A class fuzzy fractional differential equation (FFDE) involving Riemann-Liouville H-differentiability of arbitrary order q>1 is considered. Using Krasnoselskii-Krein type conditions, Kooi type conditions, and Rogers conditions we establish the uniqueness and existence of the solution after determining the equivalent integral form of the solution.
1. Introduction
In the last several years, fractional differential equations attracted more and more researchers and have proven to be very useful tools for modeling phenomena in physics, finance, and many other areas. The Riemann-Liouville formulation arises in a natural way for problems such as transport problems from the continuum random walk scheme or generalizes Chapman-Kolmogorov models [1, 2]. It was also applied for modeling the behavior of viscoelastic and viscoplastic materials under external influences [3, 4] and the continuum and statistical mechanics for viscoelasticity problems [5]. Some of the several research papers were published to consider the uniqueness of the solution for fractional differential equations under Nagumo like conditions (see, e.g., [6–13] and references therein). In [11, 12], Lakshmikantham and Leela established the uniqueness of the solution for the problem Dqx(t)=f(t,x(t)), where 0<q<1. Then, in [13], Yoruk et al. proved the uniqueness of the solution via Krasnoselskii-Krein, Rogers, and Kooi conditions, for 1<q<2.
On the other hand, in order to obtain more realistic modeling of phenomena, one has to take uncertainty; see [14–16] and the references cited therein. Many researchers have worked in the theoretical and numerical aspect of fractional and fuzzy differential equations; the reader is kindly referred to [2, 17–31] and the references therein.
In [32], Allahviranloo and Ahmadi introduced the fuzzy Laplace transform, which they used under the strongly generalized differentiability. Recently, ElJaoui et al. [33] developed it further. The newly defined fuzzy Laplace transform [32] for high order fuzzy derivatives is one of the most useful methods as mentioned by Jafarian et al. in [34]: “…, one of the important and interesting transforms in the problems of fuzzy equations is Laplace transforms. The fuzzy Laplace transform method solves fuzzy fractional differential equations and fuzzy boundary and initial value problems [35–38] ….”
Motivated by the above works, we adopted the fuzzy Laplace transform to prove the uniqueness and existence for the following initial value problems (FFDE) for arbitrary order q>1:(1)Dqxt=ft,xt,Dq-1xt,x0=y0,Dq-ix0=0~,i=1,…,q, where y0∈E and f:E0→E is a continuous fuzzy-valued function with (2)E0=t,x:0≤t≤1,dxt,0~≤b, where d is the Hausdorff distance.
Our aim is to both generalize and extend the previous uniqueness results of [13, 39].
The organization of this paper is as follows. Section 2 contains some basic definitions concerning fuzzy set theory and Riemann-Liouville generalized H-differentiability. In Section 3, using the fuzzy Laplace transform we determine the equivalent integral problem. Section 4 is devoted to the main results: a Krasnoselskii-Krein type of uniqueness theorem, a Kooi type uniqueness theorem, and a Rogers type uniqueness theorem; then we prove that the successive approximations converge to the unique solution.
2. Preliminaries
First, let us recall some basic definitions about fuzzy numbers and fuzzy sets. Here and in the rest of the paper, we denote by Γ the Gamma function and [α] the integer part of α.
As defined in [40], E={u:R→[0,1];usatisfiesA1–(A4)} is the space of fuzzy numbers:
u is normal; that is, there exists y0∈R such that u(y0)=1.
u is fuzzy convex; that is, u(λy+(1-λ)z)≥min{u(y),u(z)} whenever y,z∈R and λ∈[0,1].
u is upper semicontinuous; that is, for any y0∈R and ε>0 there exists δ(y0,ε)>0 such that u(y)<u(y0)+ε whenever |y-y0|<δ, y∈Rn.
The closure of the set {y∈R;u(y)>0} is compact.
The set [u]α={u∈R;u(y)≥α} is called α-level set of u.
It follows from A1–(A4) that the α-level sets [u]α are convex compact subsets of R for all α∈(0,1]. The fuzzy zero is defined by (3)0~=0if y≠0,1if y=0.
Definition 1 (see [40]).
A fuzzy number u in the parametric form is a pair (u_(r),u¯(r)) of functions u_(r),u¯(r), 0≤r≤1, which satisfy the following conditions:
u_(r) is a bounded nondecreasing left continuous function in (0,1] and right continuous at 0;
u¯(r) is a bounded nonincreasing left continuous function in (0,1] and right continuous at 0;
u_(r)≤u¯(r), 0≤r≤1.
Moreover, we also can present the r-cut representation of fuzzy numbers as [u]r=[u_(r),u¯(r)] for all 0≤r≤1.
According to Zadeh’s extension principle, we have the following properties of fuzzy addition and multiplication by scalar on E: (4)u⊕vx=supy∈Rminuy,vx-y,x∈R,k⊙ux=uxkif k≥0,0~if k=0. Seeking simplicity, we note ⊕,⊙ by the usual +,·. The Hausdorff distance between the fuzzy numbers is denoted by d:E×E→[0,+∞[, such that d(u,v)=supr∈[0,1]max{|u_(r)-v_(r)|,|u¯(r)-v¯(r)|}. And (d,E) is a complete metric space.
Definition 2.
Let x,y∈E. If there exists z∈E such that x=y+z, then z is called the H-difference of x and y, and it is denoted by x⊖y.
Remark 3.
Note that the sign ⊖ stands for H-difference and x⊖y≠x+(-1)y.
We denote by CF[0,a] the space of all fuzzy-valued functions which are continuous on [0,a] and LF[0,a] the space of all Lebesgue integrable fuzzy-valued functions on [0,a], where a>0. We also denote by AC(n-1)F[0,a] the space of fuzzy-valued functions f which have continuous H-derivatives up to order n-1 on [0,a] such that f(n-1) in ACF[0,a].
Definition 4 (see [41]).
Let f∈CF[0,1]∩LF[0,1]. The fuzzy fractional integral of the fuzzy-valued function f is defined by (5)Iβfx;r=Iβf_x;r,Iβf¯x;r,0≤r≤1, where (6)Iβf_x;r=1Γβ∫0xx-sβ-1f_s;rds,Iβf¯x;r=1Γβ∫0xx-sβ-1f¯s;rds.
Definition 5 (see [41, Definition 6]).
Let f∈C(n)F[0,1]∩LF[0,1], x0∈(0,1), and Φ(x)=(1/Γ(n-β))∫0x(f(t)dt/(x-t)β-n+1), where n=[β]+1. One says that f is fuzzy Riemann-Liouville fractional differentiable of order β at x0, if there exists an element D0βf(x0)∈E, such that, for all h>0 sufficiently small, one has
Denote by C(n-1)F([0,a]) the space of fuzzy-valued functions f on the bounded interval [0,a] which have continuous H-derivative up to order n-2 such that f(n-1)∈CF[0,a]. C(n-1)F([0,a]) is a complete metric space endowed by the metric D such that for every g,h∈C(n-1)F([0,a])(9)Dg,h=∑i=0n-1supt∈0,adgit,hit.
In the rest of the paper, we say that a fuzzy-valued function f is [(i)-β]RL-differentiable if it is differentiable as in Definition 5 case (i) and is [(ii)-β]RL-differentiable if it is differentiable as in Definition 5 case (ii).
Definition 6 (see [41, Theorem 7]).
Let f∈C(n)F[0,1]∩LF[0,1], x0∈(0,1), and Φ(x)=(1/Γ(n-β))∫0x(f(t)dt/(x-t)β-n+1), where n=[β]+1 such that 0≤r≤1; then one has the following:
if f is [(i)-β]RL-differentiable fuzzy-valued function, then (10)D0βfx0;r=D0βf_x0;r,D0βf¯x0;r,
or
if f is [(i)-β]RL-differentiable fuzzy-valued function, then (11)D0βfx0;r=D0βf¯x0;r,D0βf_x0;r,
where (12)D0βf_x0;r=1Γn-βddxn∫0xx-tn-β-1f_t;rdtx=x0,D0βf¯x0;r=1Γn-βddxn∫0xx-tn-β-1f¯t;rdtx=x0.
The following theorem is an important one about the fuzzy Laplace transform L of the Riemann-Liouville H-derivative for fuzzy-valued functions.
Theorem 7.
Suppose that f∈C(n)F[0,∞)∩LF[0,∞); one has the following:
if f is [(i)-β]RL-differentiable fuzzy-valued function, then(13)LD0βfx=pβLft⊖∑k=0n-1pkDβ-k-1f0
or
if f is [(i)-β]RL-differentiable fuzzy-valued function, then(14)LD0βfx=-∑k=0n-1pkDβ-k-1f0⊖-pβLft.
The proof runs along similar lines as that of [41, Theorem 16], and we omit it here.
3. Fuzzy Fractional Integral Equation
In this section, we study the relation between problem (1) and the fuzzy integral form using the well-known fuzzy Laplace transform.
In fact, by taking Laplace transform on both sides of (15)Dqxt=ft,xt,Dq-1xt≜rt,x we get (16)LDqxt=Lft,xt,Dq-1xt. Based on the type of Riemann-Liouville H-differentiability, we obtain two cases.
Case 1.
If Dqx is [(i)-q]RL-differentiable fuzzy-valued function, then(17)Lrt,x=-∑k=0n-1pkDβ-k-1x0⊖pqLxt, and based on the lower and upper functions of Dqx the above equation becomes(18)Lr_t,x;r=pqLx_t;r-∑k=0n-1pkDβ-k-1x_0;r,Lr¯t,x;r=pqLx¯t;r-∑k=0n-1pkDβ-k-1x¯0;r, where (19)Lr_t,x;r=minrt,u∣u∈x_t;r,x¯t;r,0≤r≤1,Lr¯t,x;r=maxrt,u∣u∈x_t;r,x¯t;r,0≤r≤1. In order to solve system (18), and for the sake of simplicity, we assume that(20)Lx_t;r=H1p;r,Lx¯t;r=K1p;r, where H1(p;r) and K1(p;r) are solutions of the previous system (18); it yields (21)x_t;r=L-1H1p;r,x¯t;r=L-1K1p;r.
Case 2.
If Dqx is [(ii)-q]RL-differentiable fuzzy-valued function, then (22)Lrt,x=pqLxt⊖∑k=0n-1pkDβ-k-1x0, and based on the lower and upper functions of Dqx the above equation becomes(23)Lr_t,x;r=pqLx_t;r-∑k=0n-1pkDβ-k-1x_0;r,Lr¯t,x;r=pqLx¯t;r-∑k=0n-1pkDβ-k-1x¯0;r, where (24)Lr_t,x;r=minrt,u∣u∈x_t;r,x¯t;r,0≤r≤1,Lr¯t,x;r=maxrt,u∣u∈x_t;r,x¯t;r,0≤r≤1. In order to solve system (23), and for the sake of simplicity, we assume that (25)Lx_t;r=H2p;r,Lx¯t;r=K2p;r, where H2(p;r) and K2(p;r) are solutions of the previous system (23). Then we obtain(26)x_t;r=L-1H2p;r,x¯t;r=L-1K2p;r.
Taking into account the initial conditions of problem (1) and using the linearity of the inverse Laplace transform on systems (20) and (26), we obtain the following for both cases.
x is a solution for problem (1) if and only if x is a solution for the following integral equation:(27)xt=y0+1Γq∫0tt-sq-1rs,xdsin the sense of [(i)-q]RL-differentiability, and(28)x^t=y0⊖-1Γq∫0tt-sq-1rs,xds in the sense of [(ii)-q]RL-differentiability.
4. Main Results
Now, we state the Krasnoselskii-Krein type conditions for FFDE (1).
Theorem 8.
Let f∈C(E0,E) satisfy the following Krein type conditions:
d(f(t,x,y),f(t,x¯,y¯))≤min{Γ(q),1}k+α(q-[q])/2t1-α(q-[q])d(x,x¯)+d(y,y¯), t≠0 and 0<α<1,
where δ and k are positive constants and k(1-α)<1+α(q-[q]); then in the sense of [(i)-q]RL-differentiability, the solution x is unique and in the sense of [(i)-q]RL-differentiability, the solution x^ is unique on [0,η], where η=min1,bΓ(1+q)/M1/q,d/M and M is the bound for f on E0: that is, d(f,0~)≤M.
Proof.
First we establish the uniqueness; suppose x and y are any two solutions of (1) in [(i)-q]RL-differentiability and let ϕ(t)=d(x(t),y(t)) and θ(t)=d(Dq-1x(t),Dq-1y(t)). Note that ϕ(0)=θ(0)=0.
We define R(t)=∫0tϕα(s)+sα(q-[q])θα(s)ds; clearly R(0)=0.
Using (27) and condition (H2), we get (29)ϕt≤δΓq∫0tt-sq-1ϕαs+sαq-qθαsds≤δΓqtq-1Rt,θt≤∫0tδϕsα+tαq-qθsαds≤δRt. For the sake of simplicity we use the same symbol C to denote all different constants arising in the rest of the proof.
We have(30)R′t=ϕαt+tαq-qθαt≤Ctαq-1+tαq-qRαt. Since R(t)>0 for t>0, multiplying both sides of (30) by (1-α)R-α(t) and then integrating the resulting inequality, we get(31)Rt≤Ctα/1-αq+1+tα/1-αq+1-αq/1-α. Using the fact that(32)a+b1-α≤121-α-1a1-α+b1-α for every a,b∈(0,1), (31) becomes(33)Rt≤Ctα/1-αq+1+tα/1-αq+1-αq/1-α. This leads to the following estimates on ϕ and θ, for t∈[0,η]:(34)ϕt≤Ctq/1-α+tq/1-α+α1-q/1-α,θt≤Ctα/1-αq+1+tα/1-αq+1-αq/1-α. Define the function ψ(t)=t-kmax{ϕ(t),θ(t)} for t∈(0,1]. When either t-kϕ(t) or t-kθ(t) is the maximum, we get(35)0≤ψt≤Ctq/1-α-k+tq/1-α+α1-q/1-α-k, or(36)0≤ψt≤Ctα/1-αq+1-k+tαq/1-α+1-αq/1-α-k. Since k(1-α)<1+α(q-[q]) (by assumption), we have(37)k1-α<1+αq-q⟹k1-α<qk-11-α<αqk1-α<q+α-αqk1-α<αq+1-αq. So all of the exponents of t in the above inequalities are positive. Hence, limt→0+ψ(t)=0. Therefore, if we define ψ(0)=0, the function ψ is continuous in [0,η].
We want to prove that ψ≡0. In fact, since the function ψ is continuous, if ψ does not vanish at some points t, that is, ψ(t)>0 on ]0,η], then there exists a maximum m>0 reached when t is equal to some t1: 0<t1≤η≤1 such that ψ(s)<m=ψ(t1), for s∈]0,t1). But, from condition (H1) we get for either cases(38)m=ψt1=t1-kϕt1≤minΓq,1mt1q-1+αq-q<m or(39)m=ψt1=t1-kθt1≤minΓq,1mt1αq-q<m which is a contradiction. Thus, the uniqueness of the solution is established in the sense of [(i)-q]RL-differentiability. The second part of the proof is almost completely similar to the [(i)-q]RL-differentiability; thus, we omit it.
Remark 9.
For the case 1<q<2, of the deterministic case, Theorem 8 is reduced to [13, Theorem 3.1].
Theorem 10 (Kooi’s type uniqueness theorem).
Let f satisfy the following conditions:
d(f(t,x,y),f(t,x¯,y¯))≤min{Γ(q),1}k+α(q-[q])/2t1-α(q-[q])d(x,x¯)+d(y,y¯), t≠0 and 0<α<1;
where c and k are positive constants and k(1-α)<1+α(q-[q])-β, for (t,x,y),(t,x¯,y¯)∈R0; then in the sense of [(i)-q]RL-differentiability, the solution x is unique and in the sense of [(i)-q]RL-differentiability, the solution x^ is unique.
Proof.
It is similar to that of Theorem 8; thus, we omit it.
Lemma 11.
Let ϕ and θ be two nonnegative continuous functions in the interval [0,η] for a real number a>0. Let ψ(t)=∫0tϕ(s)+sq-[q]θ(s)/2sq-[q]+2ds. Assume the following:
ϕ(t)≤tq-[q]ψ(t),
θ(t)≤ψ(t),
ϕ(t)=o(tq-[q]e-1/t),
θ(t)=o(e-1/t).
Then ϕ≡θ≡0.
Proof.
Let ψ(t)=∫0tϕ(s)+sq-[q]θ(s)/2sq-[q]+2ds. After differentiating ψ and using (ii), we obtain, for t>0, ψ′(t)≤1/t2ψ(t), so that e1/tψ(t) is decreasing. Now, from (iii) and (iv), if ϵ>0 then, for a small t, we have(40)e1/tψt≤e1/t∫0t12s22ϵe-1/sds=ϵ. Hence, limt→0e1/tψ(t)=0 which implies that ψ(t)≤0. Finally, ψ is nonnegative due to (i), and thus ψ≡0.
Theorem 12 (Rogers’ type uniqueness theorem).
Let the function f verify the following conditions:
df(t,x,y),0~≤min{Γ(q),1}oe-1/t/t2, uniformly for positive and bounded x and y on E,
The proof of this theorem is essentially based on Lemma 11.
Proof.
Suppose x and y are any two solutions of (1) in [(i)-q]RL-differentiability, and let ϕ(t)=d(x(t),y(t)) and θ(t)=d(Dq-1x(t),Dq-1y(t)); we get for t∈[0,η]⊂[0,1](41)ϕt≤1Γq∫0tt-sq-1drs,x,rs,y≤∫0tt-sq-12sq-q+2ϕs+sq-qθsds≤tq-1∫0t12sq-q+2ϕs+sq-1θsds≤tq-q∫0t12sq-q+2ϕs+sq-1θsds≤tq-qψt,θs≤∫0tdrs,x,rs,y≤∫0tminΓq,12sq-q+2ϕs+sq-qθsds≤∫0t12sq-q+2ϕs+sq-qθsds≤ψt, where ψ is defined as in Lemma 11.
Also, if ϵ>0, then from the condition (K1) for small t, we have (42)ϕt≤tq-1Γq∫0tdrs,x,rs,y<tq-12ϵ∫0te-1/ss2ds≤tq-1e-1/t2ϵ<ϵtq-qe-1/t2,θt≤∫0tdrs,x,rs,y<2ϵmin1,Γq∫0te-1/ss2ds≤2ϵe-1/t. By applying Lemma 11, we obtain d(x(t),y(t))=0 for every t∈[0,1], and this proves the uniqueness of the solution of the FFDE (1) in [(i)-q]RL-differentiability. The second part of the proof is almost completely similar; thus, we omit it.
Theorem 13.
Let f∈C(E0,E) satisfy the conditions of Theorem 8. Then the successive approximations (43)xnt=y0+1Γq∫0tt-sq-1rs,xn-1ds in the sense of [(i)-q]RL-differentiability or(44)x^nt=y0⊖-1Γq∫0tt-sq-1rs,xn-1ds in the sense of [(ii)-q]RL-differentiability converge to the unique solution of the FFDE (1).
Proof.
Without loss of generality, we prove Theorem 13 for the sequence {xn} in the sense of [(i)-q]RL-differentiability using Ascoli-Arzela Theorem. The convergence of the sequence {x^n} in the sense of [(ii)-q]RL-differentiability is completely similar so we omit it.
Step 1. The sequences {xj}j≥0 and {Dq-1xj}j≥0 are well defined and continuous and uniformly bounded on [0,η]; in fact(45)dxj+1t,y0≤1Γq∫0tt-sq-1drs,xj,0~ds,dDq-1xj+1t,y0≤∫0tdrs,xj,0~ds. For j=0 and t∈[0,η], we have(46)dx1t,y0≤MtqΓq+1≤b,dDq-1x1t,y0≤Mt≤d. Moreover, for every i∈{0,…,n-1} we have (47)dx1it,0~=dDiIqft,x0t,Dq-1x0t,0~=dIq-ift,x0t,Dq-1x0t,0~=1Γq-i∫0tt-sq-i-1dfs,x0s,Dq-1x0s,0~ds≤MΓq-i∫0tt-sq-i-1ds≤Mtq-iq-iΓq-i≤Mtq-iΓq-i+1. By induction, the sequences {xj+1(t)} and {Dq-1xj+1(t)} are well defined and uniformly bounded on [0,η].
Step 2. We prove that the functions y and z are continuous in [0,η], where y and z are defined by (48)yt=limsupj→∞ζj0t,zt=limsupj→∞ξjt, such that (49)ζj0t=dxjt,xj-1t,ξjt=dDq-1xjt,Dq-1xj-1t. Let us note(50)mt=∑i≤n-1limsupj→∞ζjit, where(51)ζjit=dxjit,xj-1it. For 0≤t1≤t2 and for every i∈{0,…,n-1}, we obtain (52)ζjit1-ζjit2=dxj+1it1,xjit1-dxj+1it2,xjit2≤1Γq-i∫0t1t1-sq-1-idrs,xj,rs,xj-1ds-∫0t2t2-sq-1-idrs,xj,rs,xj-1ds≤2MΓq-i∫0t1t1-sq-1-i-t2-sq-1-ids-∫t1t2t2-sq-1-ids≤2Mq-iΓq-it1q-i-t2q-i+2t2-t1q-i≤4MΓq-i+1t2-t1q-i. The right-hand side in the above inequalities is at most 4M/Γ(q-i+1)(t2-t1)q-i+ϵ for large n if ϵ>0 provided that(53)t2-t1≤η≤4MΓq-i+1t2-t1q-i, for every i≤n-1. And since ϵ is arbitrary and t1,t2 can be interchangeable, we get(54)mt1-mt2≤∑i≤n-14MΓq-i+1t2-t1q-i≤4Mn-1Γq+1t2-t1q. The same goes for z(t), and we obtain(55)zt1-zt2≤2Mt2-t1. These imply that y and z are continuous on [0,η].
Step 3. We verify that the family {Dq-1jn+1(t)} is equicontinuous in CF([0,η],E) and that the family {xj+1(t)} is equicontinuous in C(n-1)F([0,η],E).
We may prove that by using condition (H2) and the definition of successive approximations (43) we obtain (56)ζj+10t≤C∫0tt-sq-1ζj0sα+sαq-qξjsαds,ζj+1it≤C∫0tt-sq-i-1ζj0sα+sαq-qξjsαds. As a consequence, we obtain the following estimation: (57)Dxj+1,xj≤∑i≤n-1C∫011-sq-i-1xjs-xj-1sα+sαq-qDq-1xjs-Dq-1xj-1sαds. By Arzela-Ascoli Theorem, there exists a subsequence of integers {jk}, such that(58)dxjkt,xjk-1t⟶ytas jk⟶∞,dDq-1xjkt,Dq-1xjk-1t⟶ytas jk⟶∞. Let us note (59)m∗t=limsupk→∞dxjkt-xjk-1t,z∗t=limsupk→∞dDq-1xjkt-Dq-1xjk-1t. Further, if {d(xj,xj-1)}→0 and {d(Dq-1xj,Dq-1xj-1)}→0 as j→∞, then the limit of any successive approximation of xn is the solution x of (1), which was proved to be unique in Theorem 8. It follows that a selection of subsequences is unnecessary and that the entire sequence {xj} converges uniformly to x(t). For that, it is sufficient to show that y≡0 and z≡0 which will lead to m∗(t) and z∗(t) being null.
Setting(60)Rt=∫0tysα+sαq-qzsαds, and by defining ψ∗(t)=t-kmax{y(t),z(t)}, we show that limt→0+ψ∗(t)=0.
Now we shall prove that ψ∗(t)≡0. Suppose that ψ∗(t)>0 at any point in [0,η]; then there exists t1 such that 0<m¯=ψ∗(t1)=max0≤t≤ηψ∗(t). Hence, from condition (H1), we obtain(61)m¯=ψt1=t1-kyt1≤minΓq,1m¯t1q-1+αq-q<m¯ or(62)m¯=ψt1=t1-kzt1≤minΓq,1m¯t1αq-q<m¯. In both cases, we end up with a contradiction. So ψ∗(t)≡0. Therefore, iteration (43) converges uniformly to the unique solution x of (1) on [0,η].
5. Conclusion
In this paper we established the uniqueness and existence of the solution under a fuzzy version of the Krasnoselskii-Krein conditions. Our work generalizes and extends the work of Yoruk et al. to arbitrary order in the fuzzy version. We finally hope to study other classes of fuzzy fractional differential problems in future works.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
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