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JID:FSS AID:6522 /FLA [m3SC+; v 1.191; Prn:17/04/2014; 9:09] P.1(1-23) Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems ••• (••••) •••••• www.elsevier.com/locate/fss 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 46 46 47 47 48 48 49 49 50 50 51 51 52 52 Fractional calculus for interval-valued functions Vasile Lupulescu Constantin Brancusi University, Targu-Jiu, Romania Received 8 February 2013; received in revised form 6 April 2014; accepted 7 April 2014 Abstract We use a generalization of the Hukuhara difference for closed intervals on the real line to develop a theory of the fractional calcu- lus for interval-valued functions. The properties of Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative and Caputo fractional derivative for interval-valued functions are investigated. Several examples are presented to illustrate the concepts and results. © 2014 Published by Elsevier B.V. Keywords: Interval-valued function; Generalized Hukuhara derivative; Interval-valued Riemann–Liouville fractional integral; Interval-valued Riemann–Liouville fractional derivative; Interval-valued Caputo fractional derivative 1. Introduction The theory of fractional calculus, which deals with the investigation and applications of derivatives and integrals of arbitrary order has a long history. Although for almost three centuries the theory of fractional calculus developed mainly as a pure theoretical field of mathematics, in the last decades it has been used in various fields as rheology, viscoelasticity, electrochemistry, diffusion processes, etc. First book, devoted exclusively to the subject of fractional calculus, is the book by Oldham and Spanier [54]. A rigorous study of fractional calculus can be found in Samko et al. [63]. The most recent books and papers which deal with the fractional calculus, fractional differential equations and their applications are Kilbas et al. [28], Miller and Ross [49], Podlubny [58], Diethelm [15], Das [20], N’Doye et al. [21], Si et al. [66] and Wang et al. [70]. The concept of solution for fractional differential equations with uncertainty was introduced by Agarwal, Lak- shmikantham and Nieto [1]. This concept has been studied and developed in several papers, such as: Arshad and Lupulescu [8,9], Agarwal et al. [2,3], Allahviranloo et al. [7], Salahshour et al. [60,61], Mazandarani and Kamyad [48]. The concepts of fractional derivatives for a fuzzy function are either based on the notion of Hukuhara derivative (H -derivative) or on the notion of strongly generalized derivative (G-derivative). The concept of Hukuhara derivative is old and well known (Hukuhara [25], Banks and Jacobs [13], Puri and Ralescu [59]), but the concept of G-derivative * Corresponding author. Tel. +40 253225881. E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.fss.2014.04.005 0165-0114/© 2014 Published by Elsevier B.V.

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Page 1: Fractional calculus for interval-valued functions

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ScienceDirect

Fuzzy Sets and Systems ••• (••••) •••–•••www.elsevier.com/locate/fss

Fractional calculus for interval-valued functions

Vasile Lupulescu ∗

Constantin Brancusi University, Targu-Jiu, Romania

Received 8 February 2013; received in revised form 6 April 2014; accepted 7 April 2014

Abstract

We use a generalization of the Hukuhara difference for closed intervals on the real line to develop a theory of the fractional calcu-lus for interval-valued functions. The properties of Riemann–Liouville fractional integral, Riemann–Liouville fractional derivativeand Caputo fractional derivative for interval-valued functions are investigated. Several examples are presented to illustrate theconcepts and results.© 2014 Published by Elsevier B.V.

Keywords: Interval-valued function; Generalized Hukuhara derivative; Interval-valued Riemann–Liouville fractional integral; Interval-valuedRiemann–Liouville fractional derivative; Interval-valued Caputo fractional derivative

1. Introduction

The theory of fractional calculus, which deals with the investigation and applications of derivatives and integralsof arbitrary order has a long history. Although for almost three centuries the theory of fractional calculus developedmainly as a pure theoretical field of mathematics, in the last decades it has been used in various fields as rheology,viscoelasticity, electrochemistry, diffusion processes, etc. First book, devoted exclusively to the subject of fractionalcalculus, is the book by Oldham and Spanier [54]. A rigorous study of fractional calculus can be found in Samkoet al. [63]. The most recent books and papers which deal with the fractional calculus, fractional differential equationsand their applications are Kilbas et al. [28], Miller and Ross [49], Podlubny [58], Diethelm [15], Das [20], N’Doyeet al. [21], Si et al. [66] and Wang et al. [70].

The concept of solution for fractional differential equations with uncertainty was introduced by Agarwal, Lak-shmikantham and Nieto [1]. This concept has been studied and developed in several papers, such as: Arshad andLupulescu [8,9], Agarwal et al. [2,3], Allahviranloo et al. [7], Salahshour et al. [60,61], Mazandarani and Kamyad [48].The concepts of fractional derivatives for a fuzzy function are either based on the notion of Hukuhara derivative(H -derivative) or on the notion of strongly generalized derivative (G-derivative). The concept of Hukuhara derivativeis old and well known (Hukuhara [25], Banks and Jacobs [13], Puri and Ralescu [59]), but the concept of G-derivative

* Corresponding author. Tel. +40 253225881.E-mail addresses: [email protected], [email protected].

50

51

52

http://dx.doi.org/10.1016/j.fss.2014.04.0050165-0114/© 2014 Published by Elsevier B.V.

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was recently introduced by Bede and Gal [16]. Using this new concept of derivative, the class of fuzzy differentialequations has been extended and studied in some papers such as: Bede and Stefanini [17,68], Chalco-Cano et al. [19],Khastan et al. [26,27] and Li et al. [30]. The connection between the fuzzy analysis and the interval analysis is verywell known (Zadeh [73], Moore and Lodwick [50], Pedrycz and Gomide [55]). Interval analysis and fuzzy analysiswere introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer modelsof some deterministic real-world phenomena. The main theoretical and practical results in the fields of fuzzy analy-sis and the interval analysis can be found in several works (Moore [51,52], Alefeld and Herzberger [5], Kolev [29],Alefed and Mayer [6], Baker Kearfott and Kreinovich [12], and Nguyen et al. [53]). In recent years the interval dif-ferential equations and fuzzy differential equations began to be intensively studied (Abbas et al. [4], Plotnikova [56],Stefanini [67], Bede and Stefanini [17,68], Malinowski [35–44], and Lupulescu [31–34]). The major shortcomingsof the H -derivative are well known. To eliminate these shortcomings, several notions of derivative of an interval-valued function were introduced. Some concepts of derivatives for an interval-valued function, like G-derivative (Bedeand Gal [16]), gH-derivative (Markov [45], Bede and Stefanini [17,68]) and π -derivative (Banks and Jacobs [13],Schröder [65], Plotnikova [57]), were analyzed in Chalco-Cano, Román-Flores and Jiménez-Gamero [18,19]. It isknown that, if the set of switching points is finite, then the notions of G-differentiability, gH-differentiability andπ -differentiability coincide for an interval-valued function (see Chalco-Cano et al. [18], and Stefanini and Bede [68]).For this reason, in this paper we use only the notion of gH-differentiability.

The purpose of this paper is to develop a theory of fractional calculus for interval-valued functions. This approach ismotivated since this topic has not yet been addressed, and by the fact that this calculus is a powerful tool for the study ofthe interval fractional differential equations and the fuzzy fractional differential equations. From this perspective, webelieve that the presented results will be useful for the development of the theory of the interval differential equationsand the fuzzy differential equations. A similar calculus can be developed for functions having values compact andconvex subsets in Rn. But in this case there are some difficulties, such as the fact that gH-difference A �g B does notexist for every pair of subsets A and B in Rn, and on the other hand, it is very difficult to find examples to illustrateconcepts and results obtained. Taking into account the specific difficulties that may be encountered, we think thistopic deserves a separate approach. The paper is organized as follows. In Section 2 we present the basic notationson the integral and differential calculus for interval-valued functions. In Section 3 we study the Riemann–Liouvillefractional integral for interval-valued functions. Main properties of Riemann–Liouville fractional derivative and theCaputo derivative for an interval-valued function are presented in Sections 4 and 5, respectively.

2. Preliminaries

Let us denote by K the set of all nonempty compact intervals of the real line R. If A = [a−, a+], B = [b−, b+] ∈ K,then the usual interval operations, i.e. Minkowski addition and scalar multiplication, are defined by

A + B = [a−, a+]+ [

b−, b+]= [a− + b−, a+ + b+]

and

λA = λ[a−, a+]=

⎧⎨⎩

[λa−, λa+] if λ > 0{0} if λ = 0[λa+, λa−] if λ < 0,

respectively. If λ = −1, scalar multiplication gives the opposite

−A := (−1)A = (−1)[a−, a+]= [−a+,−a−].

In general, A + (−A) �= {0}; that is, the opposite of A is not the inverse of A with respect to the Minkowski addition(unless A = {a} is a singleton). Minkowski difference is A − B = A + (−1)B = [a− − b+, a+ − b−]. With respect tothe above operations, K is a quasilinear space (Markov [46], Mayer [47], Schneider [64]).

We recall that a nonempty set Q, together with operations of addition (+) and multiplication by real scalars (·), iscalled a quasilinear space if the following laws hold (Markov [46]):

(q1) (Associativity of addition) (A + B) + C = A + (B + C) for all A,B,C ∈ Q.(q2) (Neutral element) There exists an element θ ∈ Q such that A + θ = θ + A = A for all A ∈ Q.

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(q3) (Commutativity of addition) A + B = B + A for all A,B ∈Q.(q4) (Cancellative law) A,B,C ∈ Q and A + C = B + C implies A = B .(q5) (Associative law) λ · (μ · A) = (λμ) · A for all λ,μ ∈ R and A ∈Q.(q6) (Unit law) 1 · A = A for all A ∈Q.(q7) (First distributive law) λ · (A + B) = λ · A + λ · B for all λ ∈ R and A,B ∈Q.(q8) (Second distributive law) (λ + μ) · A = λ · A + μ · A for all A ∈ Q and λ,μ ∈R with λμ ≥ 0.

Remark 1. There is no precise terminology in literature that has been generally adopted for the algebraic structure(Q,+, ·). In the paper [69] has been used the term of R-semigroup with cancellation law. If the cancellative law (q4)does not hold and the second distributive law (q8) is true only for λ,μ ≥ 0, in [22] is used the notion of almost linearspace. Without the cancellative law and with the property:

(q8′) 0 · A = θ for all A ∈Q,

instead of second distributive law, in the work [23] is used the term of semilinear space. The term of semilinear spaceis also used in the paper [72] where the domain R of real scalars is replaced by the set of positive reals R+, and theexamples can be continued.

The generalized Hukuhara difference (or gH-difference) of two intervals [a−, a+], [b−, b+] ∈ K is defined asfollows (Markov [45] and Stefanini [67]):[

a−, a+]�g

[b−, b+]= [

min{a− − b−, a+ − b+},max

{a− − b−, a+ − b+}]. (1)

We denote the width of an interval A = [a−, a+] by w(A) = a+ − a−. Then, for A = [a−, a+] and B = [b−, b+], wehave

A �g B ={ [a− − b−, a+ − b+], if w(A) ≥ w(B)

[a+ − b+, a− − b−], if w(A) < w(B).(2)

If A,B,C ∈K then it is easy to see that

A �g B = C ⇐⇒{

A = B + C, if w(A) ≥ w(B)

B = A + (−C), if w(A) < w(B).(3)

If A,B ∈K and w(A) ≥ w(B), then the gH-difference A�g B will be denoted by A�B and it is called the Hukuharadifference (or H -difference) of A and B . For other properties involving the operations on K, see Markov [45], andStefanini and Bede [68]. If A ∈ K, let us define the norm of A by ‖A‖ := max{|a−|, |a+|}. Then it is easy to seethat ‖ · ‖ is a norm on K, and therefore (K,+, ·,‖ · ‖) is a normed quasilinear space. A metric structure on K isgiven by the Hausdorff–Pompeiu distance H : K × K → [0,∞) defined by H(A,B) = max{|a− − b−|, |a+ − b+|},where A = [a−, a+] and B = [b−, b+]. Obviously, the metric H is associated with the norm ‖ · ‖ by ‖A‖ = H(A, {0})and H(A,B) = ‖A �g B‖. It is well known that (K,H) is a complete and separable metric space (Debreu [14]).Since K is a normed quasilinear space, the continuity and the limits of an interval-valued function F : [a, b] → Kare understood in the sense of the norm ‖ · ‖. We recall that if F : [a, b] → K is an interval-valued function suchthat F(t) = [f −(t), f +(t)], then limt→t0 F(t) exists, if and only if limt→t0 f −(t) and limt→t0 f +(t) exist as finitenumbers. In this case, we have

limt→t0

F(t) =[

limt→t0

f −(t), limt→t0

f +(t)].

In particular, F is continuous if and only if f − and f + are continuous. If F,G : [a, b] → K are two interval-valuedfunctions, then we define the interval-valued function F �g G : [a, b] → K by (F �g G)(t) := F(t) �g G(t), for allt ∈ [a, b]. If there exist limt→t0 F(t) = A and limt→t0 G(t) = B , then limt→t0(F �g G)(t) exists, and

limt→t0

(F �g G)(t) = A �g B.

In particular, if F,G : [a, b] → K are continuous, then the interval-function F �g G is a continuous interval-valuedfunction. Let C([a, b],K) denote the set of continuous interval-valued functions from [a, b] into K. Then C([a, b],K)

is a complete normed space with respect to the norm ‖F‖c := supa≤t≤b ‖F(t)‖.

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Definition 1. (See Markov [45] and Stefanini [67].) Let F : [a, b] → K be an interval-valued function and let t0 ∈[a, b]. We define F ′(t0) ∈ K (provided it exists) as

F ′(t0) = limh→0

F(t0 + h) �g F (t0)

h. (4)

We call F ′(t0) the generalized Hukuhara derivative (gH-derivative for short) of F at t0. Also, we define the leftgH-derivative F ′+(t0) ∈K (provided it exists) as

F ′−(t0) = limh→0−

F(t0 + h) �g F (t0)

h,

and the right gH-derivative F ′−(t0) ∈ K (provided it exists) as

F ′+(t0) = limh→0+

F(t0 + h) �g F (t0)

h.

We say that F is generalized Hukuhara differentiable (gH-differentiable for short) on [a, b] if F ′(t) ∈ K exists ateach point t ∈ [a, b]. At the end points of [a, b] we consider only the one sided gH-derivatives. The interval-valuedfunction F ′ : [a, b] → K is then called the gH-derivative of F on [a, b].

Proposition 1. (See Markov [45] and Stefanini [67].) Let F : [a, b] → K be such that F(t) = [f −(t), f +(t)], t ∈[a, b]. If the real-valued functions f − and f + are differentiable at t ∈ [a, b], then F is gH-differentiable at t ∈ [a, b]and

F ′(t) =[

min

{d

dtf −(t),

d

dtf +(t)

},max

{d

dtf −(t),

d

dtf +(t)

}]. (5)

The converse of Theorem 1 is not true, that is, the gH-differentiability of F does not imply the differentiability off − and f + (see Chalco-Cano et al. [18] and Markov [45]).

We say that an interval-valued function F : [a, b] →K is w-increasing (w-decreasing) on [a, b] if the real functiont �→ wF (t) := w(F(t)) is increasing (decreasing) on [a, b]. If F is w-increasing or w-decreasing on [a, b], then wesay that F is w-monotone on [a, b].

Proposition 2. (See Markov [45].) Let F : [a, b] → K be such that F(t) = [f −(t), f +(t)], t ∈ [a, b]. If F isw-monotone and gH-differentiable on [a, b], then d

dtf −(t) and d

dtf +(t) exist for all t ∈ [a, b]. Moreover, we have

that:

(i) F ′(t) = [ ddt

f −(t), ddt

f +(t)] for all t ∈ [a, b], if F is w-increasing;

(ii) F ′(t) = [ ddt

f +(t), ddt

f −(t)] for all t ∈ [a, b], if F is w-decreasing.

Remark 2. Let F : [a, b] → K be such that the both one sided derivatives d±dt

f −(τ ) and d±dt

f +(τ ) exist and are finite,where τ ∈ (a, b) is a given point. If F is w-increasing on [a, τ ] and w-decreasing on [τ, b], then it is easy to see that

F ′−(τ ) =[d−

dtf −(τ ),

d−

dtf +(τ )

]and F ′+(τ ) =

[d+

dtf +(τ ),

d+

dtf −(τ )

].

If F is w-decreasing on [a, τ ] and w-increasing on [τ, b], then

F ′−(τ ) =[d−

dtf +(τ ),

d−

dtf −(τ )

]and F ′+(τ ) =

[d+

dtf −(τ ),

d+

dtf +(τ )

].

Therefore, in the both cases we have that F ′−(τ ) = F ′+(τ ) (that is, F is gH-differentiable at τ ) if and only if (seeTheorem 6 in Chalco-Cano et al. [18])

d−

dtf −(τ ) = d+

dtf +(τ ) and

d−

dtf +(τ ) = d+

dtf −(τ ).

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Example 1. Consider the interval-valued function F : [0,1] →K given by F(t) = [−t2 − 1, t2 − 2t]. Since wF (t) =2t2 − 2t + 1, it follows that F is w-decreasing on [0, 1

2 ] and w-increasing on [ 12 ,1]. Since f −(t) = −t2 − 1 and

f +(t) = t2 − 2t are differentiable on [0,1], then by Proposition 1 and Remark 2, we obtain that

F ′(t) =

⎧⎪⎨⎪⎩

[2t − 2,−2t] if t ∈ [0, 12 )

{−1} if t = 12

[−2t,2t − 2] if t ∈ ( 12 ,1].

If we consider the interval-valued function G : [0,2] →K given by G(t) = [2t − 3, |t2 − 1|], then G is w-decreasingon [0,1] and w-increasing on [1,2]. Also g− and g+ are differentiable on [0,2]�{1}, d−

dtg−(1) = d+

dtg−(1) = 2,

d−dt

g+(1) = −2 and d+dt

g+(1) = 2. It follows that G′−(1) = [−2,2], G′+(1) = {2}, G is gH-differentiable on [0,2]�{1}and

G′(t) ={ [−2t,2] if t ∈ [0,1)

[2,2t] if t ∈ (1,2].

Proposition 3. (See Markov [45].) Let F : [a, b] →K be w-monotone and gH-differentiable on [a, b]. The followingproperties are then true:

(a) For all C ∈ K and for all λ ∈ R, the interval-valued functions F + C, F �g C and λF are gH-differentiable on[a, b], and (F + C)′ = F ′, (F �g C)′ = F ′ and (λF )′ = λF ′, respectively.

(b) If F and G are equally w-monotonic (that is, both are w-increasing or both are w-decreasing), then (F + G)′ =F ′ + G′ and (F �g G)′ = F ′ �g G′.

(c) If F and G are differently w-monotonic (that is, one is w-increasing and the other is w-decreasing), then (F +G)′ = F ′ �g (−G′) and (F �g G)′ = F ′ + (−G′).

In the following, denote by C1([a, b],K) the space of interval-valued functions which are continuouslygH-differentiable on [a, b]. For γ ∈ [0,1), we introduce the space Cγ ([a, b],K) of interval-valued functionsF : [a, b] → K such that the function (· − a)γ F (·) ∈ C([a, b],K). Also, C1

γ ([a, b],K) denotes the space of interval-valued functions F ∈ C([a, b],K) which have the gH-derivative F ′ on (a, b] such that F ′ ∈ Cγ ([a, b],K).

The Lebesgue integral for interval-valued functions is a special case of the Lebesgue integral for set-valued map-pings (Aumann [11]).

Let F : [a, b] → K be an interval-valued function such that F(t) = [f −(t), f +(t)] and f − and f + are measurableand Lebesgue integrable on [a, b]. Then we define

∫ b

aF (t)dt by

b∫a

F (t)dt =[ b∫

a

f −(t)dt,

b∫a

f +(t)dt

], (6)

and we say that F is Lebesgue integrable on [a, b].We recall that an interval-valued function F : [a, b] → K is called a step interval-valued function if there exists a

partition {Jk; k = 1,2, ..., n} of disjoint Lebesgue measurable subsets in [a, b], ⋃nk=1 Jk = [a, b] such that F is con-

stant on each set Jk , k = 1,2, ..., n. Since (K,H) is a complete and separable metric space, then we can consider thefollowing definition for the measurability of the interval-valued functions. An interval-valued function F : [a, b] → Kis said to be measurable if there exists a sequence Fn: [a, b] →K of step interval-valued functions such that

limn→∞H

(Fn(t),F (t)

)= 0 for a.e. t ∈ [a, b].

Given that limn→∞ H(Fn(t),F (t)) = 0 if and only if limn→∞ |f ±n (t) − f ±(t)| = 0, then it is clear that an interval-

valued function F : [a, b] → K is measurable if and only if f − and f + are measurable. Also, it is easy to show thatF : [a, b] → K is integrable on [a, b] if and only if F is measurable and the real function t �→ ‖F(t)‖ is Lebesgueintegrable on [a, b] (see Aumann [11], Artstein [10]).

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For 1 ≤ p ≤ ∞, let Lp([a, b],K) be the set of all interval-valued functions F : [a, b] → K such that the realfunction t �→ ‖F(t)‖ belongs to Lp[a, b]. Then Lp([a, b],K) is a complete metric space (see Hiai and Umegaki [24],Salinetti [62]) with respect to the metric Hp defined by Hp(F,G) := ‖F �g G‖p , where

‖F‖p :={

(∫ b

a‖F(t)‖pdt)1/p if 1 ≤ p < ∞,

ess supt∈[a,b] ‖F(t)‖ if p = ∞.

An interval-valued function F : [a, b] → K is said to be absolutely continuous if, for each ε > 0, there exists δ > 0such that, for each family {(sk, tk); k = 1,2, ..., n} of disjoint open intervals in [a, b] with

∑nk=1(tk − sk) < δ, we have∑n

k=1 H(F (tk),F (sk)) < ε. Let AC([a, b],K) denote the set of all absolutely continuous interval-valued functionsfrom [a, b] into K.

Proposition 4. An interval-valued function F : [a, b] → K is absolutely continuous if and only if f − and f + are bothabsolutely continuous.

Proof. Assume that F is absolutely continuous. If ε > 0 is arbitrary, there exists δ > 0 such that, for each family{(sk, tk); k = 1,2, ..., n} of disjoint open intervals in [a, b] with

∑nk=1(tk − sk) < δ, we have

∑nk=1 H(F (tk),F (sk)) <

ε. Since∣∣f ±(tk) − f ±(sk)∣∣≤ max

{∣∣f −(tk) − f −(sk)∣∣, ∣∣f +(tk) − f +(sk)

∣∣}=H

(F(tk),F (sk)

)for all k = 1,2, ..., n, it follows that

n∑k=1

∣∣f ±(tk) − f ±(sk)∣∣≤ n∑

k=1

H(F(tk),F (sk)

)< ε

and so f − and f + are both absolutely continuous. Conversely, if f − and f + are both absolutely continuous, thenhere exists δ > 0 such that, for each family {(sk, tk); k = 1,2, ..., n} of disjoint open intervals in [a, b] with

∑nk=1(tk −

sk) < δ, we have∑n

k=1 |f ±(tk) − f ±(sk)| < ε/2. Then we have that

n∑k=1

H(F(tk),F (sk)

)=n∑

k=1

max{∣∣f −(tk) − f −(sk)

∣∣, ∣∣f +(tk) − f +(sk)∣∣}

≤n∑

k=1

[∣∣f −(tk) − f −(sk)∣∣+ ∣∣f +(tk) − f +(sk)

∣∣]< ε.

This proves that F is absolutely continuous. �Proposition 5. (See Markov [45].) If F : [a, b] → K is Lebesgue integrable on [a, b], then the interval-valued functionG : [a, b] → K, defined by G(t) := ∫ t

aF (s)ds for all t ∈ [a, b], is absolutely continuous and G′(t) = F(t) for a.e.

t ∈ [a, b].

Proposition 6. If F ∈ AC([a, b],K), then F is gH-differentiable a.e. on [a, b] and F ′ ∈ L1([a, b],K). Moreover, if F

is w-monotone on [a, b], then

F(t) �g F (a) =t∫

a

F ′(s)ds for all t ∈ [a, b]. (7)

Proof. If F = [f −, f +] ∈ AC([a, b],K), then from Proposition 4 it follows that f − and f + are absolutely contin-uous. Hence, by Corollary 7.23 in Wheeden and Zygmund [71], it follows that d

dtf − and d

dtf + exist a.e. on [a, b]

and ddt

f −, ddt

f + ∈ L1[a, b]. Therefore, Propositions 1 and 4 imply that F is gH-differentiable a.e. on [a, b] andF ′ ∈ L1([a, b],K). If F ∈ AC([a, b],K) is w-increasing on [a, b], then F ′(t) = [ d f −(t), d f +(t)] for a.e. t ∈ [a, b].

dt dt

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Then by Theorem 7.29 in Wheeden and Zygmund [71] we have that f ±(t) = f ±(a) + ∫ t

adds

(f ±)(s)ds for all

t ∈ [a, b]. If follows that F(t) = F(a)+∫ t

aF ′(s)ds for all t ∈ [a, b]. Since F is w-increasing on [a, b] then, by (3), the

last equality and (7) are equivalent. If F ∈ AC([a, b],K) is w-decreasing on [a, b], then F ′(t) = [ ddt

f +(t), ddt

f −(t)]for a.e. t ∈ [a, b]. Since F is w-decreasing on [a, b] then, from (2), we obtain that

t∫a

F ′(s)ds = [f +(t) − f +(a), f −(t) − f −(a)

]= [

f −(t), f +(t)]�g

[f −(a), f +(a)

]= F(t) �g F (a). �Remark 3. A similar result can be found in Markov [45] for w-increasing interval-valued functions. Also, we remarkthat if F is w-increasing on [a, b], then (7) is equivalent with

F(t) = F(a) +t∫

a

F ′(s)ds,

and if F is w-decreasing on [a, b] then (7) is equivalent with

F(t) = F(a) � (−1)

t∫a

F ′(s)ds,

for all t ∈ [a, b].

We note that the relation (7) is false if F is not w-monotone on [a, b].

Example 2. If F : [0,1] →K is the interval-valued function given in Example 1, then for all t ∈ ( 12 ,1] we have that

t∫0

F ′(s)ds =1/2∫0

[2s − 2,−2s]ds +t∫

1/2

[−2s,2s − 2]ds =[−t2 − 1

2, t2 − 2t + 1

2

]

and

F(t) �g F (0) = [−t2 − 1,2t2 − 2t]�g [−1,0] = [

t2 − 2t,−t2] �=t∫

0

F ′(s)ds

for all t ∈ ( 12 ,1]. Therefore, (7) is not true for all t ∈ [0,1].

3. Interval-valued Riemann–Liouville fractional integral

In this section we give the definition and present some properties of the interval fractional integral of fractionalorder. First, we recall that if ϕ ∈ L1[a, b], then the Riemann–Liouville fractional integral Iα

a+ϕ of order α > 0 isdefined by (see [28])

Iαa+ϕ(t) := 1

Γ (α)

t∫a

(t − s)α−1ϕ(s)ds, for all t ∈ [a, b].

Let F ∈ L1([a, b],K) and α > 0. Since, for s < t , the real-valued function s �→ (t − s)α−1 is Lebesgue integrable on[a, t], then the real-valued functions s �→ (t − s)α−1f ±(s) are also Lebesgue integrable on [0, t] for each t ∈ [a, b].It follows that the interval-valued function s �→ (t − s)α−1F(s) is Lebesgue integrable on [0, t] for each t ∈ [a, b].Hence

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Iαa+F(t) := 1

Γ (α)

t∫a

(t − s)α−1F(s)ds (8)

exists for all t ∈ [a, b].Iα

a+F is called the interval-valued Riemann–Liouville fractional integral of order α > 0. If F = [f −, f +] ∈L1([a, b],K) and α > 0, then it is obvious that

Iαa+F(t) = [

Iαa+f −(t), Iα

a+f +(t)], for all t ∈ [a, b]. (9)

Lemma 1. Let ϕ ∈ L1[a, b] be a positive and increasing real function on [a, b] and let α ∈ (0,1). Then the realfunction t �→ φ(t) := ∫ t

a(t − s)−αϕ(s)ds is also increasing on [a, b].

Proof. Let τ, t ∈ [a, b] be such that τ ≤ t . Since

t∫τ

(t − s)−αds ≥τ∫

a

[(τ − s)−α − (t − s)−α

]ds

and maxs∈[a,τ ] ϕ(s) = ϕ(τ), then we have

t∫τ

(t − s)−αϕ(s)ds ≥t∫

τ

(t − s)−αϕ(τ)ds

≥τ∫

a

[(τ − s)−α − (t − s)−α

]ϕ(τ)ds

≥τ∫

a

[(τ − s)−α − (t − s)−α

]ϕ(s)ds.

Therefore

φ(t) − φ(τ) =t∫

a

(t − s)−αϕ(s)ds −τ∫

a

(τ − s)−αϕ(s)ds

= −τ∫

a

[(τ − s)−α − (t − s)−α

]ϕ(s)ds +

t∫τ

(t − s)−αϕ(s)ds ≥ 0,

and thus φ is an increasing function. �Remark 4. From the previous lemma it follows that if F ∈ Lp([a, b],K) (1 ≤ p ≤ ∞) is w-increasing on [a, b], thenthe interval-valued functions t �→ Iα

a+F(t) and t �→ I1−αa+ F(t) are w-increasing on [a, b] if α ∈ (0,1). If ϕ ∈ L1[a, b]

is positive and decreasing on [a, b], then the real function t �→ φ(t) is not decreasing on [a, b], in general. For example,if ϕ : [0,2] → [0,2] is given by ϕ(t) = 1 − t , then the function

φ(t) =t∫

0

(t − s)−αϕ(s)ds = t1−α

1 − α

(1 − t

2 − α

)

is increasing on [0,2 − α], and decreasing on [2 − α,2], for α ∈ (0,1).

Remark 5. If F , G ∈ Lp([a, b],K) (1 ≤ p ≤ ∞) and α, β > 0 then it is obvious that:

(a) Iα+Iβ

+F(t) = Iα+β

+ F(t) for a.e. t ∈ [a, b],

a a a
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(b) Iαa+(F + G)(t) = Iα

a+F(t) + Iαa+G(t) for all t ∈ [a, b],

(3) w(Iαa+F(t)) = Iαw(F (t)) for all t ∈ [a, b].

If α + β > 1, then the relation in (a) holds for all t ∈ [a, b] (see Lemma 2.3 in Kilbas et al. [28])

Theorem 1. If F , G ∈ L1([a, b],K) and α > 0, then

Iαa+(F �g G)(t) ⊇ Iα

a+F(t) �g Iαa+G(t), for all t ∈ [a, b]. (10)

Moreover, if the difference w(F(t)) − w(G(t)) has a constant sign on [a, b], then

Iαa+(F �g G)(t) = Iα

a+F(t) �g Iαa+G(t), for all t ∈ [a, b]. (11)

Proof. Let ϕ− := f − − g− and ϕ+ := f + − g+. The inequalities

t∫a

(t − s)α−1 min{ϕ−(s), ϕ+(s)

}ds

≤ min

{ t∫a

(t − s)α−1ϕ−(s)ds,

t∫a

(t − s)α−1ϕ+(s)ds

}

≤ max

{ t∫a

(t − s)α−1ϕ−(s)ds,

t∫a

(t − s)α−1ϕ+(s)ds

}

≤t∫

a

(t − s)α−1 max{ϕ−(s), ϕ+(s)

}ds, (12)

imply that[min

{Iαa+ϕ−(t), Iα

a+ϕ+(t)},max

{Iαa+ϕ−(t), Iα

a+ϕ+(t)}]

⊆ [Iαa+ min

{ϕ−(t), ϕ+(t)

}, Iα

a+ max{ϕ−(t), ϕ+(t)

}],

for all t ∈ [a, b]. Therefore, for all t ∈ [a, b] we have

Iαa+F(t) �g I

αa+G(t) = [

Iαa+f −(t), Iα

a+f +(t)]�g

[Iαa+g−(t), Iα

a+g+(t)]

= [min

{Iαa+ϕ−(t), Iα

a+ϕ+(t)},max

{Iαa+ϕ−(t), Iα

a+ϕ+(t)}]

⊆ [Iαa+ min

{ϕ−(t), ϕ+(t)

}, Iα

a+ max{ϕ−(t), ϕ+(t)

}]= Iαa+(F �g G)(t);

that is (10). If the difference w(F(t)) − w(G(t)) has a constant sign on [a, b], then (F �g G)(t) = [ϕ−(t), ϕ+(t)] ifw(F(t)) − w(G(t)) ≥ 0 on [a, b], or (F �g G)(t) = [ϕ+(t), ϕ−(t)] if w(F(t)) − w(G(t)) < 0 on [a, b]. In the firstcase, since f − − g− ≤ f + − g+ on [a, b], then 0 ≤ (t − s)α−1ϕ−(s) ≤ (t − s)α−1ϕ+(s) for a ≤ s < t ≤ b, and soIαa+ϕ−(t) ≤ Iα

a+ϕ+(t) for all t ∈ [a, b]. Therefore, we have that

Iαa+F(t) �g I

αa+G(t) = [

Iαa+f −(t), Iα

a+f +(t)]�g

[Iαa+g−(t), Iα

a+g+(t)]

= [min

{Iαa+ϕ−(t), Iα

a+ϕ+(t)},max

{Iαa+ϕ−(t), Iα

a+ϕ+(t)}]

= [Iαa+ϕ−(t), Iα

a+ϕ+(t)]= Iα

a+(F �g G)(t).

Since a similar reasoning applies for the second case, the equality (11) is proved. �In the following example we show that the inclusion in (10) is strict.

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Example 3. Let F,G : [0,3] → K be given by F(t) = [0, t] and G(t) = [−2,0], respectively. First, we remark thatw(F(t)) − w(G(t)) = t − 2 has not a constant sign on [0,3]. We have

I120+F(t) = 1

Γ (1/2)

t∫0

(t − s)−1/2[0, s]ds

=[

0,4

3√

πt3/2

], t ∈ [0,3],

I120+G(t) = 1

Γ (1/2)

t∫0

(t − s)−1/2[−2,0]ds

=[− 4√

πt1/2,0

], t ∈ [0,3],

and thus

I12 F(t) �g I

12 G(t) =

[min

{4√π

t1/2,4

3√

πt3/2

},max

{4√π

t1/2,4

3√

πt3/2

}]

=[

4

3√

πt3/2,

4√π

t1/2]

= 4

3√

π

[t3/2,3t1/2]

for all t ∈ [0,3]. On the other hand, we have that

(F �g G)(t) = [min{2, t},max{2, t}]=

{ [t,2] if t ∈ [0,2][2, t] if t ∈ (2,3].

For t ∈ [0,2], the difference w(F(t)) − w(G(t)) = t − 2 has a constant sign on [0,2], and we have

I120+(F �g G)(t) = 1

Γ (1/2)

t∫0

(t − s)−1/2[s,2]ds

= 4

3√

π

[t3/2,3t1/2];

that is, I12a+(F �g G)(t) = I

12a+F(t) �g I

12a+G(t), for t ∈ [0,2]. But, for t ∈ (2,3], we have

I120+(F �g G)(t) = 1

Γ (1/2)

2∫0

(t − s)−1/2[s,2]ds + 1

Γ (1/2)

t∫2

(t − s)−1/2[2, s]ds

= 4

3√

π

[t3/2 − (t − 2)

√t − 2,3t1/2 + (t − 2)

√t − 2

]⊃ 4

3√

π

[t3/2,3t1/2];

that is, I12a+(F �g G)(t) ⊃ I

12a+F(t) �g I

12a+G(t) for t ∈ (2,3]. It follows that the inclusion in (10) is strict on [0,3].

The following result can be obtained by a standard argument (see Theorem 2.6 and Theorem 3.5 in Samkoet al. [63]).

Theorem 2. The interval Riemann–Liouville integral of order α > 0 is a bounded operator from Lp([a, b],K) intoLp([a, b],K) (1 ≤ p ≤ ∞):

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∥∥Iαa+F

∥∥p

≤ (b − a)α

Γ (α + 1)‖F‖p.

Moreover, if α ∈ (0,1) and 1 < p < 1/α, then Iαa+ is a bounded operator from Lp([a, b],K) into Lq([a, b],K), where

q = p/(1 − αp). �4. Interval-valued Riemann–Liouville fractional derivative

First, we recall that the Riemann–Liouville derivative of order α ∈ (0,1] for a real function ϕ ∈ AC[a, b] is definedfor a.e. t ∈ [a, b] by (see [28])

Dαa+ϕ(t) = d

dtI 1−αa+ ϕ(t) = 1

Γ (1 − α)

d

dt

t∫a

(t − s)−αϕ(s)ds.

In particular, when α = 1, then D1a+ϕ(t) = d

dtϕ(t) for a.e. t ∈ [a, b]. For a given interval-valued function F =

[f −, f +] ∈ L1([a, b],K) and α ∈ (0,1], we define the interval-valued function F1−α : [a, b] → K by

F1−α(t) = I1−αa+ F(t) := 1

Γ (1 − α)

t∫a

(t − s)−αF (s)ds for all t ∈ [a, b].

If the gH-derivative F ′1−α(t) exists for a.e. t ∈ [a, b], then F ′

1−α(t) is called the interval-valued Riemann–Liouvillefractional derivative (or Riemann–Liouville gH-fractional derivative) of order α ∈ (0,1]. The Riemann–LiouvillegH-fractional derivative of F will be denoted by Dα

a+F . Therefore,

Dαa+F(t) := (

I1−αa+ F

)′(t) for a.e. t ∈ [a, b].

Theorem 3. Let F = [f −, f +] ∈ AC([a, b],K). Then:

(a) F1−α ∈ AC([a, b],K) and

Dαa+F(t) = [

min{Dα

a+f −(t),Dαa+f +(t)

},max

{Dα

a+f −(t),Dαa+f +(t)

}], (13)

for a.e. t ∈ [a, b]. In particular, when α = 1, then D1a+F(t) = F ′(t) for a.e. t ∈ [a, b].

(b) If either F is w-increasing on [a, b] or F is w-decreasing and F1−α is w-increasing on [a, b], then

Dαa+F(t) = [

Dαa+f −(t),Dα

a+f +(t)]

for a.e. t ∈ [a, b]. (14)

(c) If F1−α is w-decreasing on [a, b], then

Dαa+F(t) = [

Dαa+f +(t),Dα

a+f −(t)]

for a.e. t ∈ [a, b]. (15)

Proof. Since F = [f −, f +] ∈ AC([a, b],K), then (see Lemma 2.1 in Samko et al. [63]) the real functions t �→I 1−αa+ f −(t) and t �→ I 1−α

a+ f +(t) are absolutely continuous on [a, b]. It follows that F1−α ∈ AC([a, b],K), and thusF ′

1−α(t) exists for a.e. t ∈ [a, b]. Therefore, Dαa+F exists for a.e. t ∈ [a, b], and by Proposition 1 we have

Dαa+F(t) = F ′

1−α(t) =[

min

{d

dtf −

1−α(t),d

dtf +

1−α(t)

},max

{d

dtf −

1−α(t),d

dtf +

1−α(t)

}]= [

min{Dα

a+f −(t),Dαa+f +(t)

},max

{Dα

a+f −(t),Dαa+f +(t)

}],

for a.e. t ∈ [a, b]. Next, suppose F is w-increasing on [a, b]. Since w(I1−αa+ F(t)) = I 1−α

a+ w(F(t)) and t �→ w(F(t))

is increasing on [a, b], then by Lemma 1 it follows that t �→ F1−α(t) is w-increasing on [a, b]. Therefore,

Dαa+F(t) = F ′

1−α(t) =[

d

dtI 1−αa+ f −(t),

d

dtI 1−αa+ f +(t)

]= [

Dαa+f −(t),Dα

a+f +(t)]

for a.e. t ∈ [a, b]. If F1−α is w-decreasing on [a, b], then

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Dαa+F(t) = F ′

1−α(t) =[

d

dtI 1−αa+ f +(t),

d

dtI 1−αa+ f −(t)

]= [

Dαa+f +(t),Dα

a+f −(t)]

for a.e. t ∈ [a, b]. �The following result is a direct consequence of Proposition 3.

Theorem 4. Let F,G ∈ AC([a, b],K) be w-monotone on [a, b], and let α ∈ (0,1].

(a) If F1−α and G1−α are equally w-monotonic on [a, b], then

Dαa+(F + G)(t) =Dα

a+F(t) +Dαa+G(t) for a.e. t ∈ [a, b]

and

Dαa+(F �g G)(t) =Dα

a+F(t) �g Dαa+G(t) for a.e. t ∈ [a, b].

(b) If F1−α and G1−α are differently w-monotonic on [a, b], then

Dαa+(F + G)(t) =Dα

a+F(t) �g

(−Dαa+G

)(t) for a.e. t ∈ [a, b]

and

Dαa+(F �g G)(t) =Dα

a+F(t) + (−Dαa+G

)(t) for a.e. t ∈ [a, b].

Theorem 5. If F ∈ Lp([a, b],K) (1 ≤ p ≤ ∞), then

Dαa+Iα

a+F(t) = F(t) for a.e. t ∈ [a, b]. (16)

Proof. Indeed, by the definition of the Riemann–Liouville gH-fractional derivative, Proposition 4 and Remark 5, wehave

Dαa+Iα

a+F(t) = (I

1−αa+ Iα

a+F)′(t) = (

I1a+F

)′(t)

=( t∫

a

F (s)ds

)′= F(t),

for a.e. t ∈ [a, b]. �Theorem 6. Let F ∈ Lp([a, b],K) (1 ≤ p ≤ ∞) be such that F1−α ∈ AC([a, b],K). If there exists an interval-valuedfunction G ∈ Lp([a, b],K) such that F = Iα

a+G, then

Iαa+Dα

a+F(t) = F(t) for a.e. t ∈ [a, b]. (17)

Proof. Indeed, we have that

Iαa+Dα

a+F(t) = Iαa+(I

1−αa+ F

)′(t) = Iα

a+(I

1−αa+ Iα

a+G)′(t)

= Iαa+(I1

a+G)′(t) = Iα

a+G(t) = F(t),

for a.e. t ∈ [a, b]. �Lemma 2. Let F ∈ L1([a, b],K) be such that F1−α ∈ AC([a, b],K).

(a) If ddt

w(F1−α(t)) ≥ 0 for a.e. t ∈ [a, b], then

w(F(t)

)≥ (t − a)α−1

Γ (α)w(F1−α(a)

)for a.e. t ∈ [a, b].

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(b) If ddt

w(F1−α(t)) ≤ 0 for a.e. t ∈ [a, b], then

w(F(t)

)≤ (t − a)α−1

Γ (α)w(F1−α(a)

)for a.e. t ∈ [a, b].

Proof. It is known that (see Lemma 2.5 in Kilbas et al. [28]) Iαa+Dα

a+w(F(t)) = w(F(t)) − (t−a)α−1

Γ (α)w(F1−α(a)) for

a.e. t ∈ [a, b]. Since

Iαa+Dα

a+w(F(t)

)= Iαa+

d

dtI 1−αa+ w

(F(t)

)= Iαa+

d

dtw(F1−α(t)

)for a.e. t ∈ [a, b],

it follows that

w(F(t)

)− (t − a)α−1

Γ (α)w(F1−α(a)

)= Iαa+

d

dtw(F1−α(t)

)for a.e. t ∈ [a, b].

Noting that Iαa+

ddt

w(F1−α(t)) ≥ 0 for a.e. t ∈ [a, b] if ddt

w(F1−α(t)) ≥ 0 for a.e. t ∈ [a, b], and Iαa+

ddt

w(F1−α(t)) ≤ 0

for a.e. t ∈ [a, b] if ddt

w(F1−α(t)) ≤ 0 for a.e. t ∈ [a, b], the proof is completed. �Theorem 7. Let F ∈ L1([a, b],K) be such that F1−α ∈ AC([a, b],K). If either d

dtw(F1−α(t)) ≥ 0 for a.e. t ∈ [a, b]

or ddt

w(F1−α(t)) ≤ 0 for a.e. t ∈ [a, b], then the gH-difference F(t) �g(t−a)α−1

Γ (α)F1−α(a) exists for a.e. t ∈ [a, b], and

Iαa+Dα

a+F(t) = F(t) �g

(t − a)α−1

Γ (α)F1−α(a) for a.e. t ∈ [a, b]. (18)

Proof. The existence of the difference F(t)�g(t−a)α−1

Γ (α)F1−α(a) follows from Lemma 2. Further, it is known that (see

Lemma 2.5 in Kilbas et al. [28]) for a real-valued function ϕ ∈ L1[a, b] such that ϕ ∈ AC[a, b], we have Iαa+Dα

a+ϕ(t) =ϕ(t)− (t−a)α−1

Γ (α)ϕ1−α(a) for a.e. t ∈ [a, b]. If d

dtw(F1−α(t)) ≥ 0 for a.e. t ∈ [a, b], then F1−α is w-increasing on [a, b].

Thus

Iαa+Dα

a+F(t) = Iαa+F ′

1−α(t) = Iαa+[Dα

a+f −,Dαa+f +](t)

= [Iαa+Dα

a+f −(t), Iαa+Dα

a+f +(t)]

=[f −(t) − (t − a)α−1

Γ (α)f −

1−α(a), f +(t) − (t − a)α−1

Γ (α)f +

1−α(a)

]

= F(t) �g

(t − a)α−1

Γ (α)F1−α(a),

for a.e. t ∈ [a, b]. By a similar reasoning we obtain (18) if ddt

w(F1−α(t)) ≤ 0 for a.e. t ∈ [a, b]. �Remark 6. In the conditions of Theorem 5, the relation (18) can be rewritten as

F(t) = (t − a)α−1

Γ (α)F1−α(a) + Iα

a+Dαa+F(t) for a.e. t ∈ [a, b]

if ddt

w(F1−α(t)) ≥ 0 for a.e. t ∈ [a, b], and as

F(t) = (t − a)α−1

Γ (α)F1−α(a) � (−Iα

a+Dαa+F(t)

)for a.e. t ∈ [a, b]

if ddt

w(F1−α(t)) ≤ 0 for a.e. t ∈ [a, b].

The following assertions are a direct consequence of Lemma 2.9 from Kilbas [28] and of the properties of interval-valued fractional operators.

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Theorem 8. Let α ∈ (0,1) and γ ∈ [0,1). The following properties are then true:

(a) If F ∈ Cγ ([a, b],K), then the relations in Remark 5 and the relation (16) hold for all t ∈ [a, b].(b) If F ∈ Cγ ([a, b],K) and F1−α ∈ C1

γ ([a, b],K), then the relation (18) holds for all t ∈ [a, b]. �Example 4. Let us consider the interval-valued function F : [0,1] → K, given by F(t) = [t1/2, t−1/2] if t ∈ (0,1],and F(0) = [0,1]. Then F ∈ L1([0,1],K), F is w-decreasing on (0,1], and

F1− 12(t) := I

1− 12

0+ F(t) = 1√π

2t, π

]for all t ∈ [0,1].

It follows that F1− 12

∈ AC([0,1],K) and ddt

w(F1− 12(t)) ≤ 0 for all t ∈ [0,1]. Since the interval-valued function F1− 1

2is w-decreasing on [0,1], then we have that

D120+F(t) := F ′

1− 12(t) =

[0,

√π

2

]for all t ∈ [0,1],

and so I120+D

120+F(t) = [0, t1/2] for all t ∈ [0,1]. On the other hand,

F(t) �g

t12 −1

Γ (1/2)F1− 1

2(0) = [

t1/2, t−1/2]�g

[0, t−1/2]= [

0, t1/2],that is, I

120+D

120+F(t) = F(t) �g

t12 −1

Γ (1/2)F1− 1

2(0) for all t ∈ [0,1]. Here, we note that F ∈ C 1

2([0,1],K) and F1− 1

2∈

C112([0,1],K).

The following example shows us that the relation (18) can be false if ddt

w(F1−α(t)) has not a constant sign on[a, b].

Example 5. Consider the interval-valued function F : [0,1] → K, given by F(t) = [0,1− t1/2] for all t ∈ [0,1]. ThenF ∈ L1([0,1],K), F is w-decreasing on [0,1], and

F1− 12(t) := I

1− 12

0+ F(t) = 1√π

[0,2t1/2 − π

2t

]for all t ∈ [0,1].

It follows that F1− 12

is gH-differentiable on (0,1], ddt

w(F1− 12(t)) ≥ 0 for all t ∈ (0,4/π2], and d

dtw(F1− 1

2(t)) ≤ 0 for

all t ∈ [4/π2,1]. Thus F1− 12

is w-increasing on (0,4/π2], w-decreasing on [4/π2,1], and we have that

D120+F(t) := F ′

1− 12(t) =

⎧⎨⎩

1√π[0, t−1/2 − π

2 ] if t ∈ (0,4/π2]1√π[t−1/2 − π

2 ,0] if t ∈ (4/π2,1].

Now, it is easy to see that F1− 12

∈ C112([0,1],K). Also, it is easy to check that I

120+D

120+F(t) = F(t) for all t ∈ [0,4/π2].

If t ∈ (4/π2,1], then we have

I120+D

120+F(t) = 1

Γ (1/2)

t∫0

(t − s)−1/2F ′1− 1

2(s)ds = 1

π

4/π2∫0

(t − s)−1/2[

0, s−1/2 − π

2

]ds

+ 1

π

t∫4/π2

(t − s)−1/2[s−1/2 − π

2,0

]ds = [

ξ−(t), ξ+(t)] �= F(t) �g

t12 −1

Γ (1/2)F1− 1

2(0),

where

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ξ−(t) = 1

π

2− arcsin

π2t − 8

π2t−√

π2t − 4

)

ξ+(t) = 1

π

2+ arcsin

π2t − 8

π2t+√

π2t − 4 − π√

t

),

for all t ∈ (4/π2,1]. Therefore, the relation (18) is not true for all t ∈ [0,1].

5. Interval-valued Caputo fractional derivative

In this section we give the definition and present some properties of the Caputo fractional derivative for interval-valued functions. First, we recall that if ϕ ∈ AC[a, b], then the Caputo fractional derivative Dα

a+ϕ of order α ∈ (0,1)

is defined for a.e. t ∈ [a, b] by

CDαa+ϕ(t) := d

dtI 1−αa+ ϕ′(t) = 1

Γ (1 − α)

t∫a

(t − s)−α d

dsϕ(s)ds.

If F ∈ AC([a, b],K) and α ∈ (0,1] then, by Proposition 4,

CDαa+F(t) := 1

Γ (1 − α)

t∫a

(t − s)−αF ′(s)ds, (19)

exists for a.e. t ∈ [a, b]. Obviously, CDαa+F(t) = I

1−αa+ F ′(t) for a.e. t ∈ [a, b]. CDα

a+F is called the interval-valuedCaputo fractional derivative (or Caputo gH-fractional derivative) of order α ∈ (0,1).

Remark 7. If F = [f −, f +] ∈ AC([a, b],K) and α ∈ (0,1], then it is obvious that

CDαa+F(t) = [

min{CDα

a+f −(t), CDαa+f +(t)

},max

{CDα

a+f −(t), CDαa+f +(t)

}],

for a.e. t ∈ [a, b]. If F is w-monotone, then

(i) CDαa+F(t) = [CDα

a+f −(t), CDαa+f +(t)] for a.e. t ∈ [a, b], if F is w-increasing;

(ii) CDαa+F(t) = [CDα

a+f +(t), CDαa+f −(t)] for a.e. t ∈ [a, b], if F is w-decreasing.

Theorem 9. Let F,G ∈ AC([a, b],K) be w-monotone on [a, b], and let α ∈ (0,1).

(a) If F and G are equally w-monotonic on [a, b], then

CDαa+(F + G)(t) = CDα

a+F(t) + CDαa+G(t) for a.e. t ∈ [a, b] (20)

and

CDαa+(F �g G)(t) ⊇ CDα

a+F(t) �gCDα

a+G(t) for a.e. t ∈ [a, b]. (21)

Moreover, if the difference w(F ′(t)) − w(G′(t)) has a constant sign for a.e. t ∈ [a, b], then

CDαa+(F �g G)(t) = CDα

a+F(t) �gCDα

a+G(t) for a.e. t ∈ [a, b]. (22)

(b) If F and G are differently w-monotonic on [a, b], then

CDαa+(F �g G)(t) = CDα

a+F(t) + (−CDαa+G

)(t) for a.e. t ∈ [a, b] (23)

and

CDαa+(F + G)(t) ⊇ CDα

a+F(t) �g

(−CDαa+G

)(t) for a.e. t ∈ [a, b]. (24)

Moreover, if the difference w(F ′(t)) − w(G′(t)) has a constant sign for a.e. t ∈ [a, b], then

CDαa+(F + G)(t) = CDα

a+F(t) �g

(−CDαa+G

)(t) for a.e. t ∈ [a, b]. (25)

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Proof. The equality (20) can be obtained by a simple calculation. Next, if both F and G are w-increasing orw-decreasing then, by Proposition 3, we have that (F �g G)′(t) = F ′(t) �g G′(t) for a.e. t ∈ [a, b]. Hence, fora.e. t ∈ [a, b], by Theorem 1 we have

CDαa+(F �g G)(t) = I

1−αa+ (F �g G)′(t) = I

1−αa+

(F ′ �g G′)(t)

⊇ I1−αa+ F ′(t) �g I

1−αa+ G′(t) = CDα

a+F(t) �gCDα

a+G(t);that is (21). If w(F ′(t)) − w(G′(t)) has a constant sign for a.e. t ∈ [a, b], then by Theorem 1 we obtain (22). Supposethat F and G are such that one is w-increasing and the other is w-decreasing. Then, by Proposition 3, we have that(F �g G)′(t) = F ′(t) + (−G′)(t) and (F + G)′(t) = F ′(t) �g (−G′)(t) for a.e. t ∈ [a, b]. The equality (23) followsby a straightforward calculation. Further, for a.e. t ∈ [a, b], by Theorem 1 we have

CDαa+(F + G)(t) = I

1−αa+ (F + G)′(t) = I

1−αa+

(F ′ �g

(−G′))(t)⊇ I

1−αa+ F ′(t) �g I

1−αa+

(−G′)(t) = CDαa+F(t) �g

(−CDαa+G

)(t);

that is (24). If w(F ′(t)) − w(G′(t)) has a constant sign for a.e. t ∈ [a, b], then by Theorem 1 we obtain (25). �In the following example we show that the relations (20)–(25) can be false if the assumptions of Theorem 2 are not

satisfied.

Example 6. Consider the interval-valued functions F,G,H : [0,2] → K, given by F(t) = [0,−t2 + 2t], G(t) =[0,2t2 − 4t + 3] and H(t) = [0,−t2 + 3t], respectively. We have that wF (t) = −2t2 + 2t and wG(t) = 2t2 − 4t + 3for all t ∈ [0,2]. It follows that F is w-increasing on [0,1] and w-decreasing on [1,2], and G is w-decreasing on [0,1]and w-increasing on [1,2]. Then we have that (F + G)(t) = [0, t2 − 2t + 3] and (F �g G)(t) = [−3t2 + 6t − 3,0]for all t ∈ [0,2]. Also, it is easy to check that

F ′(t) ={ [0,2 − 2t] if t ∈ [0,1)

{0} if t = 1[2 − 2t,0] if t ∈ (1,2],

G′(t) ={ [4t − 4,0] if t ∈ [0,1)

{0} if t = 1[0,4t − 4] if t ∈ (1,2].

(F + G)′(t) ={ [2t − 2,0] if t ∈ [0,1)

{0} if t = 1[0,2t − 2] if t ∈ (1,2],

and

(F �g G)′(t) ={ [0,6 − 6t] if t ∈ [0,1)

{0} if t = 1[6 − 6t,0] if t ∈ (1,2].

We see that w(F ′(t)) − w(G′(t)) has a constant sign on each intervals [0,1] and [1,2], but it has not a constant signon the interval [0,2]. For all t ∈ [0,1], we obtain that

CD120+F(t) = 4√

π

[0, t1/2 − 2

3t3/2

],

CD120+G(t) = 8√

π

[−t1/2 + 2

3t3/2,0

],

CD120+(F + G)(t) = 4√

π

[−t1/2 + 2

3t3/2,0

],

CD120+(F �g G)(t) = 12√

π

[−t1/2 + 2

3t3/2,0

],

and

CD120+F(t) +C D

120+G(t) = 4√

π

[−t1/2 + 2

3t3/2, t1/2 − 2

3t3/2

]�= CD

120+(F + G)(t),

CD120+F(t) �C

g D120+G(t) = 4√

[t1/2 − 2

t3/2,2t1/2 − 4t3/2

]� CD

120+(F �g G)(t).

π 3 3

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Also

CD120+(F �g G)(t) � CD

120+F(t) �C

g D120+G(t),

but

CD120+F(t) + (−CD

120+)(t) = 12√

π

[−t1/2 + 2

3t3/2,0

]= CD

120+(F �g G)(t),

and

CD120+F(t) �g

(−CD120+G

)(t) = 4√

π

[−t1/2 + 2

3t3/2,0

]= CD

120+(F + G)(t).

Further, for all t ∈ (1,2], we have

CD120+F(t) = 1

Γ (1/2)

t∫0

(t − s)−1/2F ′(s)ds

= 1√π

1∫0

(t − s)−1/2[0,2 − 2s]ds + 1√π

t∫1

(t − s)−1/2[2 − 2s,0]ds

= 8

3√

π

[(1 − t)

√t − 1, (t − 1)

√t − 1 +

(3

2− t

)t1/2

],

and similar we obtain that

CD120+G(t) = 16

3√

π

[(1 − t)

√t − 1 +

(t − 3

2

)t1/2, (t − 1)

√t − 1

],

CD120+(F + G)(t) = 8

3√

π

[(1 − t)

√t − 1 +

(t − 3

2

)t1/2, (t − 1)

√t − 1

]

CD120+(F �g G)(t) = 8√

π

[(1 − t)

√t − 1, (t − 1)

√t − 1 +

(3

2− t

)t1/2

].

It follows that

CD120+F(t) + (−CD

120+G(t)

)= 8

3√

π

[3(1 − t)

√t − 1 +

(t − 3

2

)t1/2,3(t − 1)

√t − 1 +

(3

2− t

)t1/2

]

�= CD120+(F �g G)(t)

and

CD120+F(t) �g

(−CD120+G

)(t)

= 8√π

[(1 − t)

√t − 1, (t − 1)

√t − 1 +

(3

2− t

)t1/2

]

�= CD120+(F + G)(t)

But

CD120+F(t) �g

(−CD120+G

)(t) ⊃ CD

120+(F + G)(t)

for t ∈ ( 3 ,2]. Next, the function H is w-increasing on [0,1], and for t ∈ [0,1] we have

2
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CD120+F(t) + CD

120+H(t) = 1√

π

[0,−16

3t3/2 + 10t1/2

],

CD120+F(t) + (−CD

120+H

)(t) = 1√

π

[8

3t3/2 − 6t1/2,−8

3t3/2 + 4t1/2

],

CD120+F(t) �g

(−CD120+H

)(t) = 1√

π

[−8

3t3/2 + 4t1/2,−8

3t3/2 + 6t1/2

],

CD120+(F + H)(t) = 1√

π

[0,−16

3t3/2 + 10t1/2

],

CD120+(F �g H)(t) = 1√

π

[−2t1/2,0].

Therefore, for t ∈ [0,1], it follows that

CD120+(F �g H)(t) �= CD

120+F(t) + (−CD

120+H

)(t),

CD120+(F + H)(t) � CD

120+F(t) �g

(−CD120+H

)(t).

Theorem 10. Let F ∈ AC([a, b],K). The following properties are then true:

(a) If either F is w-increasing on [a, b] or F is w-decreasing and F1−α is w-increasing on [a, b], then

CDαa+F(t) =Dα

a+F(t) �g

(t − a)−α

Γ (1 − α)F (a) (26)

for a.e. t ∈ [a, b].(b) If both F and F1−α are w-decreasing on [a, b], then

Dαa+F(t) = CDα

a+F(t) �g

(t − a)−α

Γ (1 − α)

(−F(a))

(27)

for a.e. t ∈ [a, b].

Proof. It is well known (see Lemma 2.2 in Samko et al. [63]) that, for a real-valued function ϕ ∈ AC[a, b], we haveDα

a+ϕ(t) = CDαa+ϕ(t) + (t−a)−α

Γ (1−α)ϕ(a) for a.e. t ∈ [a, b]. If F is w-increasing on [a, b], then F1−α is also w-increasing

on [a, b] and we have

CDαa+F(t) + (t − a)−α

Γ (1 − α)F (a) = [

CDαa+f −(t), CDα

a+f +(t)]+ (t − a)−α

Γ (1 − α)

[f −(a), f +(a)

]=[

CDαa+f −(t) + (t − a)−α

Γ (1 − α)f −(a), CDα

a+f +(t) + (t − a)−α

Γ (1 − α)f +(a)

]= [

Dαa+f −(t),Dα

a+f +(t)]=Dα

a+F(t),

that is, the relation (26) is true for a.e. t ∈ [a, b]. If F is w-decreasing and F1−α is w-increasing on [a, b], then

CDαa+F(t) + (−Dα

a+F(t))= [

CDαa+f +(t), CDα

a+f −(t)]+ [−Dα

a+f +(t),−Dαa+f −(t)

]=[− (t − a)−α

Γ (1 − α)f +(a),− (t − a)−α

Γ (1 − α)f −(a)

]= (t − a)−α

Γ (1 − α)

(−F(a)).

It follows that

CDαa+F(t) = (t − a)−α

Γ (1 − α)

(−F(a))�g

(−Dαa+F(t)

)=Dα

a+F(t) �g

(t − a)−α

Γ (1 − α)F (a),

for a.e. t ∈ [a, b]. If both F and F1−α are w-decreasing on [a, b], then

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Dαa+F(t) + (t − a)−α

Γ (1 − α)

(−F(a))= [

Dαa+f +(t),Dα

a+f −(t)]+ (t − a)−α

Γ (1 − α)

[−f +(a),−f −(a)]

=[Dα

a+f +(t) − (t − a)−α

Γ (1 − α)f +(a),Dα

a+f −(t) − (t − a)−α

Γ (1 − α)f −(a)

]= [

CDαa+f +(t), CDα

a+f −(t)]= CDα

a+F(t),

that is, the relation (27) is true for a.e. t ∈ [a, b]. �Theorem 11. If F ∈ AC([a, b],K) is a w-monotone interval-function and α ∈ (0,1], then

Iαa+CDα

a+F(t) = F(t) �g F (a), (28)

for a.e. t ∈ [a, b].

Proof. By Proposition 5, we have that

Iαa+CDα

a+F(t) = Iαa+I

1−αa+ F ′(t) = I1

a+F ′(t) =t∫

a

F ′(s)ds = F(t) �g F (a),

for a.e. t ∈ [a, b]; that is, (28). �Remark 8. The relation (28) can be rewritten as

F(t) = F(a) + Iαa+CDα

a+F(t)

if F is w-increasing on [a, b], and as

F(t) = F(a) � (−1)Iαa+CDα

a+F(t),

if F is w-decreasing on [a, b]. Also, we remark that the equality (28) can be false if F is not w-monotone on [a, b].Indeed, for interval-valued function F(t) = [0,−t2 + 2t], t ∈ [0,2], we have that (see Example 6)

CD120+F(t) = 8

3√

π

[(1 − t)

√t − 1, (t − 1)

√t − 1 +

(3

2− t

)t1/2

],

for all t ∈ (1,2]. Then we obtain that

I120+

CD120+F(t) = [−2t2 + 4t + 2, t2 + 2t + 2

] �= F(t) �g F (0),

for all t ∈ (1,2]. Therefore, the relation (28) is not true for all t ∈ [0,2].

Theorem 12. Let F ∈ L∞([a, b],K) be such that either F is w-increasing on [a, b], or F is w-decreasing on [a, b]and t �→ Fα(t) := Iα

a+F(t) is w-increasing on [a, b], then

CDαa+Iα

a+F(t) = F(t) for a.e. t ∈ [a, b]. (29)

Proof. It is known that (see Kilbas et al. [28]) for a real-valued function ϕ ∈ L∞[a, b], we have CDαa+Iα

a+ϕ(t) = ϕ(t)

for a.e. t ∈ [a, b]. If F is w-increasing on [a, b], then from Lemma 1 it follows that Fα is also w-increasing on [a, b].Therefore, in the both cases, Fα is w-increasing on [a, b] and we have

CDαa+Iα

a+F(t) = CDαa+[Iαa+f −, Iα

a+f +](t) = [CDα

a+Iαa+f −(t), CDα

a+Iαa+f +(t)

]= [

f −(t), f +(t)]= F(t),

for a.e. t ∈ [a, b]. �The equality (29) can be false if F is not w-monotone on [a, b].

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Example 7. Consider the interval-valued function F : [0,2] → K, given by F(t) = [0,−t2 + 2t]. Then F isw-increasing on [0,1] and w-decreasing on [1,2]. We have

F 12(t) := I

120+F(t) = 1√

π

t∫0

(t − s)−1/2[0,−s2 + 2s]ds = 1√

π

[0,

8

3t3/2 − 16

15t5/2

],

for all t ∈ [0,2]. Since ddt

w(F 12(t)) = 4√

πt1/2(1 − 2

3 t), it follows that F 12

is w-increasing on [0, 32 ] and w-decreasing

on [ 32 ,2]]. Then we have

F ′12(t) =

⎧⎪⎪⎨⎪⎪⎩

4√π[0, t1/2 − 2

3 t3/2] if t ∈ [0, 32 )

{0} if t = 32

4√π[t1/2 − 2

3 t3/2,0] if t ∈ ( 32 ,2].

For all t ∈ [0, 32 ] we have

CD120+I

120+F(t) = I

1− 12

0+ F ′1/2(t) = 4

π

t∫0

(t − s)−1/2[

0, s1/2 − 2

3s3/2

]ds

= 4

π

[0,

πt

2− πt2

4

]= [

0,2t − t2]= F(t);

that is, (29) is satisfied for all t ∈ [0, 32 ]. If t ∈ ( 3

2 ,2], we have

CD120+I

120+F(t) = 4

π

3/2∫0

(t − s)−1/2[

0, s1/2 − 2

3s3/2

]ds

+ 4

π

t∫3/2

(t − s)−1/2[s1/2 − 2

3s3/2,0

]ds = [

ξ−(t), ξ+(t)] �= F(t),

where

ξ−(t) = 1 − t

π

√3(2t − 3) + t (2 − t)

π

2+ arcsin

t − 3

t

)

ξ+(t) = t − 1

π

√3(2t − 3) + t (2 − t)

π

2− arcsin

t − 3

t

),

for all t ∈ ( 32 ,2]. Therefore, the relation (29) is not true for all t ∈ [0,1].

Also, the equality (29) can be false if F /∈ L∞([a, b],K) or if the interval-valued function t �→ Fα(t) := Iαa+F(t)

is w-decreasing on [a, b].

Example 8. Consider the interval-valued function F : [0,1] → K, given by F(t) = [t1/2, t−1/2] if t ∈ (0,1], andF(0) = [0,1]. Then F ∈ L1([0,1],K), but F /∈ L∞([0,1],K). We have that

F 12(t) := I

120+F(t) = 1√

π

2t, π

]for all t ∈ [0,1].

It follows that F 12

∈ AC([0,1],K) and ddt

w(F 12) ≤ 0 for all t ∈ [0,1]. Since the interval-valued function F 1

2= I

120+F

is w-decreasing on [0,1], then we have that

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CD120+F 1

2:= F ′

12(t) =

[0,

√π

2

]for all t ∈ [0,1],

and so CD120+I

120+F(t) = [0, t1/2] �= F(t) for all t ∈ [0,1].

6. Conclusions

In this paper, we developed the fractional calculus for interval-valued functions. As far as we know, this is thefirst paper that addresses this issue. The fractional calculus for interval-valued functions is an important tool forthe fractional calculus of fuzzy functions, and also it represents the necessary support to develop a theory of theinterval-valued fractional differential equations.

Acknowledgements

The author would like to thank the anonymous referees for their comments and suggestions that greatly improvedthe paper.

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