# Existence and Uniqueness of Mild Solution for Fractional-Order Controlled Fuzzy Evolution Equation

Authors: Naveed Iqbal; Azmat Ullah Khan Niazi; Ramsha Shafqat; Shamsullah Zaland
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, we investigated the existence and uniqueness of mild solutions for fractional-order controlled fuzzy evolution equations with Caputo derivatives of the controlled fuzzy nonlinear evolution equation of the form0cDIγxI=αxI+PI,xI+AIWI,I∈0,T,xI0=x0, in which γ∈0,1, E1 is the fuzzy metric space and I=0,T is a real line interval. With the help of few conditions on functions P:I×E1×E1⟶E1, WI is control and it belongs to E1, A∈FI,LE1, and α stands for the highly continuous fuzzy differential equation generator. Finally, a few instances of fuzzy fractional differential equations are shown.


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## Body

## 1. Introduction

A significant range of physical processes in real-world events may be modeled using dynamical equations with fractional-order derivatives. Fractional calculus authorizes the operations of differentiation and integration of fractional order. On imaginary and real values [1, 2], the fractional order can be used. The theory of fuzzy sets attracts researchers’ attention owing to its wide range of applications in domains such as signal processing, robotics and control, thermal systems, electrical engineering, and mechanics [3–9]. As a result, it has been noted that it has become a focus of rising attention among scholars in recent years.Agarwal et al. [10] proposed the notion of the fractional differential equation in 2010. However, in terms of Hukuhara differentiability, this notion was unable to achieve the large and diversified behavior of the crisp solution at the time. However, Bede and Stefanini and Allahviranloo et al. [11, 12] developed the Riemann-Liouville H-derivative in 2012, which was based on highly extended Hukuhara differentiability [13, 14]. They also defined the fractional derivative of Riemann-Liouville. See the references [15–17] for the most recent paper on fuzzy differential equations.Kaleva [18] discovers features of differentiability fuzzy set-valued mappings using H-differentiability. Puri and Ralescu [19] presented the uniqueness and existence theorem for the solution of the fuzzy differential equation as a result of it.
(1)u′I=PI,uI,u0=u0,where P:I×E1⟶E1 satisfies Lipschitz condition.Moreover, the existence and uniqueness of a mild solution in details for fuzzy differential equations were studied by Melliani et al. [20].
(2)u′I=αuI+PI,uhI,u0=u0+−1gu.With the help of the fuzzy initial condition, the work on linear time-invariant systems was done by Dubey and George [21].
(3)u′I=αuI+BcI,uI0=u0.In the above equation,cI∈E1m control and α,A are n×n,n×m real matrices, I0≥0. More results are given in references [21, 22].In 2019, Melliani et al. [20] worked on controlled fuzzy evolution equation which is given below:
(4)u′I=αuI+PI,uI,uρss+AIcI,I∈I,uI0=u0,where the intervalI=I0,I1is on real line and fuzzy metric space isE1.By the inspiration of the above works, we adopted the Caputo derivative to prove the uniqueness and existence for the below controlled fuzzy differential equation of fractional order(5)0cDIγxI=αxI+PI,xI+AIWI,I∈0,T,(6)xI0=x0,where E1 is a fuzzy metric space and I=0,T is an interval of real line. With the help of few conditions on functions P:I×E1×E1⟶E1,σ:I⟶I, where WI is control and it belongs to E1,A∈F1,LE1 and α denotes the generator of strongly continuous fuzzy differential equation. The purpose of this paper consists of the study of the existence of mild solution which depends on fuzzy differential equation of fractional order [23, 24].The following is how the research was handled. Section2 discusses several notations and useful concepts for fuzzy continuous semigroups and properties for fuzzy fractional differential equations. This study is based on a modified hypothesis in Section 3 and also works on a moderate solution for the system (5) and control continuity. Finally, in Section 4, we show some instances of our primary findings.


## 2. Preliminaries

In this section, we give basic knowledge and notation used in our work.Suppose a family of all nonempty compact convex subsets ofRn is denoted by Pkℝn and scalar multiplication and addition is also defined as Pkℝn. Let α and A be two nonempty bounded subsets of ℝn. Hausdorff metric is used to define the distance between α and A,
(7)dα,A=maxsupa∈αinfb∈Aa−b,supb∈Ainfa∈αa−b.In the above equation· indicate usual Euclidean norm in ℝn, whereas Pkℝn,d becomes separable and complete metric space [19].Denote(8)E1=x:ℝ⟶0,1∣xsatisfiesi−ivbelow.In the above equation,(i)
x is normal due to the exists an x0∈ℝ,xu0=1(ii)
x is fuzzy convex(iii)
x is upper semicontinuous function on ℝ(iv)
x0=clδ∈ℝn/xI>0 is compact0<β≤1representxβ=I∈ℝn/xI≥β. Then, from (i) to (iv), it shows, β-level set uβI∈Pkℝn for all 0≤β≤1.Using the Zadeh extension concept, one can also have scalar multiplication and addition in fuzzy number spaceE1:
(9)x⊕yβ=xβ⊕yβandkxβ=kxβ,where x,y∈E1,k∈ℝn, and 0≤β≤1.DefineD:E1×E1⟶ℝ+ by notation
(10)Dx,y=sup0≤β≤1maxxβ,yβ.In the above equation,d is Hausdorff metric for nonempty compact sets in ℝn. It is very simple to notice thatDis metric inE1. By using result [19], now
(i)
E1,D is a complete metric space(ii)
Dx⊕z,y⊕z=Dx,y∀x,y,z∈E1(iii)
Dkx,ky=∣k∣Dx,y∀x,y∈E1 and k∈ℝnIf we denoteδG=Dδ,0~,δ∈E1, then δG has properties of an usual norm on E1 [25]:
(i)
xG=0 if x=0~(ii)
ΛxG=∣Λ∣xG∀x,y∈E1(iii)
x+yG≤xG+xG∀x,y∈E1(iv)
Dβδ,γx≤∣β−γ∣Dx,0~,∀β,γ≥0 or β,γ≤0,x∈E1OnE1, we can define subtraction ⊖, called H-difference [26] as below x⊖y has sense if there exist ω∈E1, that is, x=y+z.RepresentWI,E1=P:I⟶E1;P is continuous on I, equipped with metric
(11)Hx,y=supI∈IDxI,yI.Now,W,H is a complete metric space.Supposea,b∈ℝ,g∈WI,E1 if g=Hg,0~, then g has properties of an usual norm on E1 [25]:
(i)
g=0 if g=0~(ii)
Λg=Λg∀g∈WI,E1,Λ∈ℝ(iii)
g⊕h≤g⊕h∀g,h∈I,E1(iv)
Hβg,γg≤∣β−γ∣Hg,0~,∀β,γ≥0 or β,γ≤0, g∈WI,E1Definition 1.
The mappingG:I⟶E1 is Hukuhara differentiable at t0∈IifG′I∈E1exists similar to the below limits:
(12)limi⟶0+GI+i⊖GIi,limi⟶0+GI⊖GI−ii
are equal and exist toG′I.
We can remember some properties of integrability and measurability for fuzzy set-valued mappings [19].Definition 2.
The mappingG:I∈E1is strongly measurable if for allβ∈0,1, set-valued functionGβ:I⟶Mjℝndefined byGβI=GIβis Lebesgue measurable.
The mappingG:I∈E1 is known as integrably bounded if there exists an integrable function j like x≤JI∀u∈G0I.Definition 3.
SupposeG:I∈E1. Then, integral of G over I represented by ∫IGIdt is defined
(13)∫IGIdtβ=∫IGβIdt=∫IGIdt/g:I⟶ℝnismesurableselectionforGβ∀β∈0,1.
Also, integrably bounded and strongly measurable mappingG:I⟶E1 is said to be integrable over I if and only if
(14)∫IGIdt∈E1.Proposition 4 (Aumann [27]).
IfG:I∈E1 is strongly measurable and integrable bounded, then F is integrable.Proposition 5 (see [18]).
SupposeG,H:I∈E1 is integrable and Λ∈ℝ. Now,
(i)
∫IGI⊕HIdt=∫IGIdt⊕∫1HIdt(ii)
∫IΛGIdt=Λ∫IGIdt(iii)
DG,H is integrable(iv)
D∫IGIdt,∫IHIdt≤∫IDG,HIdtDefinition 6 (fuzzy strongly continuous semigroups) [[28–30]].
α family TI,I≥0 is a fuzzy strongly continuous semigroup of operators from E1 into itself if
(i)
T0=k identity mapping on E1(ii)
TI⊕m=TITm∀I,m≥0(iii)
functionh:0,∞⟶E1, defined by hI=TIu at I=0∀u∈E1, is continuous, that is,(15)limI⟶0+TIu=u(iv)
There are two constantsR>0 and ω like(16)DTIu,TIv≤ReωIDu,v,forI≥0,u,v∈E1
Specially ifω=0 and R=1, TI,I≥0 is a contraction fuzzy semigroup.Remark 7.
The condition (iii) implies functionI⟶TIδ on 0,∞∀x∈E1 is continuous.Definition 8.
SupposeTI,I≥0 is fuzzy strongly continuous semigroup on E1 and x∈E1. If for h>0 sufficiently small, Hukuhara difference Thδ⊖δ exists,
(17)αu=limh⟶0+Tiu⊖xh.
When this limit is in metric spaceE1,D, operator α defined as
(18)Dα=x∈E1:limi⟶0+Tix⊖xiexists⊂E1is known as an infinitesimal generator of fuzzy semigroup TI,I≥0.Lemma 9.
Supposeαis a generator of the fuzzy semigroupTI,I≥0and is a fuzzy strongly continuous semigroup onE1,∀δ∈E1, that is,TIδ∈Dα∀I≥0; mapping I⟶TIδ is differentiable.
(19)ddtTIx=αTIx,∀I≥0.Example 1.
Define theE1 family of operator TI,I≥0 by
(20)TIδ=ejtδ,j∈ℝ.
Whenj≥0,TI,I≥0 is a fuzzy strongly continuous semigroup on E1, linear operator α defined by As=js is an infinitesimal generator.Proposition 10.
SupposeTI,I≥0 is a fuzzy strongly continuous semigroup on E1. Then, I≥0,∀TI∈FE1.Proof.
Letu,v∈E1; condition (iv) in definition (8) implies that there exist two constants ω and M>0 like
(21)DTIu,TIv≤ReωIDu,vor Du,v⟶0 as u⟶v. Then, DTIu,TIv⟶0 as u⟶v which implies TI∈FE1, for all I≥0.Lemma 11 (see [31, 32]).
Ifu,v,x,y∈E1, now
(i)
dx⊕u,y⊕u=dx,y(ii)
dΛx,Λ⊕y=Λdx,y, Λ∈ℝ(iii)
dx⊕u,y⊕v≤dδ,y+du,v(iv)
dΛx,μy=Λ−μdx,0^, Λ,μ≥0(v)
dδ⊖u,y⊖v≤dx,y+du,v, provided differences x⊖u and y⊖v exist(vi)
E1,d is a complete metric spaceLemma 12 (Gronwall lemma) [32].
Letζ, y∈W0,1,ℝ+. Suppose that μ is increasing. If u∈W0,1,ℝ+ is a solution to inequality
(22)xI≤ζI+∫0Iyxuxdx,I∈0,T,then
(23)xI≤ζIexp∫0Iyxuxdx,I∈0,T.


## 3. Main Results

To initiate this discussion, there is a need to make familiar concept of mild solution for problem (5), given that g:0,T×E1⟶E1, ϕ:0,T⟶0,T, A:0,T⟶FE1, α is a generator of a strongly continuous fuzzy semigroup, input cI∈E1 for each I∈0,T, and (.) is fuzzy-integrable in 0,T. As a result, it is required to comprehend the structure of equation (5).Definition 13.
x is a mild solution of equation (5) if
(i)
x∈W0,T,E1, xI∈Dα∀I∈0,T(ii)
xI=Iγ−1PγIx0+∫0II−δγ−1PγI−δPδ,xδ+AδWδds,∀I∈0,TLemma 14.
IfuI is a solution of equation (5) for x0=x0, then xI is given:
(24)xI=Iγ−1s0+1γ∫0II−δγ−1Axδ+Pδ,xδ+AδWδdsholds; then,
(25)xI=Iγ−1PγIx0+∫0II−δγ−1PγI−δPδ,xδ+AδWδds,where
(26)PγI=∫0∞qθMγθQIγθdθ.Proof.
LetΛ>0. Applying the Laplace transform, where
(27)vΛ=∫0∞e−Λδxδds,wΛ=∫0∞e−ΛδPδ,xδds,equation (24) becomes
(28)vΛ=x0+1ΛγAvΛ+1ΛγwΛ1−1ΛγαvΛ=x0+1ΛγwΛΛγI−αvΛ=x0+1ΛγwΛvΛ=x0ΛγI−α+1ΛγI−αwΛ=ΛγI−α−1x0+ΛγI−α−1wΛ=∫0∞e−ΛγxQxx0+∫0∞e−ΛγδQxwΛdsprovided that the integrals in (28) exist, where I is the identity operator defined on X.
Set(29)φγθ=γθγ+1Mγθ−γ,whose Laplace transform is
(30)∫0∞e−Λθφγθdθ=e−Λγ,whereq∈0,1.
Using (30), we get
(31)x=Iγdx=γIγ−1dt∫0∞e−ΛγxQxx0dx=∫0∞γIγ−1e−ΛIγQIγx0dt=∫0∞∫0∞γφγθe−ΛIθQIγIγ−1x0dθdt=∫0∞∫0∞e−ΛIγφγθQIγθγIγ−1θγx0dθdt=∫0∞e−ΛIγ∫0∞φγθQIγθγIγ−1θγx0dθdt,(32)∫0∞e−ΛγxQδwΛds=∫0∞∫0∞γIγ−1e−ΛIγQIγe−ΛδPδ,xδ+AδWδdsdt=∫0∞∫0∞∫0∞γφγθe−IΛθQIγe−ΛδIγ−1Pδ,xδ+AδWδdθdsdt=∫0∞∫0∞∫0∞γφγθe−ΛI+xQIγθγIγ−1θγPx,δx+AxWxdθdxdtvΛ=∫0∞e−IΛγ∫0IφγθQI−δγθγI−δγ−1θγPδ,xδ+AδWδdθdsdt.
According to equations (31) and (32),
(33)vΛ=∫0∞e−ΛIγ∫0∞φγθQIγθγIγ−1θγs0dθ+γ∫0I∫0∞φγθQI−δγθγI−δγ−1θγPδ,xδ+AδWδdθdsdtPξI=∫0∞qθMξθQIξθdθ.
Now, by invert Laplace transform,(34)xI=γ∫0∞θIq−1MγθQIγθs0dθ+γ∫0I∫0∞θI−δγ−1MγθQI−δγθPδ,xδ+AδWδdθds=Iγ−1PγIx0+∫0II−δγ−1PγI−δPδ,xδ+AδWδdt.
The proof is completed.The concept of controllability that we provide here is the same as that provided by Dubey and George in [19].Definition 15.
Equation (5) with fuzzy initial condition x0=x0∈E1 is controllable to a fuzzy-state x1∈E1 at t2>I0 if there is fuzzy-integrable control cI∈E1 for I∈0,T; similarly with this control, solution of problem (5) satisfiesδt2=x1.LetW=supI∈0,TReωI. Using the fixed point argument, we first investigate the existence and uniqueness of mild solutions. Let the assumptions be as follows:H0 On E1, the infinitesimal generator α of the strongly continuous fuzzy semigroup TI,I≥0.H1g:0,T×E1⟶E1 is continuous, and there exist two constants F1,F2>0,
(35)DPI,u,v,PI,u′,v′≤F1Du,v+F2Du′,v′,∀I∈0,T,u,v,u′,v′∈E1.H2ϕ:0,T⟶0,T is continuous, and ϕI≤I∀I∈0,T.H3 For I∈0,T,cI∈E1 is control and c. is fuzzy-integrable.H4A:0,T⟶FE1 is continuous.H5 There exist constant M>0,
(36)DAIu,AIv≤MDu,v∀u,v∈E1,I∈0,TTheorem 16.
Suppose that the conditionsH0−H4 are satisfied. Then, for all x0∈E1 such that TIx0∈E1∀I≥0, problem (5) has a unique mild solution on 0,T.Proof.
Change problem (5) into a fixed point problem. Suppose operator γ:W0,T,E1⟶W0,T,E1 defined as
(37)ΓxI=Iγ−1PγIx0+∫0II−δγ−1PγI−δPδ,xδ+AδWδds.
Γ is well defined and maps W0,T,E1 into itself. Indeed, for u∈E1,I∈I0,I1 and i is very small; now,
(38)DΓxI+h,ΓxI=DIγ−1PγI+ix0+∫0I+iI−δγ−1PγI+i−δ·Pδ,xδ+AδWδds,·∫0I+iI−δγ−1PγI−δPδ,xδ+AδWδdsDγxI+i,γxI≤Iγ−1DPγI+ix0,PγIx0+I−δγ−1D∫0I+iPγI+i−δPδ,xδ+AδWδds,·∫0I+iPγI−δPδ,xδ+AδWδdsDγxI+i,γxI≤NDPγix0,x0+∫0I+iI−δγ−1D·PγI+i−δPδ,xδ+AδWδ,0~ds+N∫0I+iI−δγ−1D·Pδ+h,xδ+h,Pδ,xδds+N∫0I+iI−δγ−1DAδ+hWδ+h,AδWδds
Definitely,(39)Dpγix0,x0⟶0∫00+iDpγI+i−δPδ,xδ+AδWδ,0~ds⟶0.
By dominated convergence theorem,(40)N∫0IDPδ+h,xδ+h,Pδ,xδds,∫0IDAδ+iWδ+i,AδWδds⟶0.
From overhead, we infer,ξδ∈W0,T,E1,∀δ∈W0,T,E1. For I∈0,T,δ,y∈E1; now,
(41)DΓxI,ΓyI=DIγ−1PγIx0+∫0II−δγ−1PγI−δPδ,xδ+AδWδds,Iγ−1PγIy0+∫0II−δγ−1PγI−δPδ,yδ+AδWδds≤∫0II−δγ−1PγI−δPδ,xδ+AδWδds,∫0II−δγ−1PγI−δPδ,yδ+AδWδds≤N∫0II−δγ−1DPδ,xδ+AδWδ,Pδ,yδ+AδWδds≤N∫0II−δγ−1DPδ,xδ,Pδ,yδds≤N∫0II−δγ−1αDx,yLDxδ,yδds≤LαNI−0Hx,y≤LαNIHx,yDΓ2xI,Γ2yI≤αN∫0II−δγ−1LDΓxI,ΓyIds≤αN∫0ILLx−0Hx,yds≤αNxL2Hx,y≤N∫0ILαDξxδ,ξyδds≤NαL2Hδ,y∫0Isds≤NαL2Hx,yI22.
Ifh>0,
(42)DΓhxI,ΓhyI≤NαLImHx,yh!,I∈0,Tlimh⟶0NLαIhh!=0,m>0,NLαIpm!<1,which represents that Γm is a contraction. So there is a unique fixed point x∈W0,T,E1, Γmx=x, which implies that ΓmΓx=ΓΓmx=Γx. Since x is unique, then Γx=x. It shows that x is a unique mild solution of problem (5).
The below theorem gives comparison between the continuity of control and solution.Theorem 17.
Suppose thatcn,n≥0⊂E1 is a sequence of controls in H3 like limn⟶∞cn=c. Let conditions H0−H5 hold. Let ucn be a mild solution of equation (5) corresponding to cn and uc be a mild solution corresponding to c. If NLαI<1, limn⟶∞ucn=uc.Proof.
I∈0,T.
(43)DxcnI,xcI=DIγ−1PγIx0+∫0II−δγ−1PγI−δPδ,xcnδ+AδWδds,Iγ−1PγIx0+∫0II−δγ−1PγI−δPδ,xcδ+AδWδds≤D∫0II−xγ−1PγI−δPδ,xcnδ+AδWδds,∫0II−δγ−1PγI−δPδ,xcδ+AδWδds≤N∫0II−δγ−1DPδ,xcnδ+AδWδds,Pδ,xcδ+AδWδds≤N∫0II−δγ−1Dxcnδ,xcδ+LDPxcnδ,xcδds+N∫0IDAδCnδ,AδWδds≤LNI−0αHxcn,xc+PNI−0Hcn,cHxcn,xc≤PNI1−NLαIHcn,c.
Sincecn⟶c, hence, xcn⟶xc,n⟶∞.We conclude the below result related to continuous dependence of a mild solution.Theorem 18.
Let the conditionsH0−H4 hold. Let δ and y be the mild solutions of equation (5) on 0,T corresponding to x0 and y0, respectively.
If(44)NαLIeLNI<1,then
(45)Hx,y≤N1−eLNINα+LDx0,y0.Proof.
(46)DxI,yI=DIγ−1PγIx0+∫0II−δγ−1PγI−δPδ,xδ+AδWδds,PγIy0+∫0II−δγ−1PγI−δPδ,yδ+AδWδds≤NDx0,y0+N∫0II−δγ−1DPδ,xδ+AδWδ,Pδ,yδ+AδWδds≤NDx0,y0+NDx0,y0+NαIHx,y+LN∫0IDxδ,yδds.
By using Gronwall inequality,(47)DxI,yI≤NDx0,y0+NIα+LHx,yeLNIHx,y≤NDx0,y0+Nα+LHx,yeLNI.
Finally, we get(48)Hx,y≤N1−eLNINα+LDx0,y0.


## 4. Examples

We present some examples of our main theorems in this section.Example 2.
Let a Cauchy problem for fuzzy fractional differential equation(49)0cD0+0.4xI=xI+0.4xIcosI+Ie−Ion 0,π, with initial condition
(50)0cD0+−o.6xI=0~,where α=1,T=π,PI,xI=−0.4xIcosI+Ie−I, and A=0,1,2,3∈E1 is a symmetric triangular fuzzy number. Take L=0.1. Apparently, conditionsH0−H4are satisfied. So, with respect to Theorem (16), the Cauchy problem has a unique and differential solution.Example 3.
Let a Cauchy problem for fuzzy fractional differential equation(51)0cD0+0.4xI=xI−I2+I+3,I∈0,π,with initial condition
(52)0cD0+−o.6xI=0~,
whereα=1,T=π,PI,xI=−I2+I+3 and x0=0,1,2,3,4∈E1 is a symmetric triangular fuzzy number. Take L=0.1. Apparently, conditionsH0−H4are satisfied. Moreover, it is very simple to prove that condition H5 holds. So, with respect to Theorem (17), the Cauchy problem has a unique and differential solution.


## 5. Conclusion

A controlled fuzzy evolution equation in fractional order will be examined in order to comprehend the random evolution of ride temperature using the Caputo derivative. We may achieve uniqueness and existence findings using the Caputo derivative. Future work might also involve broadening the notion proposed in this mission, adding observability, and generalizing other tasks [33, 34]. This is a promising field with a number of research efforts that might lead to a variety of applications and theories. This is a path to which we want to invest considerable resources.



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