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Existence, Uniqueness and Approximate Solutions of Fuzzy Fractional Differential Equations

By Atimad Harir, Said Melliani and Lalla Saadia Chadli

Submitted: June 26th 2020Reviewed: September 11th 2020Published: November 12th 2020

DOI: 10.5772/intechopen.94000

Downloaded: 35

Abstract

In this paper, the Cauchy problem of fuzzy fractional differential equationsTγut=Ftut, ut0=u0,

Keywords

  • fuzzy conformable fractional derivative
  • fuzzy fractional differential equations
  • existence and uniqueness of solution
  • approximate solutions
  • Cauchy problem of fuzzy fractional differential equations

1. Introduction

In this paper, we will study Fuzzy solutions to

Tγut=Ftut,ut0=u0,γ01,E2

where subject to initial condition u0for fuzzy numbers, by the use of the concept of conformable fractional H-differentiability, we study the Cauchy problem of fuzzy fractional differential equations for the fuzzy valued mappings of a real variable. Several import-extant results are obtained by applying the embedding theorem in [1] which is a generalization of the classical embedding results [2, 3].

In Section 2 we recall some basic results on fuzzy number. In Section 3 we introduce some basic results on the conformable fractional differentiability [4, 5] and conformable integrability [5, 6] for the fuzzy set-valued mapping in [7]. In Section 4 we show the relation between a solution and its approximate solution to the Cauchy problem of the fuzzy fractional differential equation, and furthermore, and we prove the existence and uniqueness theorem for a solution to the Cauchy problem of the fuzzy fractional differential equation.

2. Preliminaries

We now recall some definitions needed in throughout the paper. Let us denote by RFthe class of fuzzy subsets of the real axis u:R01satisfying the following properties:

  1. uis normal: there exists x0Rwith ux0=1,

  2. uis convex fuzzy set: for all x,tRand 0<λ1, it holds that

    uλx+1λtminuxut,E3

  • uis upper semicontinuous: for any x0R, it holds that

    ux0limxx0ux,E4

  • u0=clxRux>0is compact.

  • Then RFis called the space of fuzzy numbers see [8]. Obviously, RRF. If uis a fuzzy set, we define uα=xRuxαthe α-level (cut) sets of u, with 0<α1. Also, if uRFthen α-cut of udenoted by uα=u1αu2α.

    Lemma 1 see [9] Letu,v:RF01be the fuzzy sets. Thenu=vif and only ifuα=vαfor allα01.

    For u,vRFand λRthe sum u+vand the product λuare defined by

    u+vα=u1α+v1αu2α+v2α,E5
    λuα=λuα=λu1αλu2α,λ0;λu2αλu1α,λ<0,E6

    α01. Additionally if we denote 0̂=χ0, then 0̂RFis a neutral element with respert to +.

    Let d:RF×RFR+0by the following equation:

    duv=supα01dHuαvα,forallu,vRF,E7

    where dHis the Hausdorff metric defined as:

    dHuαvα=maxu1αv1αu2αv2αE8

    The following properties are well-known see [10]:

    du+wv+w=duvandduv=dvu,u,v,wRF,E9
    dkukv=kduv,kR,u,vRFE10
    du+vw+eduw+dve,u,v,w,eRF,E11

    and RFdis a complete metric space.

    Definition 1 The mapping u:0aRFfor some interval 0ais called a fuzzy process. Therefore, its α-level set can be written as follows:

    utα=u1αtu2αt,t0a,α01.E12

    Theorem 1.1 [11] Let u:0aRFbe Seikkala differentiable and denoteutα=u1αtu2αt. Then, the boundary functionu1αtandu2αtare differentiable and

    utα=u1αtu2αt,α01.E13

    Definition 2 [12] Let u:0aRF. The fuzzy integral, denoted by bcutdt,b,c0a, is defined levelwise by the following equation:

    bcutdtα=bcu1αtdtbcu2αtdt,E14

    for all 0α1. In [12], if u:0aRFis continuous, it is fuzzy integrable.

    Theorem 1.2 [13] If uRF, then the following properties hold:

    1. uα2uα1,if0α1α21;E15

  • αk01is a nondecreasing sequence which converges toαthen

  • uα=k1uαk.E16

    Conversely if Aα={u1αu2α;α(0,1]}is a family of closed real intervals verifyingiandii, thenAαdefined a fuzzy numberuRFsuch thatuα=Aα.

    From [1], we have the following theorems:

    Theorem 1.3 There exists a real Banach space Xsuch thatRFcan be the embedding as a convex coneCwith vertex0into X. Furthermore, the following conditions hold:

    1. the embedding jis isometric,

    2. addition in Xinduces addition in RF, i.e, for any u,vRF,

    3. multiplication by a nonnegative real number in Xinduces the corresponding operation in RF, i.e., for any uRF,

    4. C-C is dense in X,

    5. Cis closed.

    3. Fuzzy conformable fractional differentiability and integral

    Definition 3 [4] LetF:0aRFbe a fuzzy function.γthorder “fuzzy conformable fractional derivative” ofFis defined by

    TγFt=limε0+Ft+εt1γFtε=limε0+FtFtεt1γε.E17

    for allt>0,γ01. LetFγtstands forTγFt. Hence

    Fγt=limε0+Ft+εt1γFtε=limε0+FtFtεt1γε.E18

    IfFisγ- differentiable in some0a, andlimt0+Fγtexists, then

    Fγ0=limt0+FγtE19

    and the limits (in the metric d).

    Remark 1 From the definition, it directly follows that ifFisγ-differentiable then the multivalued mappingFαisγ-differentiable for allα01and

    TγFα=Fγtα,E20

    where TγFαis denoted from the conformable fractional derivative of Fαof order γ. The converse result does not hold, since the existence of Hukuhara difference uαvα,α01does not imply the existence of H-difference uv.

    Theorem 1.4 [4] Let γ01.

    If Fis differentiable and Fis γ-differentiable then

    TγFt=t1γF'tE21

    Theorem 1.5 [5, 14] If F:0aRFis γ-differentiable then it is continuous.

    Remark 2 If F:0aRFis γ-differentiable and Fγfor all γ01is continuous, then we denote FC10aRF.

    Theorem 1.6 [5, 14] Let γ01and if F,G:0aRFare γ-differentiable and λRthen

    TγF+Gt=TγFt+TγGtandTγλFt=λTγFt.E22

    Definition 4 [5] LetFC0aRFL10aRF,Define the fuzzy fractional.

    integral fora0andγ01.

    IγaFt=I1atγ1Ft=atFs1γsds,E23

    where the integral is the usual Riemann improper integral.

    Theorem 1.7 [5] TγIγaFt, for ta, where Fis any continuous function in the domain of Iγa.

    Theorem 1.8 [5] Let γ01and Fbe γ-differentiable in 0aand assume that the conformable derivative Fγis integrable over 0a. Then for each s0awe have

    Fs=Fa+IγaFγtE24

    4. Existence and uniqueness solution to fuzzy fractional differential equations

    In this section we state the main results of the paper, i.e. we will concern ourselves with the question of the existence theorem of approximate solutions by using the embedding results on fuzzy number space RFdand we prove the uniqueness theorem of solution for the Cauchy problem of fuzzy fractional differential equations of order γ01.

    4.1 Solution and its approximate solutions

    Assume that F:0a×RFRFis continuous C0a×RFRF. Consider the fractional initial value problem

    Tγut=Ftut,ut0=u0,E25

    where u0RFand γ01.

    From Theorems (1.5), (1.7) and (1.8), it immediately follows:

    Theorem 1.9 A mapping u:0aRFis a solution to the problem (25) if and only if it is continuous and satisfies the integral equation

    ut=u0+t0tsγ1FsusdsE26

    for all t0aand γ01.

    In the following we give the relation between a solution and its approximate solutions.

    We denote Δ0=t0t0+θ×Bu0μwhere θ,μbe two positive real numbers u0RF,Bu0μ=xRFduu0μ.

    Theorem 1.10 Let γ01and FCΔ0RF,η0θ, unC1t0t0+ηB(u0μ)such that

    junγt=jFtunt+Bnt,unt0=u0,BntεnE27
    tt0t0+η,n=1,2,.

    where εn>0,εn0, BntCt0t0+ηX, and js the isometric embedding from RFdonto its range in the Banach space X. For each tt0t0+ηthere exists an β>0such that the H-differences unt+εt1γuntand untuntεt1γexist for all 0ε<βand n=1,2,.If we have

    duntut0E28

    uniform convergence (u.c) for all tt0t0+η,n, then uC1t0t0+ηB(u0μ)and

    Tγut=Ftut,ut0=u0,tt0t0+η.E29

    Proof: By (28) we know that utCt0t0+ηB(u0μ). For fixed t1t0t0+ηand any tt0t0+η,t>t1,denote ε=ht1γ1and γ01

    Gtn=junt1+εt11γjunt1εjFt1unt1Bnt1.E30
    =junt1+hjunt1ht1γ1jFt1unt1Bnt1.E31
    =t11γjuntjunt1tt1jFt1unt1Bnt1.E32

    It is well know that

    limtt1Gtn=junγt1jFt1unt1Bnt1E33
    =junγt1jFt1unt1Bnt1=ΘXE34
    limnGtn=t11γjutjut1tt1jFt1ut1E35

    From FC1Δ0RF,is know that for any ε>0, there exists β1>0such that

    dFtvF(t1ut1)<ε4E36

    whenever t1<t<t1+β1and dvut1<β1with vBu0μTake natural number N>0such hat

    εn<ε4,duntut<β12foranyn>N,tt0t0+ηE37

    Take β>0such that β<β1and

    dutut1<β12E38

    whenever t1<t<t1+β.

    By the definition of Gtnand (27), we have γ01

    junt1+εt11γjunt1εjunγt1=εjFt1unt1E39
    t11γjuntjunt1tt1t11γjun't1=tt1jFt1unt1E40

    We choose ψXsuch that ψ=1and for all γ01

    ψt11γjuntjunt1tt1t11γjunt1E41
    =t11γjuntjunt1tt1t11γjunt1E42

    Let t11γφt=t11γψjunttt1t11γjunt1,consequently

    t11γφ't=t11γψjuntt11γjunt1E43

    hence

    t11γjuntjunt1tt1t11γjunt1E44
    =t11γφtφt1=t11γφt̂tt1E45
    =ψt11γjunt̂junt1tt1E46
    ψt11γjun't̂jun't1tt1E47
    =t11γjunt̂junt1tt1,E48

    where t1t̂t.In view of (40), we have

    Gtnt11γjunt̂junt1,t1t̂t.E49

    From (37) and (38) we know that

    dut̂ut1<β12E50

    and

    dunt̂ut1dunt̂ut̂+dut̂ut1E51
    <β12+β12=β1E52

    Hence by (36) and (49) we have for all γ01.

    Gtnt11γjun't̂jun't1E53
    =jFt̂unt̂+Bnt̂jFt1unt1Bnt1E54
    jFt̂unt̂jFt1ut1E55
    +jFt1ut1jFt1unt1+2εnE56
    djF(t̂unt̂)jF(t1ut1)E57
    +djF(t1ut1)jF(t1unt1)+2εnE58
    <ε4+ε4+2εn<εE59

    whenever n>Nand t1<t<t1+β.

    Let n,and applying (35), we have

    t11γjutjut1tt1jFt1ut1ε,t1<t<t1+β.E60

    On the other hand, from the assumption of Theorem (1.9), there exists an βt10βsuch that the H-differences untunt1exist for all tt1t1+βt1and n=1,2,.

    Now let vnt=untunt1we verify that the fuzzy number-valued sequence vntuniformly converges on t1t1+βt1. In fact, from the assumption duntut0u.c. for all tt0t0+η, we know

    dvntvmt=dvnt+unt1vmt+unt1E61
    duntumt+dumtvmt+unt1E62
    =duntumt+dvmt+umt1vmt+unt1E63
    =duntumt+dumt1unt1E64
    u.ctt1t1+βt1n,m.E65

    Since RFdis complete, there exists a fuzzy number-valued mapping vtsuch that vntu.c to vton t1t1+βt1as n.

    In addition, we have

    dut1+vtutdut1+vtun(t1+vnt1+dun(t1+vntutE66
    dut1+vtut1+vntE67
    + dut1+vntunt1+vnt+duntutE68
    =dvntut+dunt1ut1+duntutE69
    tt1t1+βt1.

    Let n.It follows that

    ut1+vtutforalltt1t1+βt1.E70

    Hence the H-difference utut1exist for all tt1t1+βt1.

    Thus from (60) we have for all γ01.

    dut1+t11γεut1εF(t1ut1)ε,tt1t1+βt1.E71

    So, limε0+ut1+t11γεut1/ε=Ft1ut1.Similarty, we have

    limε0ut1+t11γεut1ε=Ft1ut1.

    Hence uγt1exists and

    uγt1=Ft1ut1.E72

    from t1t0t0+ηis arbitrary, we know that Eq. (29) holds true and uC1t0t0+ηB(u0μ).The proof is concluded.

    Lemma 2 For all tt0t0+η, n=1,2,and γ01.

    If we replace Eq. (27) by

    jun+1t=jFtunt+Bnt,unt0=u0,Bntεn,E73

    retain other assumptions, then the conclusions also hold true.

    Proof: This is completely similar to the proof of Theorem (1.10), hence itis omitted here.

    4.2 Uniqueness solution

    In this section, by using existence theorom of approximate solutions, and the embedding results on fuzzy number space RFd, we give the existence and uniqueness theorem for the Cauchy problem of the fuzzy fractional differential equations of order γ.

    Theorem 1.11

    1. Let FCΔ0RFand dFtu0̂σfor all tuΔ0.

    2. GCt0t0+θ×[0μ]R,Gt00,and 0Gtyσ1,for all tt0t0+θ,0yμsuch that Gtyis noncreasing on ythe fractional initial value problem

      Tγyt=Gtyt,yt0=0E74

    has only the solution yt0on t0t0+θ.

  • dFtuF(tv)Gtduvfor all tu,tvΔ0,and duvμ.

  • Then the Cauchy problem (29) has unique solution uC1t0t0+ηB(u0μ)on t0t0+η,where η=minθμ/σμ/σ1,and the successive iterations

    un+1t=u0+t0tsγ1FsunsdsE75

    uniformly converge to uton t0t0+η.

    Proof: In the proof of Theorem 4.1 in [15], taking the conformable derivative uγfor all γ01, using theorem (1.4) and properties (10), then we obtain the proof of Theorem (1.11).

    Example 1 Let L>0is a constant, taking Gty=Lyin the proof of Theorem (4.2), then obtain the proof of Corollary 4.1 in [15] where σ1=, hence η=minθμ/σ1/L. Then the Cauchy problem (29) has unique solution uC1t0t0+ηB(Δ0μ), and the successive iterations (75) uniformly converge to uton t0t0+η.

    5. Conclusion

    In this work, we introduce the concept of conformable differentiability for fuzzy mappings, enlarging the class of γ-differentiable fuzzy mappings where γ01. Subsequently, by using the γ-differentiable and embedding theorem, we study the Cauchy problem of fuzzy fractional differential equations for the fuzzy valued mappings of a real variable. The advantage of the γ-differentiability being also practically applicable, and we can calculate by this derivative the product of two functions because all fractional derivatives do not satisfy see [4].

    On the other hand, we show and prove the relation between a solution and its approximate solutions to the Cauchy problem of the fuzzy fractional differential equation, and the existence and uniqueness theorem for a solution to the problem (2) are proved.

    For further research, we propose to extend the results of the present paper and to combine them the results in citeref for fuzzy conformable fractional differential equations.

    Conflict of interest

    The authors declare no conflict of interest.

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    Atimad Harir, Said Melliani and Lalla Saadia Chadli (November 12th 2020). Existence, Uniqueness and Approximate Solutions of Fuzzy Fractional Differential Equations [Online First], IntechOpen, DOI: 10.5772/intechopen.94000. Available from:

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