# Notes on Local and Nonlocal Intuitionistic Fuzzy Fractional Boundary Value Problems with Caputo Fractional Derivatives

Authors: Ali El Mfadel; Said Melliani; M’hamed Elomari
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we investigate the existence and uniqueness results of intuitionistic fuzzy local and nonlocal fractional boundary value problems by employing intuitionistic fuzzy fractional calculus and some fixed-point theorems. As an application, we conclude this manuscript by giving an example to illustrate the obtained results.


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## Body

## 1. Introduction

Fuzzy fractional calculus has become a powerful tool with more accurate and successful results in modeling several complex and fuzzy physical phenomena in numerous seemingly diverse and widespread fields of science and engineering. Recently, the theory of fuzzy fractional differential equations was proposed to handle uncertainty due to incomplete information that appears in many mathematical or computer models of some deterministic real-world phenomena. When a real physical phenomenon is modeled by a fractional initial value problem, we cannot usually be sure that the model is perfect. For example, the initial value of this problem may not be known precisely. In order to get a perfect model under a precise initial condition, Agarwal et al., in [1], proposed the concept of fuzzy solutions for fractional differential equations with uncertainty. Arshad and Lupulescu, in [2], proved some results on the existence and uniqueness of solutions for fuzzy fractional differential equations. Later, Alikhani and Bahrami [3] proved the existence and uniqueness results for nonlinear fuzzy fractional integral and integrodifferential equations by using the method of upper and lower solutions. The authors in [4, 5] discussed the concepts about generalized Hukuhara fractional Riemann–Liouville and Caputo differentiability of fuzzy-valued functions. The equivalence between fuzzy fractional differential equation and fuzzy fractional integral equation was discussed in [6]. Ngo et al., in [7], proved the existence and uniqueness results of solutions for initial value problem under fuzzy Caputo–Katugampola fractional derivatives. For many basic works related to the nonlinear ordinary differential equations and the fuzzy fractional differential equations, we refer the readers to [8–13] and references therein.Motivated by the results mentioned above and by using the intuitionistic fuzzy sets theory introduced by Atanassov, in [14], we study the existence and uniqueness results for the following intuitionistic fuzzy local and nonlocal fractional boundary value problems:(1)cDαXt=Ft,Xt,t∈0,T,X0=aXT,(2)cDαXt=Ft,Xt,t∈0,T,X0=GX,where Dαc is the Caputo derivative of Xt at order 0<α<1 and F:0,T×IF1⟶IF1 and G:C0,T,IF1⟶IF1 are intuitionistic fuzzy continuous functions.The spacesIF1 and C0,T,IF1 will be defined after.Our paper is organized as follows.Section 2 gives some basic definitions, lemmas, and theorems as preliminaries of intuitionistic fuzzy sets theory. The existence results for the intuitionistic fuzzy local and nonlocal fractional boundary value problems are given in Section 3 and Section 4. Illustrative example is presented in Section 5, followed by conclusion and future works in Section 6.


## 2. Preliminaries

Fuzzy set theory was introduced by Zadeh [15], and it is an extension of the classical crisp logic into a multivariate form. Atanassov generalizes this concept to intuitionistic fuzzy sets (IFSs) [14], and later, there has been much progress in the study of IFSs. As a special case of intuitionistic fuzzy sets, intuitionistic fuzzy numbers were introduced by Xu [16].We denote by(3)IFℝ=u,v:ℝ⟶0,12,0≤ux+vx≤1.Definition 1 (see [17]).
An elementu,v∈IFℝ is called an intuitionistic fuzzy number if it satisfies the following conditions:(1)
u,v is normal, i.e., there exists x0,x1∈ℝ such that ux0=1 and vx1=1(2)
u is fuzzy convex and v is fuzzy concave(3)
u is upper semicontinuous and v is lower semicontinuous(4)
suppu,v=x∈ℝ:vx<1¯ is bounded
We denote byIF1 the collection of all intuitionistic fuzzy numbers.
Letα∈0,1 and u,v∈IF1, and we define the upper and lower α−cuts of u,v, respectively, by(4)u,vα=x∈ℝ:vx≤1−α,u,vα=x∈ℝ:ux≥α.
We define also the intuitionistic fuzzy zero denoted by0IF∈IF1 as follows:(5)0IFt=1,0ift=0,0,1ift≠0.Definition 2 (see [17]).
Letu1,v1, u2,v2∈IF1, λ∈ℝ, and α∈0,1; then,(1)
u1,v1⊕u2,v2z=supz=x+yminu1x,u2y,infz=x+ymaxu1x,u2y(2)
λu1,v1=λu1,λv1ifλ≠0(3)
λu1,v1=0IFifλ=0(4)
u1,v1⊕u2,v2α=u1,v1α+u2,v2α(5)
〈u1,v1〉⊕〈u2,v2α=u1,v1α+u2,v2α(6)
λu1,v1α=λu1,v1α(7)
λu1,v1α=λu1,v1α
Letu,v∈IF1 and α∈0,1; then, we define the following sets:(6)u,vl+α=infx∈ℝ:ux≥α,u,vr+α=supx∈ℝ:ux≥α,u,vl−α=infx∈ℝ:vx≤1−α,u,vr−α=supx∈ℝ:vx≤1−α.Remark 1.
Letu,v∈IF1 and α∈0,1; then, we have(7)u,vα=u,vl−α,u,vr−α,u,vα=u,vl+α,u,vr+α.Definition 3 (see [17]).
Letu,v∈IF1 and α∈0,1, and we define the diameter of upper and lower α−cuts of u,v, respectively, as follows:(8)du,vα=u,vr−α−u,vl−α,du,vα=u,vr+α−u,vl+α.Proposition 1.
Letα,β∈0,1 and u,v∈IF1; then,(1)
u,vα⊂u,vα(2)
u,vαetu,vα are nonempty compact convex sets(3)
Ifα≤β, then u,vβ⊂u,vα and u,vβ⊂u,vα(4)
Ifαn↗α, then u,vα=∩nu,vαn and u,vα=∩nu,vαn.Conversely, letα∈0,1, and we put(9)Mα=x∈ℝ:ux≥α,Mα=x∈ℝ:vx≤1−α.Lemma 1 (see [17]).
LetMα:α∈0,1 and Mα:α∈0,1 be two subset of ℝ verify (1)–(4) of Proposition 1; if u and v are defined by(10)ux=0ifx∉M0,supα∈0,1:x∈Mαifx∈M0,vx=1ifx∉M0,1−supα∈0,1:x∈Mαifx∈M0,then u,v∈IF1.Lemma 2 (see [4]).
LetI be a dense subset in 0,1.
Ifu,vα=w,zα and u,vα=w,zα,∀α∈I, then u,v=w,z.Definition 4 (see [18]).
Letu1,v1,u2,v2∈IF1; if there exists w,z∈IF1 such that(11)u1,v1=u2,v2+w,z,then w,z is called Hukuhara difference of u1,v1 and u2,v2 denoted by u1,v1⊖Hu2,v2.Definition 5 (see [18, 19]).
The generalized Hukuhara difference of two intuitionistic fuzzy numberu1,v1,u2,v2∈IF1 is as follows:(12)u2,v2⊖gHu1,v1=u3,v3⟺iu2,v2=u1,v1+u3,v3,oriiu1,v1=u2,v2+−1u3,v3.Definition 6 (see [17]).
Letf:a,b⟶IF1 and t0∈a,b.
We say thatf is generalized Hukuhara differentiable at t0 if there exists f′t0∈IF1 such that(13)f′t0=limh⟶0+ft0+h⊖gHft0h=limh⟶0−ft0⊖gHft0−hh.Definition 7 (see [17]).
A functionF:a,b⟶IF1 is strongly measurable if the set-valued mappings Fα:a,b⟶PKℝ defined by Fαt=Ftα and Fα:a,b⟶PKℝ defined by Fαt=Ftα are Lebesgue measurable, for all α∈0,1, where PKℝ is the set of all closed and bounded intervals of ℝ.Definition 8 (see [17, 19]).
LetF:a,b⟶IF1. We say that F is integrable on a,b if there exists u,v∈IF1 such that, for each α∈0,1,(14)∫abFtdtα=∫abftdt|f:a,b⟶ℝis a measurable selection forFα,∫abFtdtα=∫abftdt|f:a,b⟶ℝis a measurable selection forFα,u,vα=∫abFtdtα,u,vα=∫abFtdtα,and we write ∫abFtdt=u,v.
Letdp:IF1×IF1⟶0,+∞ be a mapping defined by(15)dpu,v,w,z=14∫01u,vr+α−w,zr+αpdα+14∫01u,vl+α−w,zl+αpdα+14∫01u,vr−α−w,zr−αpdα+14∫01u,vl−α−w,zl−αpdα1/pforp∈1,+∞.andd∞u,v,w,z=14sup0≤α≤1u,vr+α−w,zr+α+14sup0≤α≤1u,vl+α−w,zl+α+14sup0≤α≤1u,vr−α−w,zr−α+14sup0≤α≤1u,vl−α−w,zl−α.
Then, we have the following result.Proposition 2 (see [20]).
The spaceIF1,dp is an intuitionistic fuzzy complete metric space ∀p∈1,+∞.
In the following sections, we will need some notations and definitions.(i)
We denote byC0,T,IF1 the space of all intuitionistic fuzzy continuous functions from 0,T to IF1(ii)
We denote byL0,T,IF1 the space of all intuitionistic fuzzy integrable functions on 0,TRemark 2.
Letx,y∈C0,T,IF1.
It is easy to see that the spaceC0,T,IF1,Dp is a Banach space, where(16)Dpx,y=sups∈0,Tdpxs,ys.


### 2.1. Fractional Integral and Fractional Derivatives of an Intuitionistic Fuzzy Function

Definition 9.
(see [20]). Let Ft∈L0,T,IF1.
The intuitionistic fuzzy fractional integral of orderq∈0,1 of F, denoted by(17)IqFt=1Γq∫0tt−sq−1Fsds,is defined by(18)IqFtα=IαFl−t;α,IqFr−t;α,IqFtα=IαFl+t;α,IqFr+t;α,where Γ. is the Euler gamma function.Proposition 3 (see [20]).
LetF,G∈L0,T,IF1, q∈0,1 and a∈IF1; then, we have(1)
IqaFt=aIqFt(2)
IqF+Gt=IqFt+IqGt(3)
Iq1Iq2Ft=Iq1+q2Ft, where q1,q2∈0,12Definition 10 (see [20]).
LetF∈C0,T,IF1∩L0,T,IF1.
The functionF is called intuitionistic fuzzy Caputo fractional differentiable of order 0<q<1 at x if there exists an element DcqFx∈IF1 such that(19)DcqFt=1Γq∫0tt−sq−1F′sds.Example 1.
Consider the intuitionistic fuzzy functionu,vt=tC, where C=a1;a2;a3;a4;a1′;a2;a3;a4′ is a trapezoidal intuitionistic fuzzy number and 0<q<1.
In this example, we calculate the intuitionistic fuzzy Caputo fractional derivative of the functionu,vt. For this purpose, we start by giving the gH-derivative of u,vt as follows:(20)u,v′t=limh⟶0u,vt+h⊖gHu,vth=limh⟶0t+hC⊖gHtCh=C.
This implies that(21)u,v′t=C.
SinceCα=a2−αa2−a1′,a3+αa4−a3 and Cα=a1+αa2−a1,a4−αa4′−a3, then we have(22)Dcqu,vtα=I1−qu,v′tα=I1−qCα,Dcqu,vtα=I1−qu,v′tα=I1−qCα,I1−qCα=1Γq∫0tt−sq−1a2−αa2−a1′,a3+αa4−a3ds,I1−qCα=1Γq∫0tt−sq−1a1+αa2−a1,a4−αa4′−a3ds,I1−qCα=tqΓq×qa2−αa2−a1′,a3+αa4−a3,I1−qCα=tqΓq×qa1+αa2−a1,a4−αa4′−a3,I1−qCα=tqΓq+1Cα,I1−qCα=tqΓq+1Cα.
Thus,(23)Dcqu,vt=tqΓq+1C,Dcqu,vt=tq−1Γq+1u,vt.Theorem 1 (Schaefer’s fixed-point theorem (see [21])).
LetP be a continuous and compact mapping of a Banach space X into itself such that the set(24)x∈X:x=λPxforsome0≤λ≤1is bounded; then, mapping P has a fixed point.


## 2.1. Fractional Integral and Fractional Derivatives of an Intuitionistic Fuzzy Function

Definition 9.
(see [20]). Let Ft∈L0,T,IF1.
The intuitionistic fuzzy fractional integral of orderq∈0,1 of F, denoted by(17)IqFt=1Γq∫0tt−sq−1Fsds,is defined by(18)IqFtα=IαFl−t;α,IqFr−t;α,IqFtα=IαFl+t;α,IqFr+t;α,where Γ. is the Euler gamma function.Proposition 3 (see [20]).
LetF,G∈L0,T,IF1, q∈0,1 and a∈IF1; then, we have(1)
IqaFt=aIqFt(2)
IqF+Gt=IqFt+IqGt(3)
Iq1Iq2Ft=Iq1+q2Ft, where q1,q2∈0,12Definition 10 (see [20]).
LetF∈C0,T,IF1∩L0,T,IF1.
The functionF is called intuitionistic fuzzy Caputo fractional differentiable of order 0<q<1 at x if there exists an element DcqFx∈IF1 such that(19)DcqFt=1Γq∫0tt−sq−1F′sds.Example 1.
Consider the intuitionistic fuzzy functionu,vt=tC, where C=a1;a2;a3;a4;a1′;a2;a3;a4′ is a trapezoidal intuitionistic fuzzy number and 0<q<1.
In this example, we calculate the intuitionistic fuzzy Caputo fractional derivative of the functionu,vt. For this purpose, we start by giving the gH-derivative of u,vt as follows:(20)u,v′t=limh⟶0u,vt+h⊖gHu,vth=limh⟶0t+hC⊖gHtCh=C.
This implies that(21)u,v′t=C.
SinceCα=a2−αa2−a1′,a3+αa4−a3 and Cα=a1+αa2−a1,a4−αa4′−a3, then we have(22)Dcqu,vtα=I1−qu,v′tα=I1−qCα,Dcqu,vtα=I1−qu,v′tα=I1−qCα,I1−qCα=1Γq∫0tt−sq−1a2−αa2−a1′,a3+αa4−a3ds,I1−qCα=1Γq∫0tt−sq−1a1+αa2−a1,a4−αa4′−a3ds,I1−qCα=tqΓq×qa2−αa2−a1′,a3+αa4−a3,I1−qCα=tqΓq×qa1+αa2−a1,a4−αa4′−a3,I1−qCα=tqΓq+1Cα,I1−qCα=tqΓq+1Cα.
Thus,(23)Dcqu,vt=tqΓq+1C,Dcqu,vt=tq−1Γq+1u,vt.Theorem 1 (Schaefer’s fixed-point theorem (see [21])).
LetP be a continuous and compact mapping of a Banach space X into itself such that the set(24)x∈X:x=λPxforsome0≤λ≤1is bounded; then, mapping P has a fixed point.


## 3. Intuitionistic Fuzzy Local Fractional Boundary Value Problems

Definition 11.
A functionX:0,T⟶IF1 is said to be a solution of problem (1) if X∈C10,T,IF1 such that X0=aXT and DαXt=Ft,Xt.Definition 12 (see [6]).
An intuitionistic fuzzy functionX:0,T⟶IF1 is called d-increasing (d-decreasing) on 0,T if, for every r∈0,1, the real function t⟶dXtr∪Xtr is nondecreasing (nonincreasing), respectively.Remark 3.
If the functionX:0,T⟶IF1 is d-increasing or d-decreasing on 0,T, then we say that Xt is d-monotone on 0,T.
For the existence of solutions for problem (1), we need the following lemma.Lemma 3 (see [6]).
Letα∈0,1 and H:0,T⟶IF1 be continuous such that t⟶IαHt is d-increasing function on 0,T.
A d-monotone intuitionistic fuzzy functionXt∈C0,T,IF1 is a solution of the fractional integral equation,(25)Xt⊖gHX0=1Γα∫0tt−sα−1Hsds,if and only if it is a solution of the following initial value problem:(26)cDαXt=Ht;t∈0,T,X0=X0.As a consequence of Lemma3, we have the following result.Lemma 4.
LetF:0,T×IF1⟶IF1 be continuous such that t⟶IαFt,Xt is d-increasing function on 0,T.
A d-monotone intuitionistic fuzzy functionXt is a solution of the fractional integral equation,(27)Xt⊖gHa1−aΓα∫0TT−sα−1Fs,Xsds=1Γα∫0tt−sα−1Fs,Xsds,if and only if is a solution of the fractional boundary value problem (1).Theorem 2.
Assume that there exists a positive constantk such that(28)dpFt,X,Ft,Y<kdpX,Y,for eachX,Y∈C0,T,IF1andt∈0,T.
In addition, if(29)kTα1+a/1−aΓα+1<1,then problem (1) has a unique solution.Proof.
For this purpose, we transform problem (1) into a fixed-point problem defined on 0,T.
LetP:C0,T,IF1⟶C0,T,IF1 be the operator defined as follows:(30)PXt⊖gHa1−aΓα∫0TT−sα−1Fs,Xsds=1Γα∫0tt−sα−1Fs,Xsds.
LetX,Y∈C0,T,IF1; then, we have(31)dpPXt,PYt≤1Γα∫0tt−sα−1dpFt,Xt,Ft,Ytds+a1−aΓα∫0TT−sα−1dpFs,Xs,Fs,Ysds,dpPXt,PYt≤kΓα∫0tt−sα−1dpXt,Ytds+ka1−aΓα∫0TT−sα−1dpXs,Ysds,dpPXt,PYt≤kΓαsups∈0,TdpXs,Ys∫0tt−sα−1ds+ka1−aΓαsups∈0,TdpXs,Ys∫0TT−sα−1ds,dpPXt,PYt≤kTα1+a/1−aαΓαsups∈0,TdpXs,Ys,dpPXt,PYt≤kTα1+a/1−aΓα+1sups∈0,TdpXs,Ys.
Thus,(32)sups∈0,TdpPXt,PYt≤kTα1+a/1−aΓα+1sups∈0,TdpXt,Yt,DpPX,PY≤kTα1+a/1−aΓα+1DpX,Y.
As a consequence of Banach fixed-point theorem, we can deduce that the operatorP has a unique fixed point X which is the solution of problem (1).Theorem 3.
Assume that(H1) The intuitionistic fuzzy function F:0,T×IF1⟶IF1 is continuous.(H2) There exists a positive constant M such that(33)dpFt,X,0IF<M∀t,X∈0,T×C0,T,IF1.Then, problem (1) has at least one solution.Proof.
To show that problem (1) has at least one solution defined on 0,T, we use Schaefer’s fixed-point theorem [22]. For this purpose, to prove that the operator P defined above has a fixed point, the proof of this theorem will be given in several steps.Step 1:
let us show thatP is continuous.
LetXnn⊂C0,T,IF1 such that Xn converges to X in C0,T,IF1 and t∈0,T; we have(34)dpPXnt,PXt≤1Γα∫0tt−sα−1dpFt,Xnt,Ft,Xtds+a1−aΓα∫0TT−sα−1dpFs,Xns,Fs,Xsds,dpPXnt,PXt≤1Γα∫0tt−sα−1sups∈0,TdpFs,Xns,Fs,Xsds+a1−aΓα∫0TT−sα−1sups∈0,TdpFs,Xns,Fs,Xsds,dpPXnt,PXt≤Tα1+a/1−aαΓαsups∈0,TdpFs,Xns,Fs,Xs,dpPXnt,PXt≤Tα1+a/1−aΓα+1sups∈0,TdpFs,Xns,Fs,Xs.
Thus,(35)sups∈0,TdpPXnt,PXt≤Tα1+a/1−aΓα+1sups∈0,TdpFt,Xnt,Ft,Xt.
SinceF is continuous, we obtain(36)limn⟶∞supt∈0,TdpPXnt,PXt≤Tα1+a/1−aΓα+1limn⟶∞supt∈0,TdpFt,Xnt,Ft,Xt=0,which implies that(37)limn⟶∞supt∈0,TdpPXnt,PXt=0.
Finally, the operatorP is continuous on C0,T,IF1.Step 2:
let us also show thatPBρ is bounded and equicontinuous on C0,T,IF1.
It is enough to prove that, for anyρ>0, there exists a positive real constant δ>0 such that, for each(38)X∈Bρ=X∈C0,T,IF1:supt∈0,TdpXt,0IF≤ρ,we have(39)supt∈0,TdpPXt,0IF<δ.
Lett∈0,T, and we have(40)dpPXt,0IF≤1Γα∫0tt−sα−1dpFt,Xt,0IFds+a1−aΓα∫0TT−sα−1dpFs,Xs,0IFds.
By usingH2, we have(41)dpPXt0IF≤MΓα∫0tt−sα−1ds+Ma1−aΓα∫0TT−sα−1ds,dpPXt0IF≤MTααΓα+MTαa1−aαΓα,supt∈0,TdpPXt0IF≤MTααΓα+MTαa1−aαΓα≔δ.
Thus, we have thatPBρ⊂Bδ, which implies that PBρ is bounded.
Lett∈0,T.
LetX∈Bρ and t1,t2∈0,T such that t1<t2; then, we have(42)dpPXt1,PXt2≤1Γαdp∫0t1t1−sα−1Fs,Xsds,∫0t2t2−sα−1Fs,Xsds.
Since(43)∫0t2t2−sα−1Fs,Xsds=∫0t1t2−sα−1Fs,Xsds+∫t1t2t2−sα−1Fs,Xsds,then(44)dpPXt1,PXt2≤1Γα∫t0t1t1−sα−1−t1−sα−1dpFs,Xs,0IF1+∫t1t2t2−sα−1dpFs,Xsds,0IF1ds.
By usingH2, it follows that(45)dpPXt1,PXt2≤MΓα∫t0t1t1−sα−1−t1−sα−1ds+MΓα∫t1t1t2t2−sα−1ds,dpPXt1,PXt2≤MαΓαt2−t1α+t1α−t2α+MαΓαt2−t1α.
Thus,(46)dpPXt1,PXt2≤MΓα+1t2−t1α+t2α−t1α.
Hence,(47)limt1⟶t2dpTXt1,PXt2=0,which shows that PBρ is equicontinuous; by using Arzela–Ascoli theorem [23], we deduce that PBρ is relatively compact, and we can conclude from Step 1 and Step 2 that the operator P is continuous and completely continuous.Step 3
: let us we show that the setEλ, defined as follows,(48)Eλ=X∈C0,T,IF1:X=λPXfor some0≤λ≤1,is bounded.
LetX∈Eλ; then, X=λPX, for some λ∈0,1. For each t∈0,T, we have(49)Xt⊖gHλa1−aΓα∫0TT−sα−1Fs,Xsds=λΓα∫0tt−sα−1Fs,Xsds.
Lett∈0,T, and by using H3, we have(50)dpPXt,0IF≤1Γα∫0tt−sα−1dpFt,Xt,0IFds+a1−aΓα∫0TT−sα−1dpFs,Xs,0IFds,dpPXt,0IF≤MΓα∫0tt−sα−1ds+Ma1−aΓα∫0TT−sα−1ds,dpPXt,0IF≤MTααΓα+MTαa1−aαΓα,supt∈0,TdpPXt,0IF≤MTαΓα+1+aMTα1−aΓα+1<∞,which shows that the set Eλ is bounded.
Finally, as a consequence of Schaefer’s fixed-point theorem, we deduce thatP has a fixed point which is a solution of problem (1).


## 4. Intuitionistic Fuzzy Nonlocal Fractional Boundary Value Problems

In this section, we present some existence and uniqueness results for nonlocal intuitionistic fuzzy fractional problem (2). The nonlocal conditions were initiated by Byszewski [24] when he proved the existence and uniqueness of mild and classical solutions of nonlocal Cauchy problems. Some physical phenomena are described the nonlocal condition; see [25, 26], for more details.Theorem 5.
Assume that(1)
The intuitionistic fuzzy functionF:0,T×IF1⟶IF1 is continuous.(2)
There exists a positive constantk such that(51)dpFt,X,Ft,Y<kdpX,YX,Y∈C0,T,IF1andt∈0,T.(3)
There exists a positive constantL such that(52)dpGU,GV≤LdpU,Vfor eachU,V∈C0,T,IF1.
In addition, if(53)L+kTαΓα+1<1,then problem (2) has a unique solution.Proof.
For this purpose, we transform problem (2) into a fixed-point problem defined on 0,T.
LetB:C0,T,IF1⟶C0,T,IF1 be the operator defined as follows:(54)BXt⊖gHGX=1Γα∫0tt−sα−1Fs,Xsds.
LetX,Y∈C0,T,IF1; then, we have(55)dpBXt,BYt≤dpGXt,GYt+1Γα∫0tt−sα−1dpFt,Xt,Ft,Ytds,dpBXt,BYt≤LdpXt,Yt+1Γα∫0tt−sα−1dpXt,Ytds,dpBXt,BYt≤LdpXt,Yt+1Γα∫0tt−sα−1dssups∈0,TdpXs,Ys,supt∈0,TdpBXt,BYt≤Lsupt∈0,TdpXt,Yt+kTααΓαdssupt∈0,TdpXt,Yt,supt∈0,TdpBXt,BYt≤L+kTαΓα+1supt∈0,TdpXt,Yt.
Thus,(56)DpBX,BY≤L+kTαΓα+1DpX,Y.
By using Banach fixed-point theorem, we can deduce that the operatorB has a unique fixed point X which is the solution of problem (2).Theorem 6.
Assume that(1)
The intuitionistic fuzzy functionF:0,T×IF1⟶IF1 is continuous.(2)
There exists a positive constantM1 such that(57)dpFt,X,0IF≤M1∀t,x∈0,T×IF1.(3)
There exists a positive constantM2 such that(58)dpGX,0IF≤M2∀x∈C0,T,IF1.Then, problem (2) has at least one solution.Proof.
The proof of this theorem is similar to that of Theorem3.


## 5. Illustrative Example

In this section, we give an example to illustrate our main results.Consider the following fractional boundary value problem:(59)cD2/3Xt=e−t9+etXt,t∈0,1,X0=−1X1,where t⟼et is the exponential function.LetFt,Xt=e−t/9+etXt and t∈0,1; then, we have(60)dpFt,Xt,Ft,Yt=dpe−t9+etXt,e−t9+etYt,dpFt,Xt,Ft,Yt≤e−t9+etdpXt,Yt,dFt,Xt,Ft,Yt≤110dpXt,Yt.Hence, the first condition in Theorem2 holds with k=1/10.It remains to check condition (29) with T=1, a=−1, and α=2/3; for this purpose, we have(61)Γα+1=Γ23+1=Γ53=0.89.Thus,(62)3/2kΓα+1=0.150.89=0.1685393<1.Finally, all the conditions of Theorem2 are satisfied; thus, we conclude that the intuitionistic fuzzy fractional boundary value problem (59) has a unique solution on 0,1.


## 6. Conclusion and Future Work

In this paper, we establish the existence and uniqueness results of intuitionistic fuzzy local and nonlocal boundary value problems involving Caputo fractional derivative. The existence theorems are proved by using Banach fixed-point theorem and Schaefer’s fixed-point theorem. As an application, an example is presented to illustrate the applicability of the obtained results.Our future work is to study the existence and uniqueness results of intuitionistic fuzzy hybrid fractional differential equations [27–31].



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