×Ivk+1⟶Iu and g=Iuk+1×Ivk+1⟶Iv are satisfying mixed monotone property and hence continuously differentiable. If there exist(72)m0≤minu−k,….,u0,v−l,…,v0≤maxu−k,…,u0,v−l,…,v0≤M0,n0≤minu−k,…,u0,v−l,…,v0≤maxu−k,…,u0,v−l,…,v0≤N0,such that(73)m0≤fm0a,M0b,n0s,N0t≤fM0a,m0b,N0s,n0t≤M0,n0≤gm0a1,M0b1,n0s1,N0t1≤gM0a1,m0b1,N0s1,n0t1≤N0.
Then, there existm,M∈m0,M02 and n,N∈n0,N02 holding(74)m=fm0a,M0b,n0s,N0t,M=fM0a,m0b,N0s,n0t,n=gm0a1,M0b1,n0s1,N0t1,N=gM0a1,m0b1,N0s1,n0t1.
Moreover, ifm=M,n=N, then equation (27) has a unique equilibrium point u¯,v¯∈m0,M0×n0,N0 and every solution of (27) converges to u¯,v¯.Proof.
Usingm0,M0,n0, and N0 as two couples of initial iteration, we construct four sequences mi,Mi,ni, and Nii=1,2,3,… from the following equations:(75)mi=fmi−1a,Mi−1b,ni−1s,Ni−1t,Mi=fMi−1a,mi−1b,Ni−1s,ni−1t,ni=gmi−1a1,Mi−1b1,ni−1s1,Ni−1t1,Ni=gMi−1a1,mi−1b1,Ni−1s1,ni−1t1.
It is obvious from the mixed monotone property off and g that the sequences mk,Mk,nk, and Nk possess the following monotone property:(76)m0≤m1≤⋯≤mi≤Mi≤⋯≤M1≤M0,n0≤n1≤⋯≤ni≤Ni≤⋯≤N1≤N0,where i=0,1,2,…, and(77)mi≤uq≤Mi,ni≤vp≤Ni,forq≥i+1k+1,p≥i+1l+1,i=0,1,2,….
Set(78)m=limi⟶∞mi,M=limi⟶∞Mi,n=limi⟶∞ni,N=limi⟶∞Ni.
Then,(79)m≤limi⟶∞infui≤limi⟶∞supui≤M,n≤limi⟶∞infvi≤limi⟶∞supvi≤N,by continuity of f and g, one has(80)m=fm0a,M0b,n0s,N0t,M=fM0a,m0b,N0s,n0t,n=gm0a1,M0b1,n0s1,N0t1,N=gM0a1,m0b1,N0s1,n0t1.
Moreover, ifm=Mandn=N, then m=M=limi⟶∞ui=u¯andn=N=limi⟶∞vi=v¯, and hence, proved.Theorem 8.
Ifd=p,e=q,f=r,g=s,andh=t, then 0,0 is the equilibrium point of system (55) is the global attractor for all conditions:(81)u−1,v−1∈0,b2a×0,b2a,j=−5,−4,−3,−2,−1,0.Proof.
Sinced=p,e=q,f=r,g=s,andh=t, hence, system (55) converts as(82)uj+1=duj−1uj−2e+fvj−3+gvj−4+hvj−5,vj+1=dvj−1vj−2e+fuj−3+guj−4+huj−5,j=0,1,2,….
Letf,g∈0,e/2d12×0,e/2d12⟶0,∞×0,∞ be a function expressed as(83)fuj,uj−i,uj−2,uj−3,uj−4,uj−5,vj,vj−i,vj−2,vj−3,vj−4,vj−5=duj−1uj−2e+fvj−3+gvj−4+hvj−5,guj,uj−i,uj−2,uj−3,uj−4,uj−5,vj,vj−i,vj−2,vj−3,vj−4,vj−5=dvj−1vj−2e+fuj−3+guj−4+huj−5.
Set(84)f=duve+fγ+gα+hβ,g=du∗v∗e+fγ∗+gα∗+hβ∗,and we can obtain that(85)fu=dve+fα+gβ+hγ>0,fv=due+fα+gβ+hγ>0,fα=−dfuve+fα+gβ+hγ2<0,fβ=aguve+fα+gβ+hγ2<0,fγ=dhuve+fα+gβ+hγ2<0,gu∗=dv∗e+fα∗+gβ∗+hγ∗>0,gv∗=du∗e+fα∗+gβ∗+hγ∗>0,gα∗=dfu∗v∗e+fα∗+gβ∗+hγ∗2<0,gβ∗=dgu∗v∗e+fα∗+gβ∗+hγ∗2<0,gγ∗=dhu∗v∗e+fα∗+gβ∗+hγ∗2<0,which indicate that f and g hold a mixed monotone characteristic:(86)M0=N0=maxu−5,u−4,u−3,u−2,u−1,u0,v−5,v−4,v−3,v−2,v−1,v0,dM0−ef+g+h<m0=n0<0,
We have(87)m0≤dm02e+fN0+gN0+hN0≤dM02e+fn0+gn0+hn0≤M0,n0≤dn02e+fM0+gM0+hM0≤dN02e+fm0+gm0+hm0≤N0.
It is obvious thatmj=njandMj=Nj,j=0,1,2,…; then, by the proceeding system (55) and Theorem 2.6, ∃m,M∈m0,M0, n=m, and N=M satisfy(88)m=dm2e+fN+gN+hN,n=dn2e+fM+gM+hM,M=dM2e+fn+gn+hn,N=dN2e+fm+gm+hm.
Thus,e−dm+Mm−M=0. In vision of 2dM0<e, we get e−dm+M>0. So, M=m,N=n. From Theorem 7, we get that 0,0 is the equilibrium point of system (55) which is a global attractor, and hence, proved.
Now, we establish stability of the fuzzy difference equation (10) in terms of positive results of standard difference equation (55). For this justification, we initiate the succeeding view of equation (10) for stability. It manifests that equation (10) has a trivial solution 0^.Theorem 9.
If the parameterD,E,F,G,andH are positive fuzzy numbers, i.e., nonnegative real numbers and the primary conditions are nonnegative fuzzy numbers with yjα⊂0,E/2D,j=−5,−4,−3,−2,−1,0,α∈0,1, then the trivial solution y=0^ of equation (10) is asymptotically stable with regard to E as j⟶∞.Proof.
This is proved by the result of Theorems3 and 8.


## 4. Numerical Problem

In this section, the numerical example is performed for confirmation of the result discussed in the previous section and for support of the theoretical discussion. These examples show the asymptotically behavior of the results of equation (10).Example 1.
Consider the following fuzzy difference equation:(89)yi+1=Dyi−1yi−2E+Fyi−3+Gyi−4+Hyi−5,i=0,1,2,…,where D,E,F,G,andH are positive trivial fuzzy numbers. By Theorem 9, we take Dα=D,D=0.2,Eα=E,E=17,Fα=F,F=2,Gα=G,G=2,andHα=H,H=3,α∈0,1; in addition, from Theorem 9, the initial conditions y−5,y−4,y−3,y−2,y−1,andy0 with yiα⊂0,E/2D, i=−5,−4,−3,−2,−1,0 and α∈0,1, are represented such that(90)y0y=y−42,4≤y≤6,−y+115,6≤y≤11,y−1y=y−35,3≤y≤8,−y+157,8≤y≤15,y−2y=y−25,2≤y≤7,−y+125,7≤y≤12,y−3y=y−27,2≤y≤9,−y+167,9≤y≤16,y−4y=y−17,1≤y≤8,−y+124,8≤y≤12,y−5y=y−73,7≤y≤10,−y+177,10≤y≤17.
From (40), the triangular fuzzy number is obtained:(91)y0α=4+2α,11−5α,y−1α=3+5α,15−7α,y−2α=2+5α,12−5α,y−3α=2+7α,16−7α,y−4α=1+7α,12−4α,y−5α=7+3α,17−7α.
From (41), the parameter D,E,F,G,andH and initial conditions y−5,y−4,y−3,y−2,y−1,andy0 satisfy the following system of nonlinear difference equation with parameter α:(92)Li+1,α=0.2Li−1,αLi−2,α17+2Ri−3,α+2Ri−4,α+3Ri−5,α,Ri+1,α=0.2Ri−1,αRi−2,α17+2Li−3,α+2Li−4,α+3Li−5,α,α∈0,1.
It is easy to prove thatyiα⊂0,E/2D,i=−5,−4,−3,−2,−1,0, for α∈0,1, namely, the condition of Theorem 9 holds. So, from Theorem 9, we have that the trivial solution y=0^ of equation (10) is asymptotically stable with respect to D as i⟶∞; Figures 4–10 shows the dynamics of system (91), where L and R is left and right reference functions, respectively.Figure 4
The dynamics of equations (91).Figure 5
The dynamics of equations (91), when α=0.Figure 6
The dynamics of equations (91), when α=0.2.Figure 7
The dynamics of equations (91), when α=0.4.Figure 8
The dynamics of equations (91), when α=0.6.Figure 9
The dynamics of equations (91), when α=0.8.Figure 10
The dynamics of equations (91), when α=1.


## 5. Conclusion

In this work, we demonstrate how to use the variational iteration technique to solve a system of fuzzy nonlinear difference equations. In physics, this is a powerful technique to solve nonlinear differential equations with fuzzy outcomes. According to the mathematical analysis, the solution is very satisfying. Variational iteration technique offers a powerful tool to drive nonlinear formations. Calculations are achieved by utilizing the package of MATLAB 2014(a). The dynamics action of high-order fuzzy nonlinear difference equation is examined in this work. Initially, we prove the existence and the uniqueness of fuzzy solutions through nonnegative fuzzy calculations. Then, using the linearization technique, we compute the nonzero equilibrium points of corresponding equation (55) that is not stable. At last, for equation (10), we compute the nonnegative solution 0^ is stable when D,E,F,G,andH are nonnegative fuzzy numbers. The computational conclusions are illustrated in a few exemplifying examples. Specifically, the conditions which we derive in the association are so easy, which enabled flexibility in investigating, experimenting, and implementing the fuzzy nonlinear difference equation.



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*Source: 9769093-2021-10-05.xml*