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 [1]. Balakrishnan [2] 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 [3]. Abdeljawad et al. [4] studied the fractional semigroup of operators. The semigroup generated by linear operators of a fuzzy-valued function was introduced by Gal and Gal [5]. Kaleva [6] 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 [7–14].

The concept of fuzzy fractional derivative was introduced by [15] and developed by [16–19], 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. [20] 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 [18].

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 q∈0,1 according to the fuzzy conformable fractional derivative which was introduced in [20].

2. Preliminaries

Let us denote by ℝℱ=u:ℝ⟶0,1 the class of fuzzy subsets of the real axis satisfying the following properties [21]:

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

u is the fuzzy convex, i.e., for x,y∈ℝ and 0<λ≤1,

(1)uλx+1−λy≥minux,uy.

u is upper semicontinuous,

u0=clx∈ℝ|ux>0 is compact.

Then, ℝℱ is called the space of fuzzy numbers. Obviously, ℝ⊂ℝℱ. For 0<α≤1, denote uα=x∈ℝ|ux≥α, then from (i) to (iv), it follows that the α-level sets uα∈PKℝ, for all 0≤α≤1, are a closed bounded interval which we denote by uα=u1α,u2α.

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

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

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

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

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

Let us define d:ℝℱ×ℝℱ⟶ℝ+∪0 by the equation(3)du,v=supα∈0,1dHuα,vα,for all u,v∈ℝℱ,where dH is the Hausdorff metric defined in PKℝ.(4)dHuα,vα=maxu1α−v1α,u2α−v2α.

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

ℝℱ,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,v≤du,w+dw,v,for all u,v,w∈ℝℱ and λ∈ℝ.

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

There exists a real Banach space X such that ℝℱ 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 addition in X induces the addition in ℝℱ,

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

C−C=a−b/a,b∈ℝℱ is dense in X,

C is closed.

Remark 1.

Let j˜:ℝℱ⟶X as j˜u=j−1u,u∈ℝℱ. It verifies the following properties: j˜u−j˜v=du,v, j˜su+tv=sj˜u+tj˜v, for all u,v∈ℝℱ, t,s≥0j˜ℝℱ−jℝℱ=C, since −1ℝℱ=ℝℱ.

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⟶ℝℱ be a fuzzy function. qth order “fuzzy conformable fractional derivative” of F is defined by (where the limit is taken in the metric space ℝℱ,d).(6)TqFt=limε⟶0+Ft+εt1−q⊖Ftϵ,=limε⟶0+Ft⊖Ft−εt1−qε,for all t>0,q∈0,1. Let Fqt stand for TqFt. Hence,(7)Fqt=limε⟶0+Ft+εt1−q⊖Ftε=limε⟶0+Ft⊖Ft−εt1−qε.

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

Definition 2.

Let F∈C0,a,ℝℱ∩L10,a,ℝℱ. Define the fuzzy fractional integral for q∈0,1.(9)IqFt=Itq−1Ft=∫0tFxx1−qdx,where the integral is the usual Riemann improper integral.

Definition 3.

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

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

Ts+t1/q=Ts1/qTt1/q, for all s,t≥0.

Definition 4.

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

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

(10)limt⟶0+Ttx=x.

There exist constants w≥0 and M≥1 such that dTtx,Tty≤Mewtq/qdx,y, for all t≥0,x,y∈ℝℱ.

Example 1.

Define on ℝℱ the linear operator Ttx=e2tx. Then, Ttt≥0 is a fuzzy 1/2-semigroup. Indeed

For t,s≥0,x∈ℝℱ,dTtx,x=de2tx,x, then e2t−1≥0, and then using Remark 1, we deduce that e2t−1x+x=e2tx. Therefore, the Hukuhara difference e2tx⊖xTtx⊖x exists and we have

For t≥0,x,y∈ℝℱ,dTtx,Tty=de2tx,e2ty=e2tdx,y. Consequently, Ttt≥0 is a fuzzy c0-q-semigroup on ℝℱ.

Definition 5.

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

We will write A for such generator.

Lemma 2.

Let A:ℝℱ⟶ℝℱ and A1=jAj−1: C⟶C tow the operator.

A is the operator of the fuzzy q-semigroup Ttt≥0 on ℝℱ if and only if A1 is the operator of the q-semigroup T1tt≥0 defining on the convex closed set C and T1=jTtj−1.

By using Definition 5, the proof is similar to the proof of Lemma 5 in [18] and is omitted.

Theorem 3.

Let Ttt≥0 be a c0-q-semigroup with infinitesimal generator A,0<q≤1. Then, for all x∈ℝℱ such that Ttx∈DA, for all t≥0; the mapping t⟶Ttx is q-differentiable and(15)Tqtx=ATtx,∀t≥0.

Proof.

Let q∈0,1 and x∈ℝℱ, for t≥0, and we have(16)Tt+s1/qx=Tt1/qTs1/qx.

Since Ttx∈DA, then(17)Tqtx=limε⟶0Tt+εt1−qx⊖Ttxε,=limε⟶0j−1T1t+εt1−qjx−j−1T1tjxε,=j−1limε⟶0T1t+εt1−qjx−T1tjxε,=j−1limε⟶0T1tq+t+εt1−qq−tq1/qjx−T1tjxε,=j−1limε⟶0T1tT1t+εt1−qq−tq1/qjx−T1tjxε,=j−1limε⟶0T1tT1t+εt1−qq−tq1/qjx−T10jxε.

Now, using Theorem 2.4 in [20], we get(18)T1t+εt1−qq−tq1/qjx−T10jxε=T1tT1qct+εt1−qq−tqqεjx,for some 0<c<t+εt1−qq−tq. If ε⟶0, then c⟶0 and limε⟶0T1qc=T1q0=A1.

Let F:ℝℱ⟶ℝℱ be continuous and consider the fractional initial value problem(24)xqt=Fxt,xt0=x0,where q∈0,1.

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 t≥0 and q∈0,1.

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 q∈0,1.

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 q∈0,1 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 q∈0,1. Suppose that a fuzzy semigroup Ttx is q-differentiable w.r.t t, for all x∈ℝℱ. 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=limε⟶0Tt+εt1−qx0⊖Ttx0ε,=limε⟶0Ttq+t+εt1−qq−tq1/qx0⊖Ttx0ε,=limε⟶0Tt+εt1−qq−tq1/qTtx0⊖Ttx0ε,=limε⟶0Tt+εt1−qq−tq1/qTtx0⊖T0Ttx0ε,=Tq0Ttx0,and T0x0=x0.

Finally, we show that the fuzzy exponential function is a generalization of the fuzzy semigroup introduced in [5].

Theorem 6.

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

Proof.

Let A:ℝℱ⟶ℝℱ be a bounded linear operator as defined by Gal and Gal in [5]. Then,(33)ϕr=supdx,y<¯rdAx,Ay=rA,and hence by [6] satisfies the condition. Consequently,(34)etq/qAx0=limn⟶∞I+tqAqnnx0is a solution to the Cauchy problem xqt=Axt,x0=x0. Define St by a power series as(35)St=∑k=0∞tkqqkk!Ak.

Now, by Theorem 3.9 in [5], pose sq/q=t with esq/qA and Ss in [5], 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 [25] that the problem xqt=Axt,x0=x0, has a unique solution. Hence, etq/qAx0=Stx0, for all x0∈ℝℱ.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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