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Existence, Uniqueness and Approximate Solutions of Fuzzy Fractional Differential Equations

Written By

Atimad Harir, Said Melliani and Lalla Saadia Chadli

Reviewed: 11 September 2020 Published: 12 November 2020

DOI: 10.5772/intechopen.94000

From the Edited Volume

Fuzzy Systems - Theory and Applications

Edited by Constantin Volosencu

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Abstract

In this paper, the Cauchy problem of fuzzy fractional differential equations Tγut=Ftut, ut0=u0, with fuzzy conformable fractional derivative (γ-differentiability, where γ∈01) are introduced. We study the existence and uniqueness of solutions and approximate solutions for the fuzzy-valued mappings of a real variable, we prove some results by applying the embedding theorem, and the properties of the fuzzy solution are investigated and developed. Also, we show the relation between a solution and its approximate solutions to the fuzzy fractional differential equations of order γ.

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,E1

where subject to initial condition u0 for 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.

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2. Preliminaries

We now recall some definitions needed in throughout the paper. Let us denote by RF the class of fuzzy subsets of the real axis u:R01 satisfying the following properties:

  1. u is normal: there exists x0R with ux0=1,

  2. u is convex fuzzy set: for all x,tR and 0<λ1, it holds that

    uλx+1λtminuxut,E2

  3. u is upper semicontinuous: for any x0R, it holds that

    ux0limxx0ux,E3

  4. u0=clxRux>0 is compact.

Then RF is called the space of fuzzy numbers see [8]. Obviously, RRF. If u is a fuzzy set, we define uα=xRuxα the α-level (cut) sets of u, with 0<α1. Also, if uRF then α-cut of u denoted by uα=u1αu2α.

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

For u,vRF and λR the sum u+v and the product λu are defined by

u+vα=u1α+v1αu2α+v2α,E4
λuα=λuα=λu1αλu2α,λ0;λu2αλu1α,λ<0,E5

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

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

duv=supα01dHuαvα,forallu,vRF,E6

where dH is the Hausdorff metric defined as:

dHuαvα=maxu1αv1αu2αv2αE7

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

du+wv+w=duvandduv=dvu,u,v,wRF,E8
dkukv=kduv,kR,u,vRFE9
du+vw+eduw+dve,u,v,w,eRF,E10

and RFd is a complete metric space.

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

utα=u1αtu2αt,t0a,α01.E11

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

utα=u1αtu2αt,α01.E12

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

bcutdtα=bcu1αtdtbcu2αtdt,E13

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

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

  1. uα2uα1,if0α1α21;E14

  2. αk01is a nondecreasing sequence which converges toαthen

uα=k1uαk.E15

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 j is isometric,

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

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

  4. C-C is dense in X,

  5. C is closed.

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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.E16

for allt>0,γ01. LetFγtstands forTγFt. Hence

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

IfFisγ- differentiable in some0a, andlimt0+Fγtexists, then

Fγ0=limt0+FγtE18

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,E19

where TγFα is denoted from the conformable fractional derivative of Fα of order γ. The converse result does not hold, since the existence of Hukuhara difference uαvα,α01 does not imply the existence of H-difference uv.

Theorem 1.4 [4] Let γ01.

If F is differentiable and F is γ-differentiable then

TγFt=t1γF'tE20

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

Remark 2 If F:0aRF is γ-differentiable and Fγ for all γ01 is continuous, then we denote FC10aRF.

Theorem 1.6 [5, 14] Let γ01 and if F,G:0aRF are γ-differentiable and λR then

TγF+Gt=TγFt+TγGtandTγλFt=λTγFt.E21

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

integral fora0andγ01.

IγaFt=I1atγ1Ft=atFs1γsds,E22

where the integral is the usual Riemann improper integral.

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

Theorem 1.8 [5] Let γ01 and F be γ-differentiable in 0a and assume that the conformable derivative Fγ is integrable over 0a. Then for each s0a we have

Fs=Fa+IγaFγtE23
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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 RFd and 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×RFRF is continuous C0a×RFRF. Consider the fractional initial value problem

Tγut=Ftut,ut0=u0,E24

where u0RF and γ01.

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

Theorem 1.9 A mapping u:0aRF is a solution to the problem (24) if and only if it is continuous and satisfies the integral equation

ut=u0+t0tsγ1FsusdsE25

for all t0a and γ01.

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

We denote Δ0=t0t0+θ×Bu0μ where θ,μ be two positive real numbers u0RF,Bu0μ=xRFduu0μ.

Theorem 1.10 Let γ01 and FCΔ0RF,η0θ, unC1t0t0+ηB(u0μ) such that

junγt=jFtunt+Bnt,unt0=u0,BntεnE26
tt0t0+η,n=1,2,.

where εn>0,εn0, BntCt0t0+ηX, and j s the isometric embedding from RFd onto its range in the Banach space X. For each tt0t0+η there exists an β>0 such that the H-differences unt+εt1γunt and untuntεt1γ exist for all 0ε<β and n=1,2,. If we have

duntut0E27

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

Tγut=Ftut,ut0=u0,tt0t0+η.E28

Proof: By (27) we know that utCt0t0+ηB(u0μ). For fixed t1t0t0+η and any tt0t0+η,t>t1, denote ε=ht1γ1 and γ01

Gtn=junt1+εt11γjunt1εjFt1unt1Bnt1.E29
=junt1+hjunt1ht1γ1jFt1unt1Bnt1.E30
=t11γjuntjunt1tt1jFt1unt1Bnt1.E31

It is well know that

limtt1Gtn=junγt1jFt1unt1Bnt1E32
=junγt1jFt1unt1Bnt1=ΘXE33
limnGtn=t11γjutjut1tt1jFt1ut1E34

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

dFtvF(t1ut1)<ε4E35

whenever t1<t<t1+β1 and dvut1<β1 with vBu0μ Take natural number N>0 such hat

εn<ε4,duntut<β12foranyn>N,tt0t0+ηE36

Take β>0 such that β<β1 and

dutut1<β12E37

whenever t1<t<t1+β.

By the definition of Gtn and (26), we have γ01

junt1+εt11γjunt1εjunγt1=εjFt1unt1E38
t11γjuntjunt1tt1t11γjun't1=tt1jFt1unt1E39

We choose ψX such that ψ=1 and for all γ01

ψt11γjuntjunt1tt1t11γjunt1E40
=t11γjuntjunt1tt1t11γjunt1E41

Let t11γφt=t11γψjunttt1t11γjunt1, consequently

t11γφ't=t11γψjuntt11γjunt1E42

hence

t11γjuntjunt1tt1t11γjunt1E43
=t11γφtφt1=t11γφt̂tt1E44
=ψt11γjunt̂junt1tt1E45
ψt11γjun't̂jun't1tt1E46
=t11γjunt̂junt1tt1,E47

where t1t̂t. In view of (39), we have

Gtnt11γjunt̂junt1,t1t̂t.E48

From (36) and (37) we know that

dut̂ut1<β12E49

and

dunt̂ut1dunt̂ut̂+dut̂ut1E50
<β12+β12=β1E51

Hence by (35) and (48) we have for all γ01.

Gtnt11γjun't̂jun't1E52
=jFt̂unt̂+Bnt̂jFt1unt1Bnt1E53
jFt̂unt̂jFt1ut1E54
+jFt1ut1jFt1unt1+2εnE55
djF(t̂unt̂)jF(t1ut1)E56
+djF(t1ut1)jF(t1unt1)+2εnE57
<ε4+ε4+2εn<εE58

whenever n>N and t1<t<t1+β.

Let n, and applying (34), we have

t11γjutjut1tt1jFt1ut1ε,t1<t<t1+β.E59

On the other hand, from the assumption of Theorem (1.9), there exists an βt10β such that the H-differences untunt1 exist for all tt1t1+βt1 and n=1,2,.

Now let vnt=untunt1 we verify that the fuzzy number-valued sequence vnt uniformly converges on t1t1+βt1. In fact, from the assumption duntut0 u.c. for all tt0t0+η, we know

dvntvmt=dvnt+unt1vmt+unt1E60
duntumt+dumtvmt+unt1E61
=duntumt+dvmt+umt1vmt+unt1E62
=duntumt+dumt1unt1E63
u.ctt1t1+βt1n,m.E64

Since RFd is complete, there exists a fuzzy number-valued mapping vt such that vnt u.c to vt on t1t1+βt1 as n.

In addition, we have

dut1+vtutdut1+vtun(t1+vnt1+dun(t1+vntutE65
dut1+vtut1+vntE66
+ dut1+vntunt1+vnt+duntutE67
=dvntut+dunt1ut1+duntutE68
tt1t1+βt1.

Let n. It follows that

ut1+vtutforalltt1t1+βt1.E69

Hence the H-difference utut1 exist for all tt1t1+βt1.

Thus from (59) we have for all γ01.

dut1+t11γεut1εF(t1ut1)ε,tt1t1+βt1.E70

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

limε0ut1+t11γεut1ε=Ft1ut1.

Hence uγt1 exists and

uγt1=Ft1ut1.E71

from t1t0t0+η is arbitrary, we know that Eq. (28) 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. (26) by

jun+1t=jFtunt+Bnt,unt0=u0,Bntεn,E72

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 FCΔ0RF and dFtu0̂σ for all tuΔ0.

  2. GCt0t0+θ×[0μ]R,Gt00, and 0Gtyσ1, for all tt0t0+θ,0yμ such that Gty is noncreasing on y the fractional initial value problem

    Tγyt=Gtyt,yt0=0E73

    has only the solution yt0 on t0t0+θ.

  3. dFtuF(tv)Gtduv for all tu,tvΔ0, and duvμ.

Then the Cauchy problem (28) has unique solution uC1t0t0+ηB(u0μ) on t0t0+η, where η=minθμ/σμ/σ1, and the successive iterations

un+1t=u0+t0tsγ1FsunsdsE74

uniformly converge to ut on t0t0+η.

Proof: In the proof of Theorem 4.1 in [15], taking the conformable derivative uγ for all γ01, using theorem (1.4) and properties (9), then we obtain the proof of Theorem (1.11).

Example 1 Let L>0 is a constant, taking Gty=Ly in 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 (28) has unique solution uC1t0t0+ηB(Δ0μ), and the successive iterations (74) uniformly converge to ut on t0t0+η.

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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 (1) 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.

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Conflict of interest

The authors declare no conflict of interest.

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Written By

Atimad Harir, Said Melliani and Lalla Saadia Chadli

Reviewed: 11 September 2020 Published: 12 November 2020