# Local and Global Existence and Uniqueness of Solution for Class of Fuzzy Fractional Functional Evolution Equation

Authors: Kinda Abuasbeh; Ramsha Shafqat; Azmat Ullah Khan Niazi; Muath Awadalla
Journal: Journal of Function Spaces
 (2022)

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## Abstract

For fuzzy fractional functional evolution equations, the concept of global and local existence and uniqueness will be presented in this work. We employ the contraction principle and successive approximations for global and local existence and uniqueness, respectively, as given0cDqHxI=fI,xI+∫0IgI,s,xsds,I≥I0,I∈0,T,xI=ψI−I0=ψ0∈Cσ,I0≥I≥I0−σ,x′I=ψ′I=ψ1, where Cσ denotes the set of fuzzy continuous mapping defined on I0−σ,T and σ>1. We also use this method to solve fuzzy fractional functional evolution equations with fuzzy population models and distributed delays using fuzzy fractional functional evolution equations. To explain these results, some theorems are given. Finally, certain fuzzy fractional functional evolution equations are illustrated.


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## Body

## 1. Introduction

In reality, to show fractional-order demeanor which can change with time and space in case of a large number of physical processes, fractional calculus authorizes the operations of differentiation and integration of fractional-order. The fractional-order can be applied to both imaginary and real numbers. Because of its wide range of applications in disciplines like mechanics, electrical engineering, signal processing, thermal systems, robotics and control, signal processing, and many others, the theory of fuzzy sets continues to attract academics’ attention [1–3]. Therefore, it has been noticed that it is the center of increasing interest of researchers during the past few years.In real-world systems, delays can be recognized everywhere, and there has been widespread interest in the study of delay differential equations for many years. Fractional differential equations are becoming more important in system models in biology, chemistry, physics, and other sciences. There is a large form of evidence about functional differential equations and their methods. On the other hand, we can seldom be certain that dynamic in a system is perfectly modeled using deterministic ordinary differential equations because the knowledge of dynamical systems is either unclear or incomplete. If the model’s underlying structure is based on subjective decisions, one way to incorporate these is to use the fuzziness aspect, which contributes to the consideration of fuzzy fractional functional evolution equations. In the context of fuzzy-valued analysis and set-valued differential equations, fuzzy differential equations were first studied as a separate subject. The analysis of fuzzy differential equations can be expressed in a variety of ways. In biology, chemistry, physics, and other sciences, fractional differential equations are becoming more significant in system models. The reader is referred to the monographs [4, 5], and the references therein, as there is a large quantity of literature dealing with delay differential equations and their applications. As a new branch of fuzzy mathematics, the study of fuzzy delay differential equations is growing in popularity. Over the last few years, both theory and applications have been widely discussed. The study of fuzzy delay fractional functional evolution equations has numerous interpretations in the literature.Puri and Ralescu definedH-differentiability for fuzzy functions using the Hukuhara derivative of multivalued functions. In the context of fuzzy differential equations in a time-dependent manner, Seikkala and Kaleva proposed and investigated this definition. The fuzzy initial value issue has a unique local solution if f is continuous and satisfies the Lipschitz condition with respect to u, as Kaleva established in [6].
(1)U′I=fI,u,u0=u0onEm,D.He proved that the Peano theorem is invalid in [6], since metric space Em,D can be locally compact. Peano’s existence theorem for FDEs on Em,D was proven by Nieto [7] if f is bounded and continuous. Buckley and Feuring [8] gave reasonable general formulation to the fuzzy first-order initial value problem. Citations [9, 10] present the existence of theorems for solutions to the fuzzy initial value problem under a wide range of assumptions. This H-differentiability-based approach has the disadvantage of having an increasing length of support for each solution of FDE. As a result, this method is inappropriate for modeling and fails to describe any of the complex properties of ordinary differential equations, that is, stability, periodicity, bifurcation, and other phenomena [11]. This problem is solved using FDE, which can be read as a family of differential inclusions [12]. We do not have a derivative for fuzzy-number-valued equations, which is a key drawback of differential inclusions.The above-mentioned method for fuzzy-number-valued functions with highly generalized differentiability was recently solved by Bede and Gal [13]. The derivative is maintained in this case, and the support length of the FDE solution may decrease, but the uniqueness is lost. On fuzzy differential equations, there is a lot of literature. In comparison, FFDEs and their implementations were only briefly mentioned in a few articles. Park and his colleagues’ [14] approximate solutions of fuzzy functional integral equations were studied. Park et al. [15] examined the presence of almost periodic and asymptotically almost periodic solutions for FFDEs. For nonlinear fuzzy neutral functional differential equations, Balasubramaniam and Muralisankar [16] investigate local uniqueness and existence theorem. Guo et al. [17] developed existence results for fuzzy impulsive functional differential equations using Hüllermeier’s level-wise method [13], which they then applied to fuzzy population models. Abbas et al. [18, 19] worked on a partial differential equation. Niazi et al. [20], Iqbal et al. [21], Shafqat et al. [22], Abuasbeh et al. [23], and Alnahdi et al.’s [24] existence and uniqueness of the FFEE were investigated.Khastan et al. proved the existence of two fuzzy solutions for fuzzy delay differential equations using the concept of generalized differentiability. Hoa et al. established the global existence and uniqueness results for fuzzy delay differential equations using the concept of generalized differentiability. Moreover, the authors have extended and generalized some comparison theorems and stability theorems for fuzzy delay differential equations with the definition of a new Lyapunov-like function. Besides that, some very important extensions of the fuzzy delay dierential equations were introduced. The author considered the FDE with the initial value(2)X′I=fI,xI,xI0=x0∈Ed,where f:0,∞×Ed⟶Ed and the symbol ′ denotes the first type of Hukuhara derivative, that is, the classical Hukuhara derivative. O. Kaleva also discussed the properties of differentiable fuzzy mappings and showed that if f is continuous and fI,x satisfies the Lipschitz condition concerning to x, then, there exists a unique local solution for the fuzzy initial value problem. V. Lupulescu proved several theorems stating the existence, uniqueness, and boundedness of solutions to fuzzy differential equations with the concept of the inner product on the fuzzy space. Guo et al. [25] and Shu et al. [26] studied the fractional differential equation.In [27], V. Lupulescu considered the fuzzy functional differential equation
(3)x′I=fI,xI,I≥I0,xI=ϕI−t0∈Ed,I0≥I≥I0−σ,where f:0,∞×Cσ⟶Ed and the symbol ′ denotes the first type Hukuhara derivative called classical Hukuhara derivative. The author studied the local and global existence and uniqueness results by using the method of successive approximations and contraction principle.We used Caputo derivative to prove the uniqueness and existence of several uniqueness and existence theorems for fuzzy fractional functional differential equations (FFFDEs) under certain conditions, inspired by the above research:(4)0cDqHxI=fI,xI+∫0IgI,s,xsds,I≥I0,I∈0,T,xI=ψI−I0=ψ0∈Cσ,I0≥I≥I0−σx′I=ψ′I=ψ1,where Cσ denotes the set of fuzzy continuous mapping defined on I0−σ,T and σ>1. xI denotes the fuzzy mapping xIs,I0−σ≤s≤T; that is, xI∈Cσ. The goal of this study is to use the method of contraction principle and consecutive approximations to show local and global uniqueness and existence theorems for the fuzzy fractional functional differential Equation (4) under certain conditions.The following is a description of the paper’s structure. As a warm-up, we will make some basic observations on fuzzy sets and the differentiability and integrability features of fuzzy functions. In Section3, we show the local uniqueness and existence theorem for the solution to the initial value problem for FFFDEs using the successive approximation method. Section 4 proves the global uniqueness and existence theorem for the initial value solution. A problem involving fuzzy fractional functional differential equations is solved using contraction theory. Finally, we apply what we have learned about FDEs to two different forms of fuzzy differential equations: FFFDEs with fuzzy population and distributed delays models.


## 2. Preliminaries

The set of all nonempty, compact convex subsets ofRm is denoted by KcRm. The Hausdorff distance between sets A,B∈KcRm is defined as
(5)dHA,B=maxsupa∈Ainfb∈Ba−b,supb∈Binfa∈Aa−b.DenoteEm=x:Rm⟶0,1;xsatisfiesa−dbelow.In the above equation,(a)
x is normal due to the exists of Rm, xu0=1(b)
x is fuzzy convex, for Rm, 0≤λ≤1, xλu+1−λv≥minxu,xv(c)
x is upper semicontinuous function on Rm(d)
x0=cls∈Rm/xI>0 is compact1<β≤2, represent xβ=u∈Rm/xI≥β. Then, from (a) to (b), it shows, β-level set xβI∈KcRm∀1≤β≤2. We define 0~∈Em as 0~u=1 if u=0 and 0~u=0 if u≠0 for later purposes.Using Zadeh’s extension theorem, we can have scalar multiplication and addition in fuzzy number spaceEm as shown in
(6)x⊕yβ=xβ⊕yβ,kxβ=kxβ,where x,y∈Em,k∈Rm and 1≤β≤2.DefineD:Em×Em⟶R+ by notation
(7)Dx,y=sup1≤β≤2dHxβ,yβ.where D is Hausdorff a metric for nonempty compact sets in Rm and Em,D is a complete metric space [28].It is very simple to notice thatD is a metric in Em. By using the properties of Dx,y:
(a)
Em,D is a complete metric space(b)
Dx⊕z,y⊕z=Dx,y and Dx,y=Dx,y∀x,y,z∈Em(c)
Dλx,λy=λDx,y∀x,y∈Em and λ∈Rm(d)
Dx,y≤Dx,z+Dz,yIf we denotexG=Dx,0~,x∈Em, then, xG has properties of an usual norm onEm [29]:
(i)
xG=0 if x=0~(ii)
λxG=λxG∀x,y∈Em(iii)
x+yG≤xG+yG∀x,y∈Em(iv)
Dβx,γx≤β−γDx,0~,∀β,γ≥1orβ,γ≤1,x∈EmOnEm, we can describe subtraction !, also known as H-difference [30], as follows: s⊖v has significance if ω∈Em, x=y+z exists.Supposea,b∈Rm,f∈CI,Em, if we represent f=Hf,0~, then, f has properties of an usual norm on Em [29],
(i)
f=0 if f=0~(ii)
λf=λf∀f∈ℂI,Em,λ∈Rm(iii)
f⊕h≤f⊕h∀f,h∈I,Em(iv)
Hβf,γf≤β−γHf,0~,∀β,γ≥0 or β,γ≤0, f∈ℂI,EmDefinition 1.
The mappingF:I⟶Em is Hukuhara differentiable at I∈I if exists G′I∈Em similar to the limits:
(8)limh⟶0+GI0+h⊖GI0handlimh⟶0+GI0⊖GI0−hh,and is equal and exists to G′I.We can remember some properties of integrability and measurability for fuzzy set-valued mappings [28].Definition 2.
IfG:I⟶Em is fuzzy function, that is
(9)GIβ=G1βI,G2βI,β∈1,2,
and there existsG′I0 for some I0∈I, and now
(10)G′I0β=G1β′I0,G2β′I0,β∈1,2.Definition 3.
The mappingG:I∈Em is strongly measurable if for all β∈1,2, then, the set-valued function Gβ:I⟶MjRmdefine by GβI=GIβ is Lebesgue measurable.
The mappingG:I∈Em is known as integrably bounded if there exists an integrable function j like
(11)x≤JI∀x∈G0I.Definition 4.
SupposeG:I∈Em. Then, the equation defines integral of G over I, which is expressed by ∫IGIdt, ∫IGIdtβ=∫IGβIdI=∫IGIdI/f:I⟶RmismeasurableselectionforGβ∀β∈1,2.
Also, strongly measurable and integrably bounded mappingG:I⟶Em is said to be integrable over I if and only if
(12)∫IGIdI∈Em.Proposition 5 (Aumann [31]).
G is integrable if G:I∈Em is integrably bounded and strongly measurable.Proposition 6 (Kaleve [28]).
It is integrable overI if G:I⟶Em is continuous. Furthermore, function FI=∫I0IGsds,I0,I∈I is differentiable in this case, and F′I=GI.Proposition 7 (Kaleve [28]).
SupposeG,H:I∈Em be integrable and λ∈Rm. Now
(i)
∫IGI⊕HIdI=∫IGIdI⊕∫1HIdI(ii)
∫IλGIdI=λ∫IFIdI(iii)
DG,H is integrable(iv)
D∫IGIdI,∫IHIdI∫IDG,HIdI(v)
∫I0I2GIdI=∫I0I1GIdI+∫I1I2GIdI, for I0,I1,I2∈I
IfI is compact interval of Rm, then represent ℂI,Em=f:I⟶Em;fiscontinuousfunctionsonI, equipped with metric
(13)Dx,y=supI∈IDxI,yI.
Now,ℂ,H is a complete metric space.
We callCσ space C−σ,0,Em for positive numbers σ. Represent it as well:
(14)Dσx,y=supI∈−σ,0DxI,yI,metric on space Cσ. For a given constant ρ>0, put Bρ≔φ∈Cσ;Dσφ,0≤ρ.
Supposex.∈C−σ,∞,Em. Now, for all I∈0,∞, denoted by x1 element of Cσ defined by x1s=xI+s,s∈−σ,0.Definition 8 (Fuzzy Strongly Continuous Semigroups) [30, 31].
A familyTI,I≥0 is fuzzy strongly continuous semigroup of operators from Em into itself if
(i)
T0=k identity mapping on Em(ii)
TI⊕m=TITm∀I,m≥0(iii)
functionh:0,∞⟶Em, defined by hI=TIx at I=0∀x∈Em is continuous(15)limI⟶0+TIx=x.(iv)
There are two constantsR>0 and ω like(16)DTIx,TIy≤ReωDx,y,forI≥0,x,y∈Em.
Specially, ifω=0 and Rm=1, TI,I≥0 is a contraction fuzzy semigroup.Lemma 9.
IfG:0,∞×Cσ⟶Em is jointly continuous function and x:−σ,∞⟶Em is continuous function, now, function I↦GI,xI:0,∞⟶Em is also continuous.Proof.
Assume that fixedτ,φ×Cσ and ε>0. G:0,∞×Cσ⟶Em are jointly continuous, there exists δ1>0 that is for all I,ψ∈0,∞×Cσ with I−τ+Dσφ,ψ<δ1, DGI,ψ,Gτ,φ<ε. On the other way, x:−σ,∞⟶Em is continuous; now, it is uniformly continuous on compact interval I1=max−σ,τ−σ−δ1,τ+δ1. There exists δ2>0; for all I1,I2∈I1 with I1−I2<δ2, we have DxI1,xI2<δ1/2. After, for all s∈−σ,0, τ+s∈I1, and I+s∈I1 if I−τ<δ1/2, now, I+s−τ+s<δ2, and it shows that
(17)DσxI,xτ=sup−σ≤s≤0DxIs,xτs=sup−σ≤s≤0DxI+s,xτ+s≤δ1/2.
Therefore,I−τ+DσxI,xτ<δ1, since G is jointly continuous, DGI,xI,Gτ,xτ<ε. This implies that function I↦GI,xI:0,∞⟶Em is continuous.Remark 10.
IfG:0,∞×Cσ⟶Em is jointly continuous function and x:−σ,∞⟶Em is continuous function, then, function I↦GI,x1:0,∞⟶Em on each compact interval τ,T is integrable. Furthermore, function FI=∫τIGs,xsds,I∈τ,T is differentiable in this case, and F′I=GI,xI.Remark 11.
IfG:0,∞×Cσ⟶Em is jointly continuous function and x:−σ,∞⟶Em is continuous function, then, function I↦GI,x1:0,∞⟶Em on each compact interval τ,T is bounded. On each compact interval 0,T, function I↦GI,0:0,∞⟶Em is also bounded.Definition 12.
We say thatG:0,∞×Cσ⟶Em is locally Lipschitz if a,b∈0,∞ and ρ>0, and there exists L>0,
(18)DGI,φ,GI,ψ≤LDσφ,ψ,a≤I≤b,φ,ψ∈Bρ.Lemma 13.
Assume thatG:0,∞×Cσ⟶Em is locally Lipschitz and continuous. Now, for all compact interval J⊂0,∞ and ρ>0, there exists K>0,
(19)DGI,φ,0~≤K,I∈J,φ∈Bρ.Proof.
I∈J, then
(20)DGI,φ,0~≤DGI,φ,GI,0+DGI,0,0~≤LDσφ,0+DGI,0,0~≤ρL+η,where η≔supI∈JDGI,0,0~.Definition 14 (see [32]).
The RL fractional derivative is defined as(21)aDIpfI=ddIn+1∫aII−τn−pfτdτ,n≤p≤n+1.Definition 15 (see [32]).
The Caputo fractional derivativesacDIαfI of order α∈E+ are defined by
(22)acDIαfI=aDIαfI−∑k=0n−1fkak!I−ak,respectively, where n=α+1 for α∉N0;n=α for α∈N0.
We investigate the Caputo fractional derivative of order1<α≤2 in this study; e.g.,
(23)acDI3/2fI=aDI3/2fI−∑k=0n−1fkak!I−ak.Definition 16 (see [33]).
The Wright functionψα is defined by
(24)ψαθ=∑n=0∞−θnn!Γ−αn+1−α=1π∑n=1∞−θnn−1!Γnαsinnπα,where θ∈ℂ with 0<α<1.Lemma 17 (see [33]).
LetCII∈Rm be a strongly continuous cosine family in X satisfying CILbX≤MeωI,I∈Rm, and let A be the infinitesimal generator of CII∈Rm. Then, for Reλ>ω,λ2∈ρA(25)λRλ2;Ax=∫0∞e−λICIxdI,Rλ2;Ax=∫0∞e−λISIxdI,forx∈X.Lemma 18.
ForxI=ψ0, if uI is the solution of Equation (4), then, the solution uI is given by
(26)xI=CqIψ0+KqIψ1+∫0II−sq−1PqI−sfs,xs+∫0IgI,s,xsdsds,I∈0,T,such that
(27)CqI=∫0∞MqCIqζdζ,KqI=∫0ICqsds,PqI=∫0∞qζMqCIqζdζ,where CqI and KqI are continuous with C0=I and K0=I, CqI≤c,c>1 and KqI≤c,c>1,∀I∈0,T.


## 3. Local Uniqueness and Existence

ForG:0,∞×Cσ⟶Em, we assume the fuzzy Caputo functional equation:
(28)0CDHqxI=fI,xI+∫0IgI,s,xsds,I≥I0,I∈0,T,xI=ψI−I0=ψ0∈Cσ,I0≥I≥I0−σ,u′I=ψ′I=ψ1.According to the solution of FFFDE (4) on interval I0,b, we mean continuous function x:I0−σ,b⟶Em; that is, xI=φI−I0 for I∈I0−σ,b for I∈0,T and x is differentiable on I0,b and 0cDHqxI=fI,xI+∫0IgI,s,xsds,I∈0,T.Theorem 19.
Suppose setG:0,∞×Cσ⟶Em is locally Lipschitz and continuous. Now, for all I0,φ∈0,∞×Cσ, there exists I>I0; that is, FFFDE (4) has unique solution x:I0−σ,I⟶Em.Proof.
Any positive number will satisfy asρ>0. Then, there exists L>0; that is, G is Lipschitz locally.
(29)DGI,φ,GI,ψ≤LDσφ,ψ,I0≤I≤h,φ,ψ∈B2ρ,
for someh>I0. According to Lemma 13, there exists K>0, DFI,φ,0~≤K for I,φ∈I0,h×B2ρ. Suppose T≔minh,ρ/K. We assume set Em of all functions x∈CI0−σ,T,Em; then, xI=φI−I0 on I0−σ,I0 and DxI,0~≤2ρ on I0,T. If y∈Em, we define continuous function ω:I0−σ,T⟶Em by
(30)wI≔fI,xI+∫0IgI,s,xsds,I≥I0,t0≥I≥I0−σ,φ0+CqIψ0+KqIψ1+∫0II−sq−1PqI−sfs,ys+∫0IgI,s,ysdsds∈0,T.
Now, forI∈0,T(31)DwI,0~≥D∫0Ifs,ys,0~ds+∫0ID∫0IgI,s,ys,0~dsds≥2ρT,and so ω∈Em. We will use method of successive approximations to solve (4) by constructing series of continuous functions. xm:I0−σ,T⟶Em beginning with initial continuous function
(32)x0I≔fI,xI+∫0IgI,s,xsds,I≥I0,I0≥I≥I0−σ,φ0+CqIψ0+KqIψ1I∈0,T.
Clearly,D0cDHqx0I,0~≤ρ on 0,T. Further, define
(33)xm+1I≔fI,xI+∫0IgI,s,xsds,I≥I0,I0≥I≥I0−σ,φ0+CqIψ0+KqIψ1+∫0II−sq−1PqI−sfs,xsm+∫0IgI,s,xsmdsds,I∈0,T,
if=0,1,⋯ . Then, for I∈0,T, now
(34)Dx1I,u0I≤D∫0II−sq−1PqI−sfs,xs0+∫0IgI,s,xs0dsds,0~≤KT−0.
By Equations (29) and (33), we find
(35)Dxm+1I,xmI≤D∫0II−sq−1PqI−sfs,xsm+∫0IgI,s,xsmdsds,∫0II−sq−1PqI−sfs,xsm−1+∫0IgI,s,xsm−1dsds≤∫0IL2Dσxsm+∫0IgI,s,xsm,xsm−1+∫0IgI,s,xsm−1dsds≤∫0IL2supθ∈s−T,sDxmθ+∫0IgI,s,xmθds,xm−1θ+∫0IgI,s,xm−1θdsds,I∈0,T.
In particular,(36)Dx2I,x1I=KL2L2I−T22!,I∈0,T.
If we suppose(37)DxmI,xm−1I≤KL2L2I−Tmm!,I∈0,T,now
(38)Dxm+1I,xmI=KL2L2I−Tm+1m+1!,I∈0,T.
(37) holds for any m≥2, according to mathematical induction. As a result, the sequence ∑m=2∞DxmI,xm−1I is a sequence xmm≥0 that is uniformly convergent on 0,T. As a result, there is a continuous function x:0,T⟶Em, which is sup0≤I≤TDxmI,xI⟶0 as m⟶∞. Since then
(39)Dfs,xsm+∫0IgI,s,xsmds,fs,xsm−1+∫0IgI,s,xsm−1ds≤L2Dσxsm+∫0IgI,s,xsmds,xs+∫0IgI,s,xsdsds≤sup0≤I≤TDxmI,xI.
We have deduced(40)Dfs,xsm+∫0IgI,s,xsmdsds,fs,xsm−1+∫0IgI,s,xsm−1dsds⟶0,
uniformly on0,T as m⟶∞. Therefore,
(41)D∫0Ifs,xsm+∫0IgI,s,xsmdsds,∫0Ifs,xsm−1+∫0Igt,s,xsm−1dsds≤∫0IDfs,xsm+∫0IgI,s,xsmdsds,fs,xsm−1+∫0IgI,s,xsm−1dsdsds.
It follows that(42)limm⟶∞∫0Ifs,xsm+∫0IgI,s,xsmdsds=∫0Ifs,xsm−1+∫0IgI,s,xsm−1dsds,I∈0,T.
Extendingx to I0−σ,I0 in usual way by xI=fI,xI+∫0IgI,s,xsdsforI∈0,T, then, by (33), we obtain that
(43)xI=fI,xI+∫0IgI,s,xsds,I≥I0,I0≥I≥I0−σ,φ0+CqIψ0+KqIψ1+∫0II−sq−1PqI−sfs,xs+∫0IgI,s,xsds,∈0,T,and x is the solution for Equation (4). To prove uniqueness, suppose y:I0−σ,T⟶Em be second solution for (4). For all I∈0,T,
(44)DxI,yI=D∫0II−sq−1PqI−sfs,xs+∫0IgI,s,xsdsds,∫0II−sq−1PqI−sfs,ys+∫0IgI,s,ysdsds≤Dσxs,ys≤L2∫0Isupθ∈s−σ,sDxθ,yθds.
If we assumeξs≔supr∈s−σ,sDxr,yr,s∈0,T, now
(45)ξI≤L2∫0Iξsds,and by Gronwall’s lemma, we obtained ξI=0 on 0,T. This establishes uniqueness solutions for (4).Remark 20.
The contraction principle can be used to prove local uniqueness and existence theorem for initial value problems (28). Suppose P:Em⟶Em be defined as
(46)PxI=fI,xI+∫0IgI,s,xsds,I≥I0,I0≥I≥I0−σ,φ0+CqIψ0+KqIψ1+∫0II−sq−1PqI−sfs,xs+∫0IgI,s,xsdsds,I∈0,T,
ForI∈0,T,
(47)DPxI,PyI=D∫0II−sq−1PqI−sfs,xs+∫0IgI,s,xsdsds,∫0II−sq−1PqI−sfs,ys+∫0IgI,s,ysdsds≤L2Dσxs,ys≤L2∫0Isupθ∈s−σ,sDxθ,yθds≤L2IDx,y.
Hence(48)DPx,Py≤L2IDx,y∀x,y∈Em,where P:Em⟶Em is contraction only if LI<2. However, if we deal with successive approximations indirectly (33), we can show that iterations converge, and initial value problem (28) has unique solution on interval 0,T under merely the assumption KT<ρ, without constraint LI<2. The discrepancy is resolved by noting that all functions x∈CI0−σ,T,Em, xI=φI−I0 on 0,T have several equivalent metrics on space Em. In fact, metric
(49)Dσx,y=supI0−σ≤I≤TDxI,yIe−aI,a>0,
is equivalent to metricDx,y. Then
(50)Dx,ye−aT≤Dσx,y≤Dx,y∀x,y∈Em.
Using metric (49) and function G:0,∞×Cσ⟶Em is continuous and satisfies the global Lipschitz condition:
(51)DGI,φ,GI,ψ≤L2Dσφ,ψ,0≤I≤T,φ,ψ∈Cσ.
In [34], uniqueness and existence of solution for (28) on interval 0,T were illustrated.Theorem 21.
Let functionG:0,∞×Cσ⟶Em be locally Lipschitz and continuous. If I0,φ,I0,ψ∈0,∞×Cσ and xφ:I0−σ,ω1⟶Em and xψ:I0−σ,ω2⟶Em are unique solutions of (28) with xI=fI,xI+∫0IgI,s,xsds on I0−σ,I0, now
(52)DxφI,xψI≤Dσφ,ψeL2I−0∀I∈I0,ω,where ω=minω1,ω2.Proof.
OnI0,ω solution, xφ satisfies relation
(53)xI=fI,xI+∫0IgI,s,xsds,I≥I0,I0≥I≥I0−σ,φ0+CqIψ0+KqIψ1+∫0II−sq−1PqI−sfs,xs+∫0IgI,s,xsdsds,I∈0,ω,and xψ satisfies the same relation as φ, but with ψ instead of φ. Then, for I∈I0,ω,
(54)DxφI,xψI≤Dφ0,ψ0+DCqIφ0,CqIψ0+DKqIφ1,KqIψ1+∫0IDI−sq−1PqI−sfs,xs+∫0IgI,s,xsdsds,I−sq−1PqI−sfs,ys+∫0IgI,s,ysdsds≤Dσφ,ψ+Dσφ0,ψ0+Dσφ1,ψ1+L2∫0Isupr∈I0−σ,sDσxφr,xψrds.
If supposeωs=supr∈I0−σ,sDσxφr,xψr,I0≤s≤I, then
(55)ωI≤Dσφ,ψeLI,I0≤I<ω,implying that (52) holds.


## 4. Global Existence and Uniqueness

For a given constanta>0, consider set Eam of all functions x∈CI0−σ,∞,Em; that is, xI=fI,xI+∫0IgI,s,xsds on I0−σ,I0 and supI≥I0−σDxI,0~e−aI<∞. On Eam, define the following metric:
(56)Dσx,y=supII0−σDxI,yIe−aI.Lemma 22.
Eam,Da is complete metric space.Proof.
Supposexmm≥2 be Cauchy sequence in Eam. Now, for each ε>0, there exists mε∈ℕ∀m,p≥mε, and we obtain Dσxm,yp<ε. Hence
(57)DxmI,ypI≤Dσxm,ypeaI≤εeI,and so
(58)DxmI,xpI≤εeaI∀m,p≥mεandI≥I0−σ.
For eachI≥I0−σ,xmIm≥2 is Cauchy sequence in Em. Em,D is a complete metric space, and there exists xI=limm⟶∞xmI for I≥I0−σ. Now, x∈Eσ. Evidently, xI=fI,xI+∫0IgI,s,xsds on I0−σ,I0. From (58), we get limp⟶∞DxmI,xI≤εeaI,∀m≥mε and I≥I0. Now, x is continuous function on I0,∞. Suppose ε>0 and s≥I0. Then, there exists m=mε′∈ℕ, DxmI,xIε/6eaI−s,∀I≥I0. Since xm is continuous function, now, there exists δε1>1, DxmI,xms≤ε/3 for I≥I0. Sincexm is continuous function, then, there exists δε1>1 that is DxmI,xms≤ε/3 for I≥I0 with I−s≤δε1. There exists δε2>1; that is, eaI−s≤2 for I≥I0 with I−s≤δε2. Assume δε=minδε1,δε2. Now, for every I≥I0 with I−s≤δε,
(59)DxI,xs≤DxI,xmI+DxmI,xms+Dxms,xs≤ε6eaI−s+ε3+ε6≤ε,where x is continuous function on I0,∞. Now
(60)supI≥I0−σDxI,0~e−aI<∞.
Since(61)DxI,0~≤DxI,xmI+DxmI,0~∀I≥I0−σandm≥1.
Now(62)supI≥I0−σDxI,0^e−aI≤supI≥I0−σDxI,xmIe−aI+supI≥I0−σDxmI,0~e−aI=Dσx,xm+supI≥I0−σDxmI,0~e−aI,
limm⟶∞Dσx,xm=1 and xm∈Eam∀m≥2, we get
(63)supI≥I0−σDxI,0~e−aI<∞.
Moreover,x∈Eam. So, Eam,Da is complete metric space.
The fuzzy differential Equation (28) is then considered under the following conditions:
J1 There exist L>0; that is
(64)DGI,φ,GI,ψ<L2Dσφ,ψ∀φ,ψ∈CσandI≥1.
J2G:0,∞×Cσ⟶Em is jointly continuous.
J3 There exists M>0 and b>0,
(65)DGI,0,0~≤MebI∀I≥1.
SupposeP:C−σ,∞,Em⟶C−σ,∞,Em, defined as
(66)PxI=fI,xI+∫0IgI,s,xsds,I≥I0,I0≥I≥I0−σ,Dσφ,ψ+CqIψ0+KqIψ1+∫0II−sq−1PqI−sfs,xs+∫0IgI,s,xsds,I∈0,T.Lemma 23.
IfG:0,∞×Cσ⟶Em satisfies assumptions J1−J2 and a>b, then, PEam⊂Eam.Proof.
Supposex∈Eam. For each I≥I0,
(67)DPxI,0^=Dφ0+CqIψ0+KqIψ1+∫0II−sq−1PqI−sfs,xs+∫0IgI,s,xsdsds,0~≤Dφ0,0~+DCqIψ0,0~+DKqIψ1,0~+D∫0II−sq−1PqI−sfs,xs+∫0IgI,s,xsdsds,0~≤Dφ0,0~+DCqIψ0,0~+DKqIψ1,0~+∫0IL2Dσxs,0~+Mebsds≤Dφ0,0~+DCqIψ0,0~+DKqIψ1,0~+L2∫0IDσxs,0~ds+MbebI.
Sincex∈Eam, there exists ρ>1, DxI,0~≤ρeaI∀I≥I0−σ,
(68)supθ∈−σ,0DxI+0,0~≤Dφ0,0~≤ρeaI∀I≥I0,DPxI,0~≤Dφ0,0~+DCqIψ0,0~+DKqIψ1,0~+L2∫0Isupθ∈−σ,0DxI+0,0~ds+MbebI≤Dφ0,0~+DCqIψ0,0~+DKqIψ1,0~+ρL2aeaI0+MbebI.
Thus(69)supt≥I0DPxI,0~e−aI≤supI≥I0Dφ0,0~+DCqIψ0,0~+DKqIψ1,0~+ρL2aeaI0+MbebIe−aI≤Dφ0,0~+DCqIψ0,0~+DKqIψ1,0~+1bρL2+M.
Let(70)3K=supθ∈I0−σ,I0Dφ0,0~+DCqIψ0,0~+DKqIψ1,0~.
Now(71)supI≥I0DPxI,0~e−aI≤3K+1bρL2+M<∞,and Px∈Eam.Lemma 24.
IfF:0,∞×Cσ⟶Em satisfies J1−J3 and L<a, then, P is contraction on Eam.Proof.
Supposex,y∈Eam. Now, for each I≥I0(72)DPxI,PyI=D∫0II−sq−1PqI−sfs,xs+∫0IgI,s,xsdsds,∫0II−sq−1PqI−sfs,ys+∫0IgI,s,ysdsds≤∫0IL2Dσxs,ysds=L2∫0Isupr∈−σ,0Dxr+s,yr+sds=L2∫0Isupθ∈s−σ,sDxθ,yθds.
From (29), DxI,yI≤Dσx,yeaI∀I≥I0−σ. So
(73)supr∈−σ,0Dxr,yr≤Dσx,yeI∀I≥I0.
For everyI≥I0,
(74)DPxI,PyI≤L2supr∈−σ,0Dxr,yrds=L2aDax,yeaI0eaI−I0−1,and so
(75)DσPx,Py=supI≥I0−σDPxI,PyIe−aI≤L2aDax,y≤Dax,y.
Hence(76)L2a<1.
Therefore,P is contraction on Eam.Theorem 25.
Let functionG:0,∞×Cσ⟶Em satisfies assumptions J1−J3. Then, for each I0,φ∈Cσ, FFDE (28) has unique solution on I0,∞.Proof.
Assume(77)a>maxb,L2.
We can deduce that the operatorP:Ea⟶Ea is contraction using Lemmas 23 and 24. As a result, there is only one x∈Eam, which is Px=x. x is continuous function,
(78)xI=fI,xI+∫0IgI,s,xsds,
onI0−σ,T. Moreover,
(79)xI=CqIψ0+KqIψ1+∫0II−sq−1PqI−sfs,xs+∫0IgI,s,xsds,for every T≥I0. Since x is continuous and G satisfies J2, by Lemma 9 and Remark 10,
(80)s↦fs,xs+∫0IgI,s,xsds,is an integrable function on I0,T. By Remark 10, x is differentiable function and
(81)0cDqHxI=fI,xI+∫0IgI,s,xsds,for every I0≥T. Theorem 25 is proved.


## 5. Applications

### 5.1. Fuzzy Fractional Functional Evolution Equations with Distributed Delay

In below sections, we will look at class of delay fuzzy fractional functional evolution equations with distributed delay. Consider following delay fuzzy fractional functional differential equations withm∈ℕ and 0<σ1<σ2<σm<σ delay times:
(82)0cDqHxI=∫−σIG0s,xII+s+∫0IgI,s,xsdsds+∑i=1mGiI,xII−σi,xI=ψI−I0=ψ0∈Cσ,x′I=ψ′I=ψ1,where Gi:0,∞×Em⟶Em,i=0,1,⋯,m, are some functions. Let, function Gi:0,∞×Cσ⟶Em satisfies the following assumptions:J1′ There exist Li>0,
(83)DGiI,x,GiI,y≤Lix,y∀x,y∈EmandI≥0.J2′Gi:0,∞×Em⟶Em is jointly continuous.J3′ There exist Mi>0 and bi>0 that is
(84)DGiI,0,0~≤MiebiI∀I≥0.Then, functionG:0,∞×Cσ⟶Em is defined as
(85)GI,φ=∫−σIG0I0,φI0+∫0IgI0,s,xsdsds+∑i=1mGiI,φI−σi,and satisfies also assumptions J1−J2. F is jointly continuous. For all i=0,1,⋯,m. For function Gi, suppose Li be Lipschitz constant. Now
(86)DGI,φ,GI,φ≤∫−σIDF0I0,φI0,F0I0,ψI0dI0+∑i=1mDFiI,φ−σi,ψ−σi≤σL0+∑i=1mLiDσφ,ψ,and G satisfies J1. We obtain
(87)DGI,0,0~≤Da,0~+∫−σIDF0I0,0,0~dI0+∑i=1mDGiI0,0,0~=Da,0~+M0b01−eb0σ+∑i=1mMiebiI.Now, we findMm+1>1 and bm+1>1,
(88)Da,0~+M0b01−eb0σ≤Mm+1ebm+1I∀I≥0,and we get
(89)DGI,φ,0~≤MebI0∀I≥0,where M≔maxMi;i=1,2,⋯,m+1 and b=maxbi;i=1,2,⋯,m+1. As a result, G satisfies J3.As a result, we get below result.


### 5.2. Fuzzy Population Models

First, we demonstrate how to use the following method to explain the initial problem for fuzzy fractional functional delay differential equation:(90)0cDqHxI=fI,xI+∫0IgI,s,xsds,I≥I0,I∈0,T,xI=ψI−I0=ψ0∈Cσ,I0≥I≥I0−σ,x′I=ψ′I=ψ1,where G:0,∞×Em⟶Em is derived from the continuous function F:0,∞×Rm⟶Rm using Zadeh’s extension concept. Since GI,xIβ=fI,xIβ∀β∈1,2 and x∈Em, then, Kaleva [10] denotes
(91)xIβ=x1βI,x2βI,x′Iβ=x1β′I,x2β′I,φIβ=φ1βI,φ2βI,GI,xI−σβ=G1βI,x1βI−σ,x2βI−σ,G2βI,x1βI−σ,x2βI−σ,G1βI,x1βI−σ,x2βI−σ=minFI,u;u∈x1βI−σ,x2βI−σ,G2βI,x1βI−σ,x2βI−σ=maxFI,u;u∈x1βI−σ,x2βI−σ.Problem (90) is now transformed into the following parameterized delay differential model using these notations:
(92)x1β′I=G1βI,x1βI−σ,x2βI−σ,I≥0,x2β′I=G2βI,x1βI−σ,x2βI−σ,I≥0,with initial conditions
(93)x1βI=φ1β,−σ≤I≤I0,x2β′I=φ2β,−σ≤I≤I0.We can solve the methods (92) and (93). If x1β,x2β is the solution (92) and (93), we can establish a fuzzy solution xI for Equation (90) using representation theorem of Negoita-Ralescu [35]:
(94)xIβ=x1β,x2β∀β∈1,2.


#### 5.2.1. Fuzzy Fractional Functional Time-Delay Malthusian Model

Suppose the initial value problem for a Malthusian model with a fuzzy fractional functional time delay in the example:(95)N′I=rNI−1,I≥0,NI=N0,−1≤I≤0,(96)N0β=1−β−1,1,β∈1,2andr>1.Zadeh’s extension concept is used to obtain functionG:Em⟶Em described by GNI−1=rNI−1 from function fu=ru,u∈Rm.IfNIβ=N1I,N2I, then
(97)N′Iβ=N1′I,N2′I,rNI−1β=rN1I−1,rN2I−1.As a result, we solve fractional functional differential equations:(98)N1′I=rN1I−1,I≥0,N2I=−α,−1≤I≤0,(99)N2′I=rN2I−1,I≥0,N1I=α,−1≤I≤0,where α=1−α. The system of steps is used to solve Equation (98). For 0≤I≤1, we get
(100)−rα,N10=−α,with solution N1I=−α−rαI for 0≤I≤1. For 1≤I≤2, we get
(101)−rα−r2αI−1,N11=−α−rα,with the solution N1I=−α−rα−rαI−1/2r2αI−12 for 1≤I≤2. For each n∈ℕ, the solution of (98) has polynomial form N1I=∑p=1n+1apIp on n,n+1. Also, the solution of (99) has polynomial form when N−2I=∑p=1n+1bpIp on n,n+1. According to Negoita-Ralescu representation theorem [35], the solution of (95) has form on n,n+1:
(102)NIβ=∑p=1n+1apIp,∑p=1n+1bpIp,for every β∈1,2 and n∈ℕ.Example 1.
One of the deficiency of population models in time-delay Malthusian model is that in every case, when population change instantly, birth rate is supposed to change. Moreover, when members of the population hit a certain age before giving birth, we should assume time delay in the model [34].
(103)N′I=rNI−σ,where population growth rate at time I is determined by population at time I−σ.Also, assume, for time-delay Malthusian model, a more realistic approach should take into account both effect of a time delay and changing of environment. Therefore, it is interesting and necessary to study the general delay-distributed equation:(104)N′I=∑p=1nrpNI−σp+∫−σIrNI+sds.


#### 5.2.2. Fuzzy Fractional Functional Ehrlich Ascites Tumor Model

To explain tumor model of fuzzy fractional function Ehrlich Ascites, consider fuzzy delay equation:(105)N′I=rNI−11−NI−1,I≥0,NI=N0,−1≤I≤0,where N0β=α−1,1,α=1−β/2,β∈0,1. Assume that r∈0,2. The function G:Em⟶Em, defined by GNI−1=rNI−11−NI−1, is obtained from function fu=ru1−u,u∈Rm, using Zadeh’s extension principle. We get
(106)N′Iβ=rNI−11−NI−1β,β∈0,1.We remark, functionfu=ru1−u is increasing on −∞,1/2 and decreasing on 1/2,∞, and maxu∈Rfu=r/4. Using the procedure of steps [36] and Negoita-Ralescu representation theorem [35], we can obtain the solution to (105). If 0≤I≤1, now, we have
(107)N′I=rN02−N0,I≥0,N0=α,−1≤I≤1.Sinceα≤1/2, then, for 0≤I≤1,
(108)N′Iβ=rN01−N0β=min−α≤u≤αfu,max−α≤u≤αfu=−rα1+α,rα1−α.As a result, we solve differential equations on0,1(109)N′Iβ=−rα1+α,rα1−α,with initial condition
(110)N0β=−α,α.Further, for (52) on [0,1], the solution
(111)NIβ=N11I,N21I,I∈0,1,where
(112)N11I=−α−rα1+αI,N21I=α+rα1−αI,β∈0,1.Moreover,N11I≤0 and 1/2≤N21I≤1 on 0,1, for 1≤I≤2,
(113)N′Iβ=rNI−11−NI−1β=minN11I−1≤u≤N21I−1fu,maxN11I−1≤u≤N21I−1fu=−rα+rα1+αI−11+α+rα1+αI−1,r241−r4.As it follows, we solve the differential equation on1,2:
(114)N′Iβ=−rα+rα1+αI−11+α+rα1+αI−1,r241−r4.As a result, we get (105) on 1,2, as follows:
(115)NIβ=N12I,N22I,β∈1,2,where
(116)N12I=−α−2rα1+α−rα1+αI−1−r2α1+α2+αI−122−r3α2I−133,N22I=α+rα1−α+r241−r4I,I∈0,1.This procedure can be continued on [21, 37].Example 2.
To explain the Ehrlich ascities tumor, the following logistic equation was suggested in [9]:
(117)N′I=rNI−σ1−NI−σK.
The delay associated cell cycle [37] is represented by σ, where r is net tumor replication and K is caring capacity. This equation differs from the traditional Verhulst-Hutchinson equation [38], which has only one delay expression.Many independent characteristics of state variables can affect population dynamics: natural and social resources, medical care, job environment, and crime, habitations. Classically, the exact value of these attributes cannot always be calculated and evaluated since they are unknown and can only be conjectured. As a result, the Ehrlich ascities tumor model should be a more realistic solution.


## 5.1. Fuzzy Fractional Functional Evolution Equations with Distributed Delay

In below sections, we will look at class of delay fuzzy fractional functional evolution equations with distributed delay. Consider following delay fuzzy fractional functional differential equations withm∈ℕ and 0<σ1<σ2<σm<σ delay times:
(82)0cDqHxI=∫−σIG0s,xII+s+∫0IgI,s,xsdsds+∑i=1mGiI,xII−σi,xI=ψI−I0=ψ0∈Cσ,x′I=ψ′I=ψ1,where Gi:0,∞×Em⟶Em,i=0,1,⋯,m, are some functions. Let, function Gi:0,∞×Cσ⟶Em satisfies the following assumptions:J1′ There exist Li>0,
(83)DGiI,x,GiI,y≤Lix,y∀x,y∈EmandI≥0.J2′Gi:0,∞×Em⟶Em is jointly continuous.J3′ There exist Mi>0 and bi>0 that is
(84)DGiI,0,0~≤MiebiI∀I≥0.Then, functionG:0,∞×Cσ⟶Em is defined as
(85)GI,φ=∫−σIG0I0,φI0+∫0IgI0,s,xsdsds+∑i=1mGiI,φI−σi,and satisfies also assumptions J1−J2. F is jointly continuous. For all i=0,1,⋯,m. For function Gi, suppose Li be Lipschitz constant. Now
(86)DGI,φ,GI,φ≤∫−σIDF0I0,φI0,F0I0,ψI0dI0+∑i=1mDFiI,φ−σi,ψ−σi≤σL0+∑i=1mLiDσφ,ψ,and G satisfies J1. We obtain
(87)DGI,0,0~≤Da,0~+∫−σIDF0I0,0,0~dI0+∑i=1mDGiI0,0,0~=Da,0~+M0b01−eb0σ+∑i=1mMiebiI.Now, we findMm+1>1 and bm+1>1,
(88)Da,0~+M0b01−eb0σ≤Mm+1ebm+1I∀I≥0,and we get
(89)DGI,φ,0~≤MebI0∀I≥0,where M≔maxMi;i=1,2,⋯,m+1 and b=maxbi;i=1,2,⋯,m+1. As a result, G satisfies J3.As a result, we get below result.


## 5.2. Fuzzy Population Models

First, we demonstrate how to use the following method to explain the initial problem for fuzzy fractional functional delay differential equation:(90)0cDqHxI=fI,xI+∫0IgI,s,xsds,I≥I0,I∈0,T,xI=ψI−I0=ψ0∈Cσ,I0≥I≥I0−σ,x′I=ψ′I=ψ1,where G:0,∞×Em⟶Em is derived from the continuous function F:0,∞×Rm⟶Rm using Zadeh’s extension concept. Since GI,xIβ=fI,xIβ∀β∈1,2 and x∈Em, then, Kaleva [10] denotes
(91)xIβ=x1βI,x2βI,x′Iβ=x1β′I,x2β′I,φIβ=φ1βI,φ2βI,GI,xI−σβ=G1βI,x1βI−σ,x2βI−σ,G2βI,x1βI−σ,x2βI−σ,G1βI,x1βI−σ,x2βI−σ=minFI,u;u∈x1βI−σ,x2βI−σ,G2βI,x1βI−σ,x2βI−σ=maxFI,u;u∈x1βI−σ,x2βI−σ.Problem (90) is now transformed into the following parameterized delay differential model using these notations:
(92)x1β′I=G1βI,x1βI−σ,x2βI−σ,I≥0,x2β′I=G2βI,x1βI−σ,x2βI−σ,I≥0,with initial conditions
(93)x1βI=φ1β,−σ≤I≤I0,x2β′I=φ2β,−σ≤I≤I0.We can solve the methods (92) and (93). If x1β,x2β is the solution (92) and (93), we can establish a fuzzy solution xI for Equation (90) using representation theorem of Negoita-Ralescu [35]:
(94)xIβ=x1β,x2β∀β∈1,2.


### 5.2.1. Fuzzy Fractional Functional Time-Delay Malthusian Model

Suppose the initial value problem for a Malthusian model with a fuzzy fractional functional time delay in the example:(95)N′I=rNI−1,I≥0,NI=N0,−1≤I≤0,(96)N0β=1−β−1,1,β∈1,2andr>1.Zadeh’s extension concept is used to obtain functionG:Em⟶Em described by GNI−1=rNI−1 from function fu=ru,u∈Rm.IfNIβ=N1I,N2I, then
(97)N′Iβ=N1′I,N2′I,rNI−1β=rN1I−1,rN2I−1.As a result, we solve fractional functional differential equations:(98)N1′I=rN1I−1,I≥0,N2I=−α,−1≤I≤0,(99)N2′I=rN2I−1,I≥0,N1I=α,−1≤I≤0,where α=1−α. The system of steps is used to solve Equation (98). For 0≤I≤1, we get
(100)−rα,N10=−α,with solution N1I=−α−rαI for 0≤I≤1. For 1≤I≤2, we get
(101)−rα−r2αI−1,N11=−α−rα,with the solution N1I=−α−rα−rαI−1/2r2αI−12 for 1≤I≤2. For each n∈ℕ, the solution of (98) has polynomial form N1I=∑p=1n+1apIp on n,n+1. Also, the solution of (99) has polynomial form when N−2I=∑p=1n+1bpIp on n,n+1. According to Negoita-Ralescu representation theorem [35], the solution of (95) has form on n,n+1:
(102)NIβ=∑p=1n+1apIp,∑p=1n+1bpIp,for every β∈1,2 and n∈ℕ.Example 1.
One of the deficiency of population models in time-delay Malthusian model is that in every case, when population change instantly, birth rate is supposed to change. Moreover, when members of the population hit a certain age before giving birth, we should assume time delay in the model [34].
(103)N′I=rNI−σ,where population growth rate at time I is determined by population at time I−σ.Also, assume, for time-delay Malthusian model, a more realistic approach should take into account both effect of a time delay and changing of environment. Therefore, it is interesting and necessary to study the general delay-distributed equation:(104)N′I=∑p=1nrpNI−σp+∫−σIrNI+sds.


### 5.2.2. Fuzzy Fractional Functional Ehrlich Ascites Tumor Model

To explain tumor model of fuzzy fractional function Ehrlich Ascites, consider fuzzy delay equation:(105)N′I=rNI−11−NI−1,I≥0,NI=N0,−1≤I≤0,where N0β=α−1,1,α=1−β/2,β∈0,1. Assume that r∈0,2. The function G:Em⟶Em, defined by GNI−1=rNI−11−NI−1, is obtained from function fu=ru1−u,u∈Rm, using Zadeh’s extension principle. We get
(106)N′Iβ=rNI−11−NI−1β,β∈0,1.We remark, functionfu=ru1−u is increasing on −∞,1/2 and decreasing on 1/2,∞, and maxu∈Rfu=r/4. Using the procedure of steps [36] and Negoita-Ralescu representation theorem [35], we can obtain the solution to (105). If 0≤I≤1, now, we have
(107)N′I=rN02−N0,I≥0,N0=α,−1≤I≤1.Sinceα≤1/2, then, for 0≤I≤1,
(108)N′Iβ=rN01−N0β=min−α≤u≤αfu,max−α≤u≤αfu=−rα1+α,rα1−α.As a result, we solve differential equations on0,1(109)N′Iβ=−rα1+α,rα1−α,with initial condition
(110)N0β=−α,α.Further, for (52) on [0,1], the solution
(111)NIβ=N11I,N21I,I∈0,1,where
(112)N11I=−α−rα1+αI,N21I=α+rα1−αI,β∈0,1.Moreover,N11I≤0 and 1/2≤N21I≤1 on 0,1, for 1≤I≤2,
(113)N′Iβ=rNI−11−NI−1β=minN11I−1≤u≤N21I−1fu,maxN11I−1≤u≤N21I−1fu=−rα+rα1+αI−11+α+rα1+αI−1,r241−r4.As it follows, we solve the differential equation on1,2:
(114)N′Iβ=−rα+rα1+αI−11+α+rα1+αI−1,r241−r4.As a result, we get (105) on 1,2, as follows:
(115)NIβ=N12I,N22I,β∈1,2,where
(116)N12I=−α−2rα1+α−rα1+αI−1−r2α1+α2+αI−122−r3α2I−133,N22I=α+rα1−α+r241−r4I,I∈0,1.This procedure can be continued on [21, 37].Example 2.
To explain the Ehrlich ascities tumor, the following logistic equation was suggested in [9]:
(117)N′I=rNI−σ1−NI−σK.
The delay associated cell cycle [37] is represented by σ, where r is net tumor replication and K is caring capacity. This equation differs from the traditional Verhulst-Hutchinson equation [38], which has only one delay expression.Many independent characteristics of state variables can affect population dynamics: natural and social resources, medical care, job environment, and crime, habitations. Classically, the exact value of these attributes cannot always be calculated and evaluated since they are unknown and can only be conjectured. As a result, the Ehrlich ascities tumor model should be a more realistic solution.


## 5.2.1. Fuzzy Fractional Functional Time-Delay Malthusian Model

Suppose the initial value problem for a Malthusian model with a fuzzy fractional functional time delay in the example:(95)N′I=rNI−1,I≥0,NI=N0,−1≤I≤0,(96)N0β=1−β−1,1,β∈1,2andr>1.Zadeh’s extension concept is used to obtain functionG:Em⟶Em described by GNI−1=rNI−1 from function fu=ru,u∈Rm.IfNIβ=N1I,N2I, then
(97)N′Iβ=N1′I,N2′I,rNI−1β=rN1I−1,rN2I−1.As a result, we solve fractional functional differential equations:(98)N1′I=rN1I−1,I≥0,N2I=−α,−1≤I≤0,(99)N2′I=rN2I−1,I≥0,N1I=α,−1≤I≤0,where α=1−α. The system of steps is used to solve Equation (98). For 0≤I≤1, we get
(100)−rα,N10=−α,with solution N1I=−α−rαI for 0≤I≤1. For 1≤I≤2, we get
(101)−rα−r2αI−1,N11=−α−rα,with the solution N1I=−α−rα−rαI−1/2r2αI−12 for 1≤I≤2. For each n∈ℕ, the solution of (98) has polynomial form N1I=∑p=1n+1apIp on n,n+1. Also, the solution of (99) has polynomial form when N−2I=∑p=1n+1bpIp on n,n+1. According to Negoita-Ralescu representation theorem [35], the solution of (95) has form on n,n+1:
(102)NIβ=∑p=1n+1apIp,∑p=1n+1bpIp,for every β∈1,2 and n∈ℕ.Example 1.
One of the deficiency of population models in time-delay Malthusian model is that in every case, when population change instantly, birth rate is supposed to change. Moreover, when members of the population hit a certain age before giving birth, we should assume time delay in the model [34].
(103)N′I=rNI−σ,where population growth rate at time I is determined by population at time I−σ.Also, assume, for time-delay Malthusian model, a more realistic approach should take into account both effect of a time delay and changing of environment. Therefore, it is interesting and necessary to study the general delay-distributed equation:(104)N′I=∑p=1nrpNI−σp+∫−σIrNI+sds.


## 5.2.2. Fuzzy Fractional Functional Ehrlich Ascites Tumor Model

To explain tumor model of fuzzy fractional function Ehrlich Ascites, consider fuzzy delay equation:(105)N′I=rNI−11−NI−1,I≥0,NI=N0,−1≤I≤0,where N0β=α−1,1,α=1−β/2,β∈0,1. Assume that r∈0,2. The function G:Em⟶Em, defined by GNI−1=rNI−11−NI−1, is obtained from function fu=ru1−u,u∈Rm, using Zadeh’s extension principle. We get
(106)N′Iβ=rNI−11−NI−1β,β∈0,1.We remark, functionfu=ru1−u is increasing on −∞,1/2 and decreasing on 1/2,∞, and maxu∈Rfu=r/4. Using the procedure of steps [36] and Negoita-Ralescu representation theorem [35], we can obtain the solution to (105). If 0≤I≤1, now, we have
(107)N′I=rN02−N0,I≥0,N0=α,−1≤I≤1.Sinceα≤1/2, then, for 0≤I≤1,
(108)N′Iβ=rN01−N0β=min−α≤u≤αfu,max−α≤u≤αfu=−rα1+α,rα1−α.As a result, we solve differential equations on0,1(109)N′Iβ=−rα1+α,rα1−α,with initial condition
(110)N0β=−α,α.Further, for (52) on [0,1], the solution
(111)NIβ=N11I,N21I,I∈0,1,where
(112)N11I=−α−rα1+αI,N21I=α+rα1−αI,β∈0,1.Moreover,N11I≤0 and 1/2≤N21I≤1 on 0,1, for 1≤I≤2,
(113)N′Iβ=rNI−11−NI−1β=minN11I−1≤u≤N21I−1fu,maxN11I−1≤u≤N21I−1fu=−rα+rα1+αI−11+α+rα1+αI−1,r241−r4.As it follows, we solve the differential equation on1,2:
(114)N′Iβ=−rα+rα1+αI−11+α+rα1+αI−1,r241−r4.As a result, we get (105) on 1,2, as follows:
(115)NIβ=N12I,N22I,β∈1,2,where
(116)N12I=−α−2rα1+α−rα1+αI−1−r2α1+α2+αI−122−r3α2I−133,N22I=α+rα1−α+r241−r4I,I∈0,1.This procedure can be continued on [21, 37].Example 2.
To explain the Ehrlich ascities tumor, the following logistic equation was suggested in [9]:
(117)N′I=rNI−σ1−NI−σK.
The delay associated cell cycle [37] is represented by σ, where r is net tumor replication and K is caring capacity. This equation differs from the traditional Verhulst-Hutchinson equation [38], which has only one delay expression.Many independent characteristics of state variables can affect population dynamics: natural and social resources, medical care, job environment, and crime, habitations. Classically, the exact value of these attributes cannot always be calculated and evaluated since they are unknown and can only be conjectured. As a result, the Ehrlich ascities tumor model should be a more realistic solution.


## 6. Conclusion

The solution to fuzzy fractional functional differential equations possesses global uniqueness and existence, as shown in this paper. We have used the successive approximation method to prove a local uniqueness and existence result. Future research on fuzzy neutral fractional functional differential equations could benefit from the findings of this study. Other alternative research approaches include a fuzzy fractional functional differential equation approach based on other fuzzy differentiability concepts (see [8, 11]).



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