IJDEInternational Journal of Differential Equations1687-96511687-9643Hindawi10.1155/2020/88360118836011Research ArticleFuzzy Conformable Fractional Semigroups of Operatorshttps://orcid.org/0000-0003-3603-9268HarirAtimadMellianiSaidChadliLalla SaadiaScapellatoAndreaLaboratory of Applied Mathematics and Scientific ComputingSultan Moulay Slimane UniversityP.O. Box 523Beni Mellal 23000Moroccouniversitesms.com202041120202020682020161020202110202041120202020Copyright © 2020 Atimad Harir et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we introduce a fuzzy fractional semigroup of operators whose generator will be the fuzzy fractional derivative of the fuzzy semigroup at t=0. We establish some of their proprieties and some results about the solution of fuzzy fractional Cauchy problem.

1. Introduction

Fractional semigroups are related to the problem of fractional powers of operators initiated first by Bochner . Balakrishnan  studied the problem of fractional powers of closed operators and the semigroups generated by them. The fractional Cauchy problem associated with a Feller semigroup was studied by Popescu . Abdeljawad et al.  studied the fractional semigroup of operators. The semigroup generated by linear operators of a fuzzy-valued function was introduced by Gal and Gal . Kaleva  introduced a nonlinear semigroup generated by a nonlinear function.In the last few decades, fractional differentiation has been used by applied scientists for solving several fractional differential equations and they proved that the fractional calculus is very useful in several fields of applications and real-life problems such as, but certainly not limited, in physics (quantum mechanics, thermodynamics, and solid-state physics), chemistry, theoretical biology and ecology, economics, engineering, signal and image processing, electric control theory, viscoelasticity, fiber optics, stochastic-based, finance, tortoise walk, Baggs and Freedman model, normal distribution kernel, time-fractional nonlinear dispersive PDEs, fractional multipantograph system, time-fractional generalized Fisher equation and time-fractional km,n equation, and nonlinear time-fractional Schrodinger equations .

The concept of fuzzy fractional derivative was introduced by  and developed by , but these researchers tried to put a definition of a fuzzy fractional derivative. Most of them used an integral from the fuzzy fractional derivative. Two of which are the most popular ones, Riemann–Liouville definition and Caputo definition. All definitions mentioned above satisfy the property that the fuzzy fractional derivative is linear. This is the only property inherited from the first fuzzy derivative by all of the definitions. The obtained fractional derivatives in this calculus seemed complicated and lost some of the basic properties that usual derivatives have such as the product rule and the chain rule. However, the semigroup properties of these fractional operators behave well in some cases. Recently, Harir et al.  defined a new well-behaved simple fractional derivative called “the fuzzy conformable fractional derivative” depending just on the basic limit definition of the derivative. They proved the product rule and the fractional mean value theorem and solved some (conformable) fractional differential equations .

Here, we introduce the fuzzy fractional semigroups of operators associated with the fuzzy conformable fractional derivative, for proving to be a very fruitful tool to solve differential equations. Then, we show that this semigroup is a solution to the fuzzy fractional Cauchy problem xqt=fxt,x0=x0, and q0,1 according to the fuzzy conformable fractional derivative which was introduced in .

2. Preliminaries

Let us denote by =u:0,1 the class of fuzzy subsets of the real axis satisfying the following properties :

u is normal, i.e., there exists an x0 such that ux0=1,

u is the fuzzy convex, i.e., for x,y and 0<λ1,

(1)uλx+1λyminux,uy.

u is upper semicontinuous,

u0=clx|ux>0 is compact.

Then, is called the space of fuzzy numbers. Obviously, . For 0<α1, denote uα=x|uxα, then from (i) to (iv), it follows that the α-level sets uαPK, for all 0α1, are a closed bounded interval which we denote by uα=u1α,u2α.

Here, PK denotes the family of all nonempty compact convex subsets of and defines the addition and scalar multiplication in PK as usual.

Lemma 1 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

Let u,v:0,1 be the fuzzy sets. Then, u=v if and only if uα=vα, for all α0,1.

The following arithmetic operations on fuzzy numbers are well known and frequently used below. If u,v, then(2)u+vα=u1α+v1α,u2α+v2α,uvα=u1αv2α,u2αv1α,λuα=λuα=λu1α,λu2α,if λ0,λu2α,λu1α,if λ<0.

For u,v, if there exists w such that u=v+w, then w is the Hukuhara difference of u and v denoted by uv.

Let us define d:×+0 by the equation(3)du,v=supα0,1dHuα,vα,for all u,v,where dH is the Hausdorff metric defined in PK.(4)dHuα,vα=maxu1αv1α,u2αv2α.

Theorem 1 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

,d is a complete metric space.

We list the following properties of du,v:(5)du+w,v+w=du,v,du,v=dv,u,dku,kv=kdu,v,du,vdu,w+dw,v,for all u,v,w and λ.

Theorem 2 (see [<xref ref-type="bibr" rid="B24">24</xref>]).

There exists a real Banach space X such that can be the embedded as a convex cone C with vertex 0 in X. Furthermore, the following conditions hold true:

The embedding j is isometric,

The multiplication by a nonnegative real number in X induces the corresponding operation in ,

CC=ab/a,b is dense in X,

C is closed.

Remark 1.

Let j˜:X as j˜u=j1u,u. It verifies the following properties: j˜uj˜v=du,v, j˜su+tv=sj˜u+tj˜v, for all u,v, t,s0j˜j=C, since 1=.

3. Fuzzy <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M68"><mml:mi>q</mml:mi></mml:math></inline-formula>-Semigroup of OperatorsDefinition 1 (see [<xref ref-type="bibr" rid="B20">20</xref>]).

Let F:0,a be a fuzzy function. qth order “fuzzy conformable fractional derivative” of F is defined by (where the limit is taken in the metric space ,d).(6)TqFt=limε0+Ft+εt1qFtϵ,=limε0+FtFtεt1qε,for all t>0,q0,1. Let Fqt stand for TqFt. Hence,(7)Fqt=limε0+Ft+εt1qFtε=limε0+FtFtεt1qε.

If F is q-differentiable in some 0,a and limt0+Fqt exists, then(8)Fq0=limt0+Fqt.

Definition 2.

Let FC0,a,L10,a,. Define the fuzzy fractional integral for q0,1.(9)IqFt=Itq1Ft=0tFxx1qdx,where the integral is the usual Riemann improper integral.

Definition 3.

Let q0,a, for any a>0. A family Ttt0 of operators from is called a fuzzy fractional q-semigroup (or fuzzy q-semigroup) of operators if

T0=I, where I is the identity mapping on ,

Ts+t1/q=Ts1/qTt1/q, for all s,t0.

Definition 4.

A q-semigroup Tt is called a c0-q-semigroup if

The function g:0,, defined by gt=Ttx, is continuous at t=0, for all x, i.e.,

(10)limt0+Ttx=x.

There exist constants w0 and M1 such that dTtx,TtyMewtq/qdx,y, for all t0,x,y.

Example 1.

Define on the linear operator Ttx=e2tx. Then, Ttt0 is a fuzzy 1/2-semigroup. Indeed

T0=I,T0x=x, for all x,

For t,s0,x,

(11)Ts+t2x=e2s+t2x=e2s+tx=e2se2tx,=Ts2e2tx=Ts2Tt2x.

For t,s0,x,dTtx,x=de2tx,x, then e2t10, and then using Remark 1, we deduce that e2t1x+x=e2tx. Therefore, the Hukuhara difference e2txxTtxx exists and we have

(12)Ttxx=e2txx=e2t1x.

Then,

(13)dTtx,x=de2txx,0˜=de2txx,0˜,=de2t1x,0˜=e2t1dx,0˜.

Since limt0+e2t1=0, then limt0+Ttx=x.

For t0,x,y,dTtx,Tty=de2tx,e2ty=e2tdx,y. Consequently, Ttt0 is a fuzzy c0-q-semigroup on .

Definition 5.

The conformable q-derivative of Tt at t=0 is called the q-infinitesimal generator of the fuzzy q-semigroup Ttt0, with domain equals(14)DA=x:limt0+Tqtx exists.

We will write A for such generator.

Lemma 2.

Let A: and A1=jAj1: CC tow the operator.

A is the operator of the fuzzy q-semigroup Ttt0 on if and only if A1 is the operator of the q-semigroup T1tt0 defining on the convex closed set C and T1=jTtj1.

By using Definition 5, the proof is similar to the proof of Lemma 5 in  and is omitted.

Theorem 3.

Let Ttt0 be a c0-q-semigroup with infinitesimal generator A,0<q1. Then, for all x such that TtxDA, for all t0; the mapping tTtx is q-differentiable and(15)Tqtx=ATtx,t0.

Proof.

Let q0,1 and x, for t0, and we have(16)Tt+s1/qx=Tt1/qTs1/qx.

Since TtxDA, then(17)Tqtx=limε0Tt+εt1qxTtxε,=limε0j1T1t+εt1qjxj1T1tjxε,=j1limε0T1t+εt1qjxT1tjxε,=j1limε0T1tq+t+εt1qqtq1/qjxT1tjxε,=j1limε0T1tT1t+εt1qqtq1/qjxT1tjxε,=j1limε0T1tT1t+εt1qqtq1/qjxT10jxε.

Now, using Theorem 2.4 in , we get(18)T1t+εt1qqtq1/qjxT10jxε=T1tT1qct+εt1qqtqqεjx,for some 0<c<t+εt1qqtq. If ε0, then c0 and limε0T1qc=T1q0=A1.

Consequently,(19)T1qtjx=T1tA1jxlimε0t+εt1qqtqqε.

By using L’Hopital’s Rule, we get limε0t+εt1qqtq/qε=1.(20)Tqtx=j1T1tA1jx,=j1A1T1tjx,=j1A1jj1T1tjx,=ATtx.

Example 2.

Let f: be continuous on 0,1. Define(21)Ttfx=fx+1qtq,q0,1.

Then, Tt is obviously a c0-q-semigroup of contraction on .

Remark 2.

If M=1 and w=0 in Definition 4, we say that Ttt0 is a contraction fuzzy semigroup.

For q0,1,

(22)Tt+s1/qfx=fx+1qt+s1/qq,=fx+1qt+1qs,=Tt1/qTs1/qfx.

T0=I and Ttf whenever f and that

(23)dTtf,0˜df,0˜,t0.

4. Fuzzy Fractional Cauchy Problems

Let F: be continuous and consider the fractional initial value problem(24)xqt=Fxt,xt0=x0,where q0,1.

It is well known that instead of the differential equation (24), it is possible to study an equivalent fractional integral equation.(25)xt=x0+IqFxt,for all t0 and q0,1.

A solution xt of equation (24) is independent of the initial time t0. In fact, let k0<a and denote yt=xk0+1/qtq. Then,(26)yqt=xqk0+1qtq=Fxk0+1qtq=Fyt,and yt0=xk0+1/qt0q=y0. Hence, yt and xt are solutions of the same fractional differential equation with a different initial value.

Theorem 4.

Let q0,1.

If xt is a solution to the fuzzy fractional initial value problem,(27)xqt=Fxt,xt0=x0.

Then, Ttx0=xt is a fuzzy semigroup. Furthermore, Ttx0 is q-differentiable w.r.t t and Tqtx0=Fxt=FTtx0.

Proof.

Let q0,1 and k>0. As obtained above, yt=xk+1/qtq is a solution of the fractional initial value problem yqt=Fyt, y0=xk. Hence,(28)Ts+t1/qx0=x0+1qs+t1/qq=x1qs+1qt.

We set k=1/qs, then(29)Ts+t1/qx0=xk+1qt=yt1/q=Tt1/qxk,=Tt1/qx1qs=Tt1/qTs1/qx0,and T0x0=x0=x0. Being a solution to a differential equation, Ttx0 is q-differentiable with respect to t and Tqtx0=xqt=Fxt.

Theorem 5.

Let q0,1. Suppose that a fuzzy semigroup Ttx is q-differentiable w.r.t t, for all x. Then, Ttx0 is a solution to the fractional initial value problem(30)xqt=Fxt,xt0=x0,where Fxt=Tq0x0.

Proof.

By the q-semigroup property and using proof of Theorem 3, we obtain(31)Tqtx0=limε0Tt+εt1qx0Ttx0ε,=limε0Ttq+t+εt1qqtq1/qx0Ttx0ε,=limε0Tt+εt1qqtq1/qTtx0Ttx0ε,=limε0Tt+εt1qqtq1/qTtx0T0Ttx0ε,=Tq0Ttx0,and T0x0=x0.

Finally, we show that the fuzzy exponential function is a generalization of the fuzzy semigroup introduced in .

Theorem 6.

If A: is a bounded linear operator, then the fuzzy exponential function has a power series representation(32)etq/qAx=k=0tkqqkk!Akx,t0.

Proof.

Let A: be a bounded linear operator as defined by Gal and Gal in . Then,(33)ϕr=supdx,y<¯rdAx,Ay=rA,and hence by  satisfies the condition. Consequently,(34)etq/qAx0=limnI+tqAqnnx0is a solution to the Cauchy problem xqt=Axt,x0=x0. Define St by a power series as(35)St=k=0tkqqkk!Ak.

Now, by Theorem 3.9 in , pose sq/q=t with esq/qA and Ss in , so St is a fuzzy semigroup, and hence by Theorem 5, Stx0 is a solution to the problem(36)xqt=Axt,x0=x0.

Since a bounded linear operator is Lipschitzian, it follows by Theorem 6.1 in  that the problem xqt=Axt,x0=x0, has a unique solution. Hence, etq/qAx0=Stx0, for all x0.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.