Random fuzzy fractional integral equations – theoretical foundations

Accepted Manuscript Random fuzzy fractional integral equations – theoretical foundations Marek T. Malinowski

PII: DOI: Reference:

S0165-0114(14)00426-6 10.1016/j.fss.2014.09.019 FSS 6641

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Fuzzy Sets and Systems

Received date: 6 December 2013 Revised date: 9 September 2014 Accepted date: 26 September 2014

Please cite this article in press as: M.T. Malinowski, Random fuzzy fractional integral equations – theoretical foundations, Fuzzy Sets and Systems (2014), http://dx.doi.org/10.1016/j.fss.2014.09.019

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Random fuzzy fractional integral equations – theoretical foundations Marek T. Malinowski Faculty of Mathematics, Computer Science and Econometrics, University of Zielona G´ ora, ul. Prof. Z. Szafrana 4a, 65-516 Zielona G´ ora, Poland E-mail addresses: [email protected], [email protected] Abstract: This paper presents mathematical foundations for studies of random fuzzy fractional integral equations which involve a fuzzy integral of fractional order. We consider two different kinds of such equations. Their solutions have different geometrical properties. The equations of the first kind possess solutions with trajectories of nondecreasing diameter of their consecutive values. On the other hand, the solutions to equations of the second kind have trajectories with nonincreasing diameter of their consecutive values. Firstly, the existence and uniqueness of solutions is investigated. This is showed by using a method of successive approximations. An estimation of error of nth approximation is given. Also a boundedness of the solution is indicated. To show well-posedness of the considered theory, we prove that solutions depend continuously on the data of the equations. Some concrete examples of random fuzzy fractional integral equations are solved explicitly. Keywords: random fuzzy fractional integral equation, existence and uniqueness of solution, epistemic and random uncertainties, fuzzy differential equation, set differential equation, mathematical foundations. MSC: 34A07, 34F05, 34A08, 34G20, 45R05, 26A33, 26E50.

1

Introduction

The theory of fractional calculus initiated in XVII century has gained more and more attention since the derivatives and integrals of a non-integer order became good tools e.g. in description of mechanical and electrical properties of some real materials or rheological properties of rocks. The fractional calculus has been used in the theory of fractals, in physics, chemistry, engineering, seismology, in problems of viscoelasticity. Although the fractional calculus was available for a long time and applicable to different areas of research, the investigation of the theory of fractional differential equations has been developed quite recently. Currently it focuses a lot of interest because of the challenges it offers compared to the study of ordinary differential equations. The current state of this area of nonlinear analysis is given e.g. in the monographs [27, 29, 52]. Many theoretical aspects on the results for fractional differential equations have been considered e.g. in [11, 17, 19, 20, 26, 31]. Applications of the ordinary differential equations and the fractional differential equations in modeling dynamical systems are evident. On the other hand the realistic information available on a considered dynamical system is often incomplete, imprecise, imperfect, vague (in a word – uncertain) and this causes a dilemma in applications of single-valued differential equations which are suitable in the case of a perfect, precise knowledge about the considered system. To handle the systems with the uncertain initial values and the uncertain relationships between parameters, it has been proposed the theory of fuzzy differential equations (see e.g. [3, 4, 7, 8, 9, 12, 14, 15, 18, 24, 25, 30, 32, 34, 33, 48] and references therein). In these equations the fuzzy sets (see [59, 60]), which replace (and generalize) the real numbers, are used to model the uncertain values. The fuzzy differential equation has been extended to the random fuzzy differential equations [21, 36, 37, 39, 51] and stochastic fuzzy differential equations [40, 41, 44, 45, 46, 49] which incorporate into the equations two different sources of uncertainties, i.e. fuzziness and randomness. The latter equations are more appropriate in modeling uncertain dynamical systems subjected to random forces or stochastic noises. They constitute the directions of extensions to the fuzzy context of the studies on single-valued random differential/integral equations [13, 57, 55] and single-valued stochastic differential equations [6, 22, 50]. Recently, some studies of fuzzy fractional differential and integral equations have been proposed by several papers, see e.g. [1, 2, 5, 10, 47, 54, 58]. They involve the notions of the fuzzy Riemann–Liouville differential operators, fuzzy Caputo derivatives and fuzzy fractional integrals and offer a more comprehensive apparatus to process the dynamical phenomena with fuzziness. However, since the influence of

1

some random factors on the evolution of dynamical system takes place very often, this apparatus would be more and more comprehensive in the case when it allowed for consideration both the fuzziness and randomness. Some investigations without fuzziness can be found in [23], where a polynomial chaos method is applied to solve three concrete examples of crisp random fractional differential equations. In this paper we initiate some studies on the random fuzzy fractional integral equations where we use a concept of a fuzzy fractional integral. We consider two forms of the random fuzzy fractional integral equations similarly as we have done in [37, 39] for the random fuzzy differential equations. The first one leads to the solutions which possess the trajectories with nondecreasing diameter of their values. In the second form the solutions have the trajectories with nonincreasing diameter of the values. We prove the existence and uniqueness of a global solution to the first kind of the equations, but in some cases for the second kind of the equations we are able to prove this only for a local solution. Hence, here remains an open problem of establishing a method allowing to obtain the existence and uniqueness of the global solution to the equations with nonincreasing diameter of values. For the both kinds of the random fuzzy fractional integral equations we estimate a distance between the approximate solutions and the exact solution, we show that the solutions depend continuously on the initial value and nonlinearity involved in the formulation of the equations. A boundedness result of the solutions is also investigated. Some concrete examples of random fuzzy fractional integral equations are solved explicitly. The presented theory of random fuzzy differential equations can be viewed as a complement in analysis of crisp random fractional differential equations and deterministic crisp fractional differential equations. The latter were never studied in general infinite dimensional metric spaces. They were studied in Banach spaces, but these have linear structure. The space of fuzzy sets does not have the linear structure. The presented theory involves special features of fuzzy sets. In fuzzy case we consider two different equations that reduce to the same crisp equation when we allow the coefficients to be single-valued. In fuzzy case they are completely different. Their solutions have different geometrical properties. This does not appear in the crips case. Moreover the second equation considered in the paper involves, in fact, the Hukuhara differences of fuzzy sets. This causes some more delicate studies in the fuzzy sets setting, because Hukuhara differences may not exist. Such a problem does not appear in crisp case, because there the crisp differences exist always.

2

Preliminaries

Let K(Rd ) denote the family of all nonempty, compact and convex subsets of Rd . The addition and scalar multiplication in K(Rd ) are defined as usual, i.e. for A, B ∈ K(Rd ) and λ ∈ R A + B := {a + b | a ∈ A, b ∈ B},

λA := {λa | a ∈ A}.

In K(Rd ) we consider the Hausdorff metric H which for A, B ∈ K(Rd ) is defined by   H (A, B) := max sup inf a − b, sup inf a − b . a∈A b∈B

b∈B a∈A

It is known (see [28]) that K(Rd ) is a complete, separable and locally compact metric space with respect to H. Also it becomes a semilinear metric space with algebraic operations of addition and non-negative scalar multiplication. Consider u : Rd → [0, 1] and denote [u]α := {x ∈ Rd | u(x) ≥ α} for α ∈ (0, 1] and [u]0 := {x ∈ Rd | u(x) > 0}. Define F(Rd ) = {u : Rd → [0, 1] | [u]α ∈ K(Rd ) for every α ∈ [0, 1]}. The elements of F(Rd ) are called the fuzzy sets of Rd . For more informations about the fuzzy sets we refer the reader to [30, 59, 60, 61]. Note that every ordinary subset of Rd belonging to K(Rd ) is also a fuzzy set belonging to F(Rd ). It is achieved by means of the characteristic function of the ordinary sets. In particular, the set Rd can be embedded into F(Rd ) by the embedding · : Rd → F(Rd ) defined as:  1, if x = r, r(x) = 0, if x = r.

2

For u, v ∈ F(Rd ) and λ ∈ R the addition u ⊕ v and scalar multiplication λu are defined as the fuzzy sets which satisfy: [u ⊕ v]α = [u]α + [v]α , [λu]α = λ[u]α for α ∈ [0, 1], respectively. The metric D(u, v) := sup H([u]α , [v]α ), for u, v ∈ F(Rd ) α∈[0,1]

will be considered in F(R ). It is known (see [53]) that (F(Rd ), D) is a complete metric space. Also, if u, v, w, z ∈ F(Rd ) and λ ∈ R, then we have d

(P1) D(u ⊕ w, v ⊕ w) = D(u, v), (P2) D(u ⊕ v, w ⊕ z) ≤ D(u, w) + D(v, z), (P3) D(λu, λv) = |λ|D(u, v). Let u, v ∈ F(Rd ). If there exists a fuzzy set w ∈ F(Rd ) such that u = v ⊕ w, then w is called the Hukuhara difference of u and v. The Hukuhara difference of u and v will be denoted by u  v. It is known that if u  v exists, then it is unique. Also, u  v = u ⊕ (−1)v. The following properties, for u, v, w, z ∈ F(Rd ), can be verified (cf. [37, 39]): (P4) if u  v exists, then D(u  v, 0) = D(u, v); (P5) if u  v, u  w exist, then D(u  v, u  w) = D(v, w); (P6) if u  v, w  z exist, then D(u  v, w  z) = D(u ⊕ z, v ⊕ w); (P7) if u  v, u  (v ⊕ w) exist, then there exists (u  v)  w and (u  v)  w = u  (v ⊕ w); (P8) if u  v, u  w, w  v exist, then there exists (u  v)  (u  w) and (u  v)  (u  w) = w  v. In the paper we will need a notion of an integral of the measurable fuzzy-valued mappings. A fuzzy mapping F : [a, b] → F(Rd ) is said to be measurable, if for every α ∈ [0, 1] the set-valued mapping [F ]α : [a, b] → K(Rd ) is measurable, i.e. the set {t ∈ [a, b] | [F (t)]α ∩ O = ∅} for every open set O ⊂ Rd is Lebesgue measurable. A fuzzy mapping F : [a, b] → F(Rd ) is called integrably bounded , if there exists an integrable function h : [a, b] → R such that x ≤ h(t) for all x ∈ [F (t)]0 . Definition 2.1 ([53]). For a measurable and integrably bounded F : [a, b] → F(Rd ) by its integral over b [a, b] we mean a fuzzy set a F (t)dt ∈ F(Rd ) such that for every α ∈ [0, 1]    b

b

F (t)dt a

:=

[F (t)]α dt a



α

b

=

f (t)dt | f : [a, b] → R is a measurable selection for [F (·)]α d

a

For measurable and integrably bounded F, G : [a, b] → F(Rd ) and λ ∈ R we have (cf. [24]):   b

b b (P9) a F (t) ⊕ G(t) dt = a F (t)dt ⊕ a G(t)dt, b b (P10) a λF (t)dt = λ a F (t)dt,

 

 b b b (P11) t → D(F (t), G(t)) is integrable and D a F (t)dt, a G(t)dt ≤ a D F (t), G(t) dt, c b F (t)dt = a F (t)dt ⊕ c F (t)dt for c ∈ [a, b], t (P13) t → a F (s)ds is D-continuous mapping. b (P14) if F : [a, b] → F(Rd ) is D-continuous then a F (t)dt exists.

(P12)

b a

3

.

Let (Ω, A, P ) be a complete probability space. A mapping X : Ω → F(Rd ) is called a fuzzy random variable (cf. [53]), if for every α ∈ [0, 1] {ω ∈ Ω : [X(ω)]α ∩ O = ∅} ∈ A for every open set O ⊂ Rd . A mapping X : [a, b] × Ω → F(Rd ) is said to be a fuzzy stochastic process, if X(t, ·) : Ω → F(Rd ) is a fuzzy random variable for every t ∈ [a, b]. A fuzzy stochastic process X : [a, b] × Ω → F(Rd ) is called continuous, if almost all (with respect to the probability measure P ) its trajectories are D-continuous. In the paper, to abbreviate the expressions P ({ω ∈ Ω | X(ω) = Y (ω)}) = 1, P ({ω ∈ Ω | X(ω) ≤ Y (ω)}) = 1 P.1

P.1

we will use the notations X(ω) = Y (ω) and X(ω) ≤ Y (ω), respectively. Also the facts that P ({ω ∈ Ω | X(t, ω) = Y (t, ω) ∀t ∈ A ⊂ [a, b]}) = 1 and P ({ω ∈ Ω | X(t, ω) ≤ Y (t, ω) ∀t ∈ A ⊂ [a, b]}) = 1 will A P.1

A P.1

be abbreviated by the notations X(t, ω) = Y (t, ω) and X(t, ω) ≤ Y (t, ω), respectively.

3

The main results

Let β > 0. In this section we introduce studies concerning the random fuzzy integral equations of the fractional order. The investigations of the fuzzy fractional integral equations in the deterministic case are also new and can be found in [1, 10, 58]. Definition 3.1 The fuzzy fractional β-order integral of the measurable and integrably bounded fuzzy mapping F : [a, b] → F(Rd ) at t ∈ [a, b] is the fuzzy set (Iaβ F )(t) ∈ F(Rd ) defined by (Iaβ F )(t) :=

1 Γ(β)



t a

1 F (s)ds, (t − s)1−β

where Γ is the well-known gamma function. If β = 1 then the integral defined above reduces to the fuzzy integral from Definition 2.1. Also, it coincides with the classical Riemann–Liouville fractional integral in the case of crisp-valued function F . The integral from Definition 3.1 takes a particularly nice form when F : [a, b] → F(R1 ), since the α-levels become intervals. In this case    t  t Fα− (s) Fα+ (s) 1 1 ds, ds , [(Iaβ F )(t)]α = Γ(β) a (t − s)1−β Γ(β) a (t − s)1−β where Fα− , Fα+ : [a, b] → R1 are defined by [Fα− (t), Fα+ (t)] = [F (t)]α for t ∈ [a, b]. Many properties of interval-valued fractional integral have been established in [35]. To formulate the random fuzzy fractional integral equations we will use the fractional integral from Definition 3.1. Let us consider a mapping F : [a, b] × Ω × F(Rd ) → F(Rd ) which satisfies conditions: (C1) the mapping F (t, ·, u) : Ω → F(Rd ) is a fuzzy random variable for every (t, u) ∈ [a, b] × F(Rd ), (C2) with P.1 the mapping F (·, ω, ·) : [a, b] × F(Rd ) → F(Rd ) is continuous at every point (t0 , u0 ) ∈ [a, b] × F(Rd ). We shall investigate the two kinds of random fuzzy fractional integral equations involving the fuzzy fractional β-order integral, i.e.  t 1 1 [a,b] P.1 F (s, ω, X(s, ω))ds (3.1) X(t, ω) = X0 (ω) ⊕ Γ(β) a (t − s)1−β and Y0 (ω)

[a,b] P.1

=

1 Y (t, ω) ⊕ Γ(β)



t a

−1 F (s, ω, Y (s, ω))ds, (t − s)1−β

(3.2)

where X0 , Y0 : Ω → F(Rd ) are some fuzzy random variables. Definition 3.2 By a global solution to (3.1) we mean a D-continuous fuzzy stochastic process X : [a, b] ×

 ˜ ω) [a,b]=P.1 0 for any Ω → F(Rd ) that satisfies (3.1). A solution X is unique, if it holds D X(t, ω), X(t, ˜ : [a, b] × Ω → F(Rd ) that is a global solution to (3.1). fuzzy stochastic process X

4

Similarly we can define a global solution to (3.2). For clarity and completeness we include this definition. Definition 3.3 A fuzzy stochastic process Y : [a, b] × Ω → F(Rd ) is said to be a global solution to (3.2), if it is D-continuous and satisfies (3.2). A solution Y to (3.2) is unique, if it holds

 [a,b] P.1 D Y (t, ω), Y˜ (t, ω) = 0 for any fuzzy stochastic process Y˜ : [a, b] × Ω → K(R) that is a global solution to (3.2). If in Definitions 3.2 and 3.3 we replace the interval [a, b] with a subinterval [a, d] ⊂ [a, b] then we obtain definitions of the (unique) local solutions to equations (3.1) and (3.2). Although the equations (3.1) and (3.2) seem to be very similar at the first glance and they reduce to the same equation in the crisp case, their solutions have different geometrical properties. This is a consequence of the fact that the considered equations have values in the space of fuzzy sets. Theorem 3.4 (i) Let X : [a, b] × Ω → F(Rd ) be a solution to (3.1). Then for P -a.a. ω and for every α ∈ [0, 1] the function t → diam[X(t, ω)]α is nondecreasing, where diamA denotes the diameter of the set A. (ii) Let Y : [a, b] × Ω → F(Rd ) be a solution to (3.2). Then for P -a.a. ω and for every α ∈ [0, 1] the function t → diam[X(t, ω)]α is nonincreasing. Proof. To prove (i) let us observe that for the solution X to (3.1) with P.1 we have: for every t1 < t2  t2 1 1 X(t2 , ω) = X(t1 , ω) ⊕ F (s, ω, X(s, ω))ds. Γ(β) t1 (t − s)1−β Hence with P.1 we have: for every α ∈ [0, 1] and for every t1 < t2 [X(t2 , ω)]α = [X(t1 , ω)]α + Thus for any fixed x ∈

1 Γ(β)



1 Γ(β)



t2 t1

1 F (s, ω, X(s, ω))ds (t − s)1−β 

t2 1 F (s, ω, X(s, ω))ds t1 (t−s)1−β α

 . α

we have

[X(t1 , ω)]α + x ⊂ [X(t2 , ω)]α , which yields diam[X(t1 , ω)]α ≤ diam[X(t2 , ω)]α . To show (ii) assume that Y : [a, b] × Ω → F(Rd ) is a solution to (3.2) and note that with P.1 we have for every t1 < t2  t2  t1 1 1 −1 −1 Y (t2 , ω) ⊕ F (s, ω, Y (s, ω))ds = Y (ω)  F (s, ω, Y (s, ω))ds 0 Γ(β) t1 (t − s)1−β Γ(β) 0 (t − s)1−β Hence 1 Y (t2 , ω) ⊕ Γ(β)



t2 t1

−1 F (s, ω, Y (s, ω))ds = Y (t1 , ω). (t − s)1−β

This allows to infer that with P.1 for every α ∈ [0, 1] and every t1 < t2  t2  1 −1 [Y (t2 , ω)]α + F (s, ω, Y (s, ω))ds = [Y (t1 , ω)]α . Γ(β) t1 (t − s)1−β α Hence diam[Y (t1 )]α ≥ diam[Y (t2 )]α .



Remark 3.5 A scope of the random fuzzy fractional integral equations is wide. Observe that: (i) if we put β = 1 in (3.1) and (3.2) then these equations reduce to the random fuzzy integral equations which are equivalent to the random fuzzy differential equations with (L1)-derivative and (L2)derivative, respectively, studied in [37, 39];

5

(ii) if F : [a, b] × F(Rd ) → F(Rd ) and X0 , Y0 ∈ F(Rd ) then (3.1) is a deterministic fuzzy fractional integral equation considered in [1, 10, 58]. However, in this case the deterministic equation (3.2) can be used in studies of fuzzy fractional differential equations under generalized differentiability. Some studies in this direction for F(R1 ) can be found e.g. in [47, 54]; (iii) if β = 1, F : [a, b] × F(Rd ) → F(Rd ) and X0 , Y0 ∈ F(Rd ) then (3.1) and (3.2) represent the deterministic fuzzy integral equations that are equivalent to the deterministic fuzzy differential equations investigated e.g. in monograph [30] and articles [8, 14, 24, 25, 32, 48]; (iv) if F : [a, b] × Ω × K(Rd ) → K(Rd ) and X0 , Y0 : Ω → K(Rd ) then (3.1) and (3.2) become the corresponding random set-valued fractional differential equations. Up until now, these equations were not examined; (v) if F : [a, b] × K(Rd ) → K(Rd ) and X0 , Y0 ∈ K(Rd ) then (3.1) and (3.2) are the deterministic set-valued fractional integral equations. These were not sudied; (vi) if β = 1, F : [a, b] × K(Rd ) → K(Rd ) and X0 , Y0 ∈ K(Rd ) then (3.1) and (3.2) are the integral counterparts of deterministic set-valued differential equations. The latter form an independent branch of research nowadays. We refer the reader e.g. to monograph [28] and articles [38, 42, 43, 56]; P.1

(vii) if F : [a, b] × Ω × Rd → Rd , X0 : Ω → Rd , X0 = Y0 , then (3.1) and (3.2) coincide and they represent the crisp random fractional integral equations which could be applied to crisp random fractional differential equations with Riemann–Liouville or Caputo derivatives [23]; P.1

(viii) if β = 1, F : [a, b] × Ω × Rd → Rd and X0 : Ω → Rd , X0 = Y0 , then (3.1) and (3.2) are the same crisp random integral equations studied e.g. in [55, 57]; (ix) if F : [a, b] × Rd → Rd , X0 = Y0 ∈ Rd , then (3.1) and (3.2) coincide and they represent the crisp deterministic fractional integral equations that are used in the theory crisp deterministic fractional differential equations, see e.g. [17, 19, 20, 27, 29, 31, 52]; (x) if β = 1, F : [a, b] × Ω × Rd → Rd and X0 = Y0 ∈ Rd , then (3.1) and (3.2) are the same crisp deterministic ordinary integral equations which are a part of a classical mathematical analysis [16].

3.1

Equations with solutions of nondecreasing diameter

In this part of the paper we present the studies and results concerning the random fuzzy fractional integral equation (3.1). The solutions to (3.1) have, accordingly to Theorem 3.4(i), the trajectories with nondecreasing diameter of their subsequent values. We start with a presentation of an existence and uniqueness result for such the solutions. Theorem 3.6 Let X0 : Ω → F(Rd ) be a fuzzy random variable. Let F : [a, b] × Ω × F(Rd ) → F(Rd ) satisfy (C1) and (C2). Assume that there exist some positive constants L, M such that — with P.1 for every t ∈ [a, b] and for every u, v ∈ F(Rd )

 D F (t, ω, u), F (t, ω, v) ≤ LD(u, v), — with P.1 for every t ∈ [a, b] and for every u ∈ F(Rd )

 D F (t, ω, u), {0} ≤ M. Then (3.1) has a unique global solution X : [a, b] × Ω → F(Rd ). β

Lh < 1. Put c1 := a + h if a + h < b and c1 := b if a + h ≥ b. Let us define Proof. Let h > 0 satisfy Γ(β+1) a sequence of approximate solutions {Xn }∞ n=0 to (3.1) as follows: for (t, ω) ∈ [a, c1 ] × Ω X0 (t, ω) := X0 (ω),  t 1 1 Xn (t, ω) := X0 (ω) ⊕ F (s, ω, Xn−1 (s, ω))ds, n ∈ N. (3.3) Γ(β) a (t − s)1−β

It is easy to see that each Xn : [a, c1 ] × Ω → F(Rd ) is a D-continuous fuzzy stochastic process.

6

Note that for (t, ω) ∈ [a, c1 ] × Ω 

D X1 (t, ω), X0 (t, ω) ≤

1 Γ(β)



t a

 D F (s, ω, X0 (ω)), 0 ds . (t − s)1−β

By assumptions we get

 [a,c1 ] P.1 D X1 (t, ω), X0 (t, ω) ≤

1 Γ(β)



t a

M hβ M ds . ≤ 1−β (t − s) βΓ(β)

Also for (t, ω) ∈ [a, c1 ] × Ω 

D X2 (t, ω), X1 (t, ω) ≤

1 Γ(β)



t a

 D F (s, ω, X1 (s, ω)), F (s, ω, X0 (s, ω)) ds . (t − s)1−β

Due to assumptions we have

 [a,c1 ] P.1 D X2 (t, ω), X1 (t, ω) ≤

L Γ(β)



t a

 D X1 (s, ω), X0 (s, ω) ds (t − s)1−β

[a,c1 ] P.1

≤

 LM

hβ βΓ(β)

2 .

Proceeding recursively we get  [a,c1 ] P.1

≤ D Xn (t, ω), Xn−1 (t, ω) which implies

L Γ(β)



t a

 D Xn−1 (s, ω), Xn−2 (s, ω) ds , (t − s)1−β

 [a,c1 ] P.1 M

≤ D Xn (t, ω), Xn−1 (t, ω) L



Lhβ βΓ(β)

n .

Hence for n > m > 0 we have

 P.1 M sup D Xn (t, ω), Xm (t, ω) ≤ L t∈[a,c1 ] ∞

Lhβ βΓ(β)

k  n  Lhβ . βΓ(β)

k=m+1

k

˜ ∈ A such that P (Ω) ˜ =1 is convergent, we can infer that there exists Ω ˜ let X(·, ˜ ω) denote its ˜ the sequence {Xn (·, ω)} is uniformly convergent. For ω ∈ Ω and for every ω ∈ Ω limit. Let us define a mapping X (1) : [a, c1 ] × Ω → F(Rd ) as  ˜ ω) for (t, ω) ∈ [a, c1 ] × Ω, ˜ X(t, X (1) (t, ω) = ˜ 0 for (t, ω) ∈ [a, c1 ] × (Ω \ Ω). Since the series

Then

k=1

 P.1 sup D Xn (t, ω), X (1) (t, ω) −→ 0 as n → ∞

t∈[a,c1 ]

and X (1) is a D-continuous fuzzy stochastic process. In the sequel we shall show that X (1) is a local solution to (3.1) in the case c1 < b, or X (1) is a global solution in the case c1 = b. Note that    t 1 1 (1) F (s, ω, X (s, ω))ds sup D X (1) (t, ω), X0 (ω) ⊕ Γ(β) a (t − s)1−β t∈[a,c1 ]   t

P.1

 D Xn−1 (s, ω), X (1) (s, ω) ds L (1) sup ≤ sup D Xn (t, ω), X (t, ω) + Γ(β) t∈[a,c1 ] a (t − s)1−β t∈[a,c1 ] P.1

≤



 Lhβ sup D Xn−1 (s, ω), X (1) (s, ω) . sup D Xn (t, ω), X (1) (t, ω) + βΓ(β) s∈[a,c1 ] t∈[a,c1 ]

Since



 P.1 Lhβ sup D Xn−1 (s, ω), X (1) (s, ω) −→ 0, sup D Xn (t, ω), X (1) (t, ω) + βΓ(β) s∈[a,c1 ] t∈[a,c1 ]

7

as n → ∞,

we obtain  D X (1) (t, ω), X0 (ω) ⊕

1 Γ(β)



t a

1 F (s, ω, X (1) (s, ω))ds (t − s)1−β



[a,c1 ] P.1

=

0.

˜ (1) : [a, c1 ] × Ω → F(Rd ) were another solution Hence X (1) is a solution to (3.1) on the interval [a, c1 ]. If X to (3.1) on [a, c1 ] then we would have   t (1) ˜ (1) (s, ω) ds

(1)  [a,c1 ] P.1 L D X (s, ω), X (1) ˜ D X (t, ω), X (t, ω) ≤ . Γ(β) a (t − s)1−β 

˜ (1) (t, ω) Applying Gronwall’s inequality we can infer that D X (1) (t, ω), X the conclusion [a,c1 ] P.1 ˜ (1) (t, ω). = X X (1) (t, ω)

[a,c1 ] P.1

=

0 which leads us to

Hence X (1) is a unique solution to (3.1) on the interval [a, c1 ]. Next, if c1 < b, we consider the following random fuzzy fractional integral equation  t 1 1 [c1 ,c2 ] P.1 = X (1) (c1 , ω) ⊕ F (s, ω, X(s, ω))ds X(t, ω) Γ(β) c1 (t − s)1−β

(3.4)

where c2 := a + 2h if a + 2h < b and c2 := b if a + 2h ≥ b. Similarly as above we can show that this equation possesses a unique solution X (2) : [c1 , c2 ] × Ω → F(Rd ). We use then the following sequence of approximate solutions {Xn }∞ n=0 to (3.4) which is defined for (t, ω) ∈ [c1 , c2 ] × Ω as follows X0 (t, ω) := X (1) (c1 , ω),  t 1 1 Xn (t, ω) := X (1) (c1 , ω) ⊕ F (s, ω, Xn−1 (s, ω))ds, n ∈ N. Γ(β) c1 (t − s)1−β We can repeat this procedure till we reach the right boundary b of the interval [a, b]. Indeed, let k ∈ N be such that a + (k − 1)h < b ≤ a + kh. We can show the existence and uniqueness of solution X () : [c−1 , c ] × Ω → F(Rd ) to equation  t 1 1 [c−1 ,c ] P.1 = X (−1) (c−1 , ω) ⊕ F (s, ω, X(s, ω))ds, X(t, ω) Γ(β) c−1 (t − s)1−β where  = 3, 4, . . . , k −1 and c = a+h, till in the last step we get a unique solution X (k) : [ck−1 , b]×Ω → F(Rd ) to equation  t 1 1 [ck−1 ,b] P.1 = X (k−1) (ck−1 , ω) ⊕ F (s, ω, X(s, ω))ds. X(t, ω) Γ(β) ck−1 (t − s)1−β Then

⎧ (1) X (t, ω) ⎪ ⎪ ⎪ ⎨ X (2) (t, ω) X(t, ω) = . ⎪ ⎪ .. ⎪ ⎩ (k) X (t, ω)

for (t, ω) ∈ [a, c1 ] × Ω, for (t, ω) ∈ [c1 , c2 ] × Ω, .. .. . . for (t, ω) ∈ [ck−1 , b] × Ω,

which is a D-continuous fuzzy stochastic process, is a desired unique global solution to (3.1).



The successive approximations Xn defined in 3.3 approximate the exact solution X. Hence it is a natural question about a distance between Xn and X. Theorem 3.7 Let X0 : Ω → F(Rd ), F : [a, b] × Ω × F(Rd ) → F(Rd ) satisfy assumptions of Theorem 3.6. β Suppose that L(b−a) Γ(β+1) < 1. Then

 P.1 M sup D Xn (t, ω), X(t, ω) ≤ L t∈[a,b]



L(b − a)β Γ(β + 1)

8

n+1 

L(b − a)β 1− Γ(β + 1)

−1 .

Proof. Note that

 sup D Xn (t, ω), X(t, ω) = t∈[a,b]

 lim sup D Xn (t, ω), Xn+k (t, ω)

k→∞ t∈[a,b]

≤



 sup D Xn (t, ω), Xn+1 (t, ω) + sup D Xn+1 (t, ω), Xn+2 (t, ω) + · · · .

t∈[a,b]

t∈[a,b]

Similarly as in the proof of Theorem 3.6 we obtain

 P.1 M sup D Xn (t, ω), Xn−1 (t, ω) ≤ L t∈[a,b] Hence

 P.1 M sup D Xn (t, ω), X(t, ω) ≤ L t∈[a,b]



L(b − a)β βΓ(β)



L(b − a)β βΓ(β)

n+1 +

n ,



M L

n ∈ N.

L(b − a)β βΓ(β)

n+2 + ··· . 

The next result shows a boundedness of the solution X to (3.1). Theorem 3.8 Let X0 : Ω → F(Rd ), F : [a, b] × Ω × F(Rd ) → F(Rd ) satisfy assumptions of Theorem 3.6. Then for solution X : [a, b] × Ω → F(Rd ) to (3.1) it holds

 P.1

 M (b − a)β . sup D X(t, ω), 0 ≤ D X0 (ω), 0 + Γ(β + 1) t∈[a,b] Proof. For (t, ω) ∈ [a, b] × Ω we have



 D X(t, ω), 0 ≤ D X0 (ω), 0 +

1 Γ(β)



t a

 D F (s, ω, X(s, ω)), 0 ds . (t − s)1−β

Further, by assumption imposed on F , we obtain  M (t − a)β

 [a,b] P.1

. ≤ D X0 (ω), 0 + D X(t, ω), 0 βΓ(β) 

Now, the assertion follows easily. Let us consider (3.1) and a sequence of equations  t 1 1 [a,b] P.1 (n) Xn (t, ω) = X0 (ω) ⊕ F (s, ω, Xn (s, ω))ds Γ(β) a (t − s)1−β

(3.5)

for n ∈ N. Let X denote solution to (3.1) and Xn solution to (3.5). Theorem 3.9 Let X0 : Ω → F(Rd ), F : [a, b] × Ω × F(Rd ) → F(Rd ) satisfy assumptions of Theorem 3.6. Then for the solution Xn to (3.5) and the solution X to (3.1) we have  

 P.1 (n)  L(b − a)β . sup D Xn (t, ω), X(t, ω) ≤ D X0 (ω), X0 (ω) exp Γ(β + 1) t∈[a,b] Proof. Observe that for (t, ω) ∈ [a, b] × Ω 

D Xn (t, ω), X(t, ω)    t  t 1 1 1 1 (n) F (s, ω, X (s, ω))ds, X (ω) ⊕ F (s, ω, X(s, ω))ds = D X0 (ω) ⊕ n 0 Γ(β) a (t − s)1−β Γ(β) a (t − s)1−β  t

D (F (s, ω, Xn (s, ω)), F (s, ω, X(s, ω))) ds 1 (n) . ≤ D X0 (ω), X0 (ω) + Γ(β) a (t − s)1−β Now, by assumption imposed on F we get

 [a,b] P.1 (n)

≤ D X0 (ω), X0 (ω) + D Xn (t, ω), X(t, ω)

9

1 Γ(β)



t a

LD (Xn (s, ω), X(s, ω)) ds . (t − s)1−β

Due to Gronwall’s inequality we obtain

 [a,b] P.1 (n)

≤ D X0 (ω), X0 (ω) exp D Xn (t, ω), X(t, ω)



L(t − a)β βΓ(β)

 .

 

 P.1 (n)  β Thus we get the inequality supt∈[a,b] D Xn (t, ω), X(t, ω) ≤ D X0 (ω), X0 (ω) exp L(b−a) . βΓ(β)



Corollary 3.10 Under assumptions of Theorem 3.9, for the solutions to equations (3.1) and (3.5) it holds

 P.1 sup D Xn (t, ω), X(t, ω) −→ 0 as n → ∞ t∈[a,b]

(n)  P.1 provided that D X0 (ω), X0 (ω) −→ 0 as n → ∞. Due to this property we can state the continuous dependence of the solution X to (3.1) with respect to the initial value X0 . In the sequel, let us consider (3.1) and the sequence of equations Xn (t, ω)

[a,b] P.1

=

X0 (ω) ⊕

1 Γ(β)



t a

 1 F (s, ω, Xn (s, ω)) ⊕ Gn (s, ω, Xn (s, ω)) ds 1−β (t − s)

(3.6)

for n ∈ N. Let X denote solution to (3.1) and Xn solution to (3.6). Theorem 3.11 Let the assumptions of Theorem 3.6 be satisfied. Let the mapping Gn : [a, b] × Ω × F(Rd ) → F(Rd ), for each n ∈ N, satisfies the same assumptions as F in Theorem 3.6. Suppose that for every n ∈ N there exists a positive constant Mn such that with P.1 for every (t, u) ∈ [a, b] × F(Rd ) it holds D (Gn (t, ω, u), 0) ≤ Mn . Then

 P.1 (b − a)β sup D Xn (t, ω), X(t, ω) ≤ Mn exp Γ(β + 1) t∈[a,b]



L(b − a)β Γ(β + 1)

 .

Proof. Note that for (t, ω) ∈ [a, b] × Ω

 D Xn (t, ω), X(t, ω) ≤



D(Gn (s, ω, Xn (s, ω)), 0)ds (t − s)1−β a  t 1 D(F (s, ω, Xn (s, ω)), F (s, ω, X(s, ω)))ds + . Γ(β) a (t − s)1−β 1 Γ(β)

t

Accordingly to the properties of F and Gn we have  [a,b] P.1 Mn (b − a)β L + ≤ D Xn (t, ω), X(t, ω) βΓ(β) Γ(β)



t a

D(Xn (s, ω), X(s, ω))ds . (t − s)1−β

 [a,b] P.1

≤ Invoking Gronwall’s inequality we infer that D Xn (t, ω), X(t, ω) leads to the end of the proof.

Mn (b−a)β βΓ(β)

 exp

L(t−a)β βΓ(β)

 which 

Corollary 3.12 Under assumptions of Theorem 3.11, for solutions to equations (3.1) and (3.6) it holds

 P.1 sup D Xn (t, ω), X(t, ω) −→ 0

as n → ∞

t∈[a,b]

provided that Mn −→ 0 as n → ∞. This property shows that the solution X to (3.1) depends continuously on the nonlinearity F . In the sequel we present some simple examples of random fuzzy fractional integral equations, where we can deduce explicit solutions. It is worth mentioning that, in a general case of random fuzzy fractional

10

integral equations, finding explicit solution is impossible. This requires a further study of approximate solutions and numerical methods for these equations. Example A. Let β > 0. Consider F : [0, T ] × Ω × F(Rd ) → F(Rd ) defined as F (t, ω, u) = tx(ω), where x : Ω → F(Rd ) is a given fuzzy random variable. Then Eq. (3.1) takes the form X(t, ω)

[0,T ] P.1

=

X0 (ω) ⊕

1 Γ(β)



t

s x(ω)ds, (t − s)1−β

0

(3.7)

where X0 : Ω → F(Rd ), which constitutes an initial value, is a fuzzy random variable. Now we can write the solution to Eq. (3.7) as the following fuzzy stochastic process X : [0, T ] × Ω → F(Rd ) tβ+1 [0,T ] P.1 x(ω). X(t, ω) = X0 (ω) ⊕ β(β + 1)Γ(β) 

Since diam

tβ+1 x(ω) β(β + 1)Γ(β)



tβ+1 diam[x(ω)]α for α ∈ [0, 1], β(β + 1)Γ(β)

= α

we notice that for P -a.a. ω the functions t → diam[X(t, ω)]α are nondecreasing. In the next example we consider the mappings with values in F(R) := F(R1 ). Example B. Consider β ∈ (0, 1], F : [0, T ] × Ω × F(R) → F(R) defined as F (t, ω, u) = u and X0 : Ω → F(R) being a fuzzy random variable. Then Eq. (3.1) reads X(t, ω)

[0,T ] P.1

=

X0 (ω) ⊕

1 Γ(β)



t 0

1 X(s, ω)ds. (t − s)1−β

(3.8)

This equation is an extension of the classical Malthus model that can be applied, for instance, in populations growth modelling in biology. In this case, the variable X0 denotes initial number of individuals (or density on an given area) and X represents evolution of the population growth. To solve Eq. (3.8) we use method of determining the boundaries of [X(t, ω)]α for α ∈ [0, 1]. Therefore, for α ∈ [0, 1], we − + , X0,α :Ω→R consider the stochastic processes Xα− , Xα+ : [0, T ] × Ω → R and the random variables X0,α such that − + (ω), X0,α (ω)]. [X(t, ω)]α = [Xα− (t, ω), Xα+ (t, ω)] and [X0 (ω)]α = [X0,α Due to (3.8) we get

⎧ ⎨ X − (t, ω) α ⎩ X + (t, ω) α

[0,T ] P.1

=

[0,T ] P.1

=

− X0,α (ω) + + X0,α (ω) +

t

− (s,ω) Xα ds, 0 (t−s)1−β +  t Xα (s,ω) 1 Γ(β) 0 (t−s)1−β ds.

1 Γ(β)

This system is equivalent to the following system of fractional differential equations with classical Caputo’s fractional derivative C Dβ of the single-valued functions Xα− (·, ω), Xα+ (·, ω): ⎧ ] P.1 P.1 ⎨ C Dβ X − (t, ω) [0,T= − Xα− (t, ω), Xα− (0, ω) = X0,α (ω), α [0,T ] P.1 P.1 + ⎩ C Dβ X + (t, ω) + + = X (t, ω), X (0, ω) = X (ω). α

α

0,α

α

Hence we obtain that Xα− (t, ω)

[0,T ] P.1

=

− Eβ (tβ )X0,α (ω) and Xα+ (t, ω)

where Eβ denotes the Mittag–Leffler function, i.e. Eβ (t) = the solution X : [0, T ] × Ω → F(R) to (3.8) is of the form X(t, ω) Observe that diam[X(t, ω)]α

[0,T ] P.1

=

[0,T ] P.1

=

[0,T ] P.1

∞

=

+ Eβ (tβ )X0,α (ω),

tk k=0 Γ(βk+1) .

This allows us to infer that

Eβ (tβ )X0 (ω).

Eβ (tβ )diam[X0 (ω)]α for α ∈ [0, 1]

and since Eβ (·) is increasing for β ∈ (0, 1] then P -a.e. the functions t → diam[X(t, ω)]α are nondecreasing.

11

3.2

Equations with solutions of nonincreasing diameter

In this part of the paper we consider the similar problems like in Section 2.1 but in the context of the second kind of the random fuzzy fractional integral equations, i.e. we consider equation (3.2) whose solutions have trajectories with the values of nonincreasing diameter. Theorem 3.13 Let F : [a, b] × Ω × F(Rd ) → F(Rd ) satisfy assumptions of Theorem 3.6 and Y0 : Ω → F(Rd ) be a fuzzy random variable. Suppose that the sequence {Yn }∞ n=0 described as follows: for (t, ω) ∈ [a, b] × Ω Y0 (t, ω) := Y0 (ω),  t −1 1 Yn (t, ω) := Y0 (ω)  F (s, ω, Yn−1 (s, ω))ds, n∈N (3.9) Γ(β) a (t − s)1−β is well defined, i.e. the Hukuhara differences appearing above do exist. (i) If

L(b−a)β Γ(β+1)

< 1, then (3.2) has a unique global solution.

(ii) If

L(b−a)β Γ(β+1)

≥ 1, then (3.2) has a unique local solution.

Proof. (i) We shall show that equation (3.2) has a unique global solution. For (t, ω) ∈ [a, b] × Ω, by (P5) and (P3), we get  t

 1 −1 D Y1 (t, ω), Y0 (t, ω) = D Y0 (ω)  F (s, ω, Y (ω))ds, Y (ω) 0 0 Γ(β) a (t − s)1−β  t

1 1 D F (s, ω, Y (ω))ds, 0 . = 0 1−β Γ(β) a (t − s) Hence, proceeding as in the proof of Theorem 3.6, we obtain  [a,b] P.1 M (b − a)β

. ≤ D Y1 (t, ω), Y0 (t, ω) βΓ(β) For n = 2, 3, . . . we get

 D Yn (t, ω), Yn−1 (t, ω) = D Y0 (ω) 

= [a,b] P.1

≤

Hence

 t 1 −1 F (s, ω, Yn−1 (ω))ds, Γ(β) a (t − s)1−β  t

1 −1 Y0 (ω)  F (s, ω, Yn−2 (ω))ds 1−β Γ(β) a (t − s)  t  t

1 1 1 D F (s, ω, Y (ω))ds, F (s, ω, Yn−2 (ω))ds n−1 1−β 1−β Γ(β) a (t − s) a (t − s)  t L D(Yn−1 (s, ω), Yn−2 (s, ω))ds Γ(β) a (t − s)1−β

 [a,b] P.1 M D Yn (t, ω), Yn−1 (t, ω) ≤ L



L(b − a)β βΓ(β)

n .

(3.10)

Thus, like in the proof of Theorem 3.6, there exists a D-continuous fuzzy stochastic process Y (1) : [a, b] × Ω → F(Rd ) such that

 P.1 sup D Yn (t, ω), Y (1) (t, ω) −→ 0, as n → ∞. (3.11) t∈[a,b]

(1)

The process Y is a global solution to (3.2). Indeed    t 1 −1 (1) sup D Y (1) (t, ω), Y0 (ω)  F (s, ω, Y (s, ω))ds Γ(β) a (t − s)1−β t∈[a,b]   t

P.1

 D Yn−1 (s, ω), Y (1) (s, ω) ds L (1) sup ≤ sup D Yn (t, ω), Y (t, ω) + Γ(β) t∈[a,c1 ] a (t − s)1−β t∈[a,b] P.1

≤

 L(b − a)β

 sup D Yn−1 (s, ω), Y (1) (s, ω) . sup D Yn (t, ω), Y (1) (t, ω) + βΓ(β) s∈[a,b] t∈[a,b]

12

Due to (3.11)  D Y

(1)

1 (t, ω), Y0 (ω)  Γ(β)



t a

−1 F (s, ω, Y (1) (s, ω))ds (t − s)1−β



[a,b] P.1

=

0.

Assuming that Y˜ (1) : [a, b] × Ω → F(Rd ) is another global solution to (3.2) we obtain   t (1)

(1)  [a,b] P.1 L D Y (s, ω), Y˜ (1) (s, ω) ds (1) ˜ D Y (t, ω), Y (t, ω) ≤ . Γ(β) a (t − s)1−β  [a,b] P.1

= 0 which shows the uniqueness of the By Gronwall’s inequality we get D Y (1) (t, ω), Y˜ (1) (t, ω) solution Y (1) . This ends the proof of the existence of a unique global solution to (3.2) in the case when L(b−a)β Γ(β+1) < 1. (ii) Note that if

L(b−a)β Γ(β+1)

≥ 1 then (3.10) is no longer applicable to show the existence of a fuzzy stochastic β

Lh process defined on the entire interval [a, b]. Therefore we look for an h satisfying Γ(β+1) < 1. Then, (1) ˜ similarly as in the case (i), we show the existence of a unique solution Y : [a, a + h] × Ω → F(Rd ) to (3.2) on the interval [a, a + h]. Since h < b − a, this solution is local. 

In the part (ii) of the proof above we cannot proceed immediately with the continuation of the local solution. Under present setting of the equations (3.2) with the nonincreasing diameter, such a procedure is not immediate to be applied as distinct from the equations (3.1) with the nondecreasing diameter. To continue the local solution Y˜ (1) which is defined on the interval [a, a + h] we should consider the equation  t −1 1 [a+h,a+h+h∗ ] P.1 Y (t, ω) = Y˜ (1) (a + h, ω)  F (s, ω, Y (s, ω))ds, Γ(β) a+h (t − s)1−β where h∗ > 0 is a constant satisfying a + h + h∗ ≤ b. A natural method which could be proposed in derivation of existence of a solution to this equation is a usage of the following sequence of the approximate solutions {Y˜n }∞ n=0 defined as Y˜0 (t, ω) := Y˜ (1) (a + h, ω) for (t, ω) ∈ [a + h, a + h + h∗ ] × Ω, and for n ∈ N Y˜n (t, ω) := Y˜ (1) (a + h, ω) 

1 Γ(β)



t a+h

−1 F (s, ω, Y˜n−1 (s, ω))ds for (t, ω) ∈ [a + h, a + h + h∗ ] × Ω. (t − s)1−β

However, here we encounter a dilemma. Namely, we are not assured that the Hukuhara differences appearing in the definition of Y˜n do exist. They may not exist and therefore the above mentioned procedure of continuation of the solution can fail. In the sequel we present some considerations concerning the method of continuation of solution for the equation (3.2). Let the assumptions of Theorem 3.13 be satisfied. Let ε ∈ (0, 1). Let R > 1 and h > 0 be such that  a + Rh = b Rβ+1 Lhβ = 1 − ε. βΓ(β) By N we denote the integer part of the real number R. Put c := a + h for  = 1, 2, . . . , N . In the first step we consider the equation  t −1 1 [a,c1 ] F (s, ω, Y (s, ω))ds. (3.12) Y (t, ω) = Y0 (ω)  Γ(β) a (t − s)1−β Using the sequence {Yn } defined in Theorem 3.13 in (3.9) and proceeding like in the proof of part (i) of this theorem we get  n

 [a,c1 ] P.1 M M Lhβ n (1 − ε) D Yn (t, ω), Yn−1 (t, ω) ≤ < L βΓ(β) L for n ∈ N. Thus there exists a D-continuous fuzzy stochastic process Y (1) : [a, c1 ] × Ω → F(Rd ) such that

 P.1 sup D Yn (t, ω), Y (1) (t, ω) −→ 0,

t∈[a,c1 ]

13

as n → ∞.

(3.13)

Further, proceeding like in the proof of Theorem 3.13 we can see that Y (1) is a unique solution to (3.12). Now, to make a continuation of the local solution Y˜ (1) , we consider the equation  t −1 1 [c1 ,c2 ] P.1 = Y (1) (c1 , ω)  F (s, ω, Y (s, ω))ds. (3.14) Y (t, ω) Γ(β) c1 (t − s)1−β To show that this equation posesses a solution we will use the sequence {Yn } again. For (t, ω) ∈ [c1 , c2 ] × Ω, by (P5) and (P3), we get  t

 −1 1 D Y1 (t, ω), Y0 (t, ω) = D Y0 (ω)  F (s, ω, Y0 (ω))ds, Y0 (ω) 1−β Γ(β) a (t − s)  t

1 1 D F (s, ω, Y0 (ω))ds, 0 . = 1−β Γ(β) a (t − s) Hence

 [c1 ,c2 ] P.1 M (2h)β

. ≤ D Y1 (t, ω), Y0 (t, ω) βΓ(β)

Further due to (P7), (P6) and (P2) we obtain

 D Y2 (t, ω), Y1 (t, ω)  a+h  t   1 1 −1 −1 = D Y0 (ω)  F (s, ω, Y (s, ω))ds  F (s, ω, Y1 (s, ω))ds, 1 Γ(β) a (t − s)1−β Γ(β) a+h (t − s)1−β  a+h  t  

1 1 −1 −1 Y0 (ω)  F (s, ω, Y (s, ω))ds  F (s, ω, Y0 (s, ω))ds 0 1−β 1−β Γ(β) a (t − s) Γ(β) a+h (t − s)  a+h −1 1 F (s, ω, Y1 (s, ω))ds, ≤ D Y0 (ω)  Γ(β) a (t − s)1−β  a+h

−1 1 F (s, ω, Y0 (s, ω))ds Y0 (ω)  1−β Γ(β) a (t − s)  t

1  t 1 −1 −1 F (s, ω, Y (s, ω))ds, F (s, ω, Y (s, ω))ds +D 1 0 Γ(β) a+h (t − s)1−β Γ(β) a+h (t − s)1−β Hence 

D Y2 (t, ω), Y1 (t, ω)

 t L D(Y1 (s, ω), Y0 (s, ω))ds D(Y1 (s, ω), Y0 (s, ω))ds + 1−β (t − s) Γ(β) a+h (t − s)1−β a   a+h t M Lhβ M L(2h)β ds ds + 2 1−β 2 βΓ (β) a (t − s) βΓ (β) a+h (t − s)1−β

[c1 ,c2 ] P.1

L Γ(β)

≤

[c1 ,c2 ] P.1

≤



a+h

M L(2h)β hβ M Lhβ (2h)β · + · βΓ2 (β) β βΓ2 (β) β   2 M 2L(2h)β . L βΓ(β)

≤ ≤

Similarly

 D Y3 (t, ω), Y2 (t, ω)

 t L D(Y2 (s, ω), Y1 (s, ω))ds D(Y2 (s, ω), Y1 (s, ω))ds + 1−β (t − s) Γ(β) a+h (t − s)1−β a     2 2 M Lhβ L M 2L(2h)β L (2h)β hβ · + · · · Γ(β) L βΓ(β) β Γ(β) L βΓ(β) β   β 3 M 2L(2h) . L βΓ(β)

[c1 ,c2 ] P.1

L Γ(β)

≤ ≤

≤



a+h

Generally we have  [c1 ,c2 ] P.1 M ≤ D Yn (t, ω), Yn−1 (t, ω) L

14



2L(2h)β βΓ(β)

n <

M n (1 − ε) . L

Thus there exists a D-continuous fuzzy stochastic process Y (2) : [c1 , c2 ] × Ω → F(Rd ) such that

 P.1 sup D Yn (t, ω), Y (2) (t, ω) −→ 0,

t∈[c1 ,c2 ]

as n → ∞.

(3.15)

Since  sup D Y (2) (t, ω), Y (1) (c1 , ω) 

t∈[c1 ,c2 ]

P.1

≤

P.1

≤



−1 F (s, ω, Y (2) (s, ω))ds 1−β c1 (t − s)

 sup D Yn (t, ω), Y (2) (t, ω) + D(Yn (c1 , ω), Y (1) (c1 , ω)) 1 Γ(β)

t



t∈[c1 ,c2 ]

  t

D Yn−1 (s, ω), Y (2) (s, ω) ds L sup + Γ(β) t∈[c1 ,c2 ] c1 (t − s)1−β

 sup D Yn (t, ω), Y (2) (t, ω) + D(Yn (c1 , ω), Y (1) (c1 , ω))

t∈[c1 ,c2 ]

+

 Lhβ sup D Yn−1 (s, ω), Y (2) (s, ω) βΓ(β) s∈[c1 ,c2 ]

and the right-hand side converges to zero, we obtain that the process Y (2) is a solution to (3.14). The uniqueness of the solution Y (2) is inferred by using of the Gronwall inequality. It turns out that considering the equations  t 1 −1 [c ,c+1 ] P.1 () = Y (c , ω)  F (s, ω, Y (s, ω))ds (3.16) Y (t, ω) Γ(β) c (t − s)1−β for  = 1, 2, . . . , N − 1, we obtain  [c ,c+1 ]

≤ D Yn (t, ω), Yn−1 (t, ω)

P.1

M L



( + 1)L(( + 1)h)β βΓ(β)

n <

M n (1 − ε) . L

This guarantees the existence of a D-continuous fuzzy stochastic process Y (+1) : [c , c+1 ] × Ω → F(Rd ) which is a unique solution to (3.16). Then ⎧ (1) Y (t, ω) for (t, ω) ∈ [a, c1 ] × Ω, ⎪ ⎪ ⎪ (2) ⎨ Y (t, ω) for (t, ω) ∈ [c1 , c2 ] × Ω, Y (t, ω) = .. .. .. ⎪ . . . ⎪ ⎪ ⎩ (N ) Y (t, ω) for (t, ω) ∈ [cN −1 , cN ] × Ω is a unique local solution to (3.2) and this solution is defined on the interval [a, cN ]. Note that the above presented procedure does not allow to obtain the solution on the interval [cN , b], so it does not allow to obtain a global solution. To have a solution on [cN , b] we should have the numbers R > 1 and h > 0 satisfying  a + Rh = b (R+1)β+1 Lhβ = 1 − ε. βΓ(β) Unfortunately, this system may not have a desired solution. To see this, suppose that L = 2 and consider a = 0, b = 10 and β = 2. Under these data we obtain that (10 + h)3 = (1 − ε)h, where ε ∈ (0, 1), and this equation does not have any solution h > 0. This observation leads to an issue of proposing a new method of continuation of the local solution to the random fuzzy fractional integral equation (3.2) in such a way that the global solution is achieved. This issue constitutes an open problem. In the sequel we present the results concerning the estimation of a distance between the approximation Yn defined in (3.9) and the exact solution Y to (3.2), a boundedness of the solution Y as well as the continuous dependence of Y with respect to the initial value Y0 and the nonlinearity F . Theorem 3.14 Let F : [a, b] × Ω × F(Rd ) → F(Rd ) satisfy assumptions of Theorem 3.6 and Y0 : Ω → F(Rd ) be a fuzzy random variable. Suppose that the sequence {Yn }∞ n=0 described as follows: for (t, ω) ∈ [a, b] × Ω Y0 (t, ω) := Y0 (ω),

15

Yn (t, ω) := Y0 (ω) 



1 Γ(β)

−1 F (s, ω, Yn−1 (s, ω))ds, (t − s)1−β

t a

n∈N

is well defined, i.e. the Hukuhara differences appearing above do exist. Assume that for the nth approximant Yn and the global solution Y to (3.2) we have 

 P.1 M sup D Yn (t, ω), Y (t, ω) ≤ L t∈[a,b]

L(b − a)β Γ(β + 1)

n+1 

L(b − a)β 1− Γ(β + 1)

L(b−a)β Γ(β+1)

< 1. Then

−1 .

The proof is omitted since it is exactly the same as the proof of Theorem 3.7. Theorem 3.15 Let the assumptions of Theorem 3.14 be satisfied. Then for the global solution Y : [a, b]× Ω → F(Rd ) to (3.2) we have

 P.1

 M (b − a)β . sup D Y (t, ω), 0 ≤ D Y0 (ω), 0 + Γ(β + 1) t∈[a,b] Proof. For (t, ω) ∈ [a, b] × Ω we have

 D Y (t, ω), 0 = D Y0 (ω) 

1 Γ(β)



t a

−1 F (s, ω, Y (s, ω))ds, 0 . 1−β (t − s)

Applying (P4), triangle inequality for the metric D, (P3) and (P11) we arrive to the inequality   t



 [a,b] P.1

D F (s, ω, Y (s, ω)), 0 ds 1 ≤ D Y0 (ω), 0 + D Y (t, ω), 0 Γ(β) a (t − s)1−β 

and further we proceed as in the proof of Theorem 3.8. Let us consider (3.2) and a sequence of equations Yn (t, ω)

[a,b] P.1

=

(n)

Y0 (ω) 

1 Γ(β)



t a

−1 F (s, ω, Yn (s, ω))ds (t − s)1−β

(3.17)

for n ∈ N. Let Y denote the global solution to (3.2) and Yn the global solutions to (3.17). Theorem 3.16 Let the assumptions of Theorem 3.14 be satisfied. Assume that for every n ∈ N equation (3.17) possesses a unique global solution Yn . Then  

 P.1 (n)  L(b − a)β . sup D Yn (t, ω), Y (t, ω) ≤ D Y0 (ω), Y0 (ω) exp Γ(β + 1) t∈[a,b] Proof. For (t, ω) ∈ [a, b] × Ω we have 

D Yn (t, ω), Y (t, ω) (n) = D Y0 (ω)  Y0 (ω) 

1 Γ(β)

and by (P6), (P2), (P3) and (P11)

 D Yn (t, ω), Y (t, ω)

(n) ≤ D Y0 (ω), Y0 (ω) +

a



−1 F (s, ω, Yn (s, ω))ds, (t − s)1−β a

−1 F (s, ω, Y (s, ω))ds 1−β (t − s)

1 Γ(β)  t

1 Γ(β)



t a

t

D (F (s, ω, Yn (s, ω)), F (s, ω, Y (s, ω))) ds . (t − s)1−β 

Next we proceed as in the proof of Theorem 3.9.

16

(n)  P.1 Corollary 3.17 Let the assumptions of Theorem 3.16 be satisfied. Assume that D Y0 (ω), Y0 (ω) −→ 0 as n → ∞. Then for the solutions to equation (3.2) and (3.17) we have

 P.1 sup D Yn (t, ω), Y (t, ω) −→ 0

as n → ∞.

t∈[a,b]

Let us consider (3.2) and the sequence of equations Yn (t, ω)

[a,b] P.1

=

Y0 (ω) 

1 Γ(β)



t a

 −1 F (s, ω, Yn (s, ω)) ⊕ Gn (s, ω, Yn (s, ω)) ds (t − s)1−β

(3.18)

for n ∈ N. Let Y denote the global solution to (3.2) and Yn the global solution to (3.18). Theorem 3.18 Let the assumptions of Theorem 3.13 be satisfied. Let the mapping Gn : [a, b] × Ω × F(Rd ) → F(Rd ), for each n ∈ N, satisfies the same assumptions as F in Theorem 3.13. Suppose that for every n ∈ N there exists a positive constant Mn such that with P.1 for every (t, u) ∈ [a, b] × F(Rd ) D (Gn (t, ω, u), 0) ≤ Mn . Then

 P.1 (b − a)β exp sup D Yn (t, ω), Y (t, ω) ≤ Mn Γ(β + 1) t∈[a,b]



L(b − a)β Γ(β + 1)

 .

Proof. For (t, ω) ∈ [a, b] × Ω we have

 D Yn (t, ω), Y (t, ω)  t

 −1 1 F (s, ω, Yn (s, ω)) ⊕ Gn (s, ω, Yn (s, ω)) ds, = D Y0 (ω)  1−β Γ(β) a (t − s)  t

1 −1 F (s, ω, Y (s, ω))ds Y0 (ω)  1−β Γ(β) a (t − s)   t

D F (s, ω, Yn (s, ω)) ⊕ Gn (s, ω, Yn (s, ω)), F (s, ω, Y (s, ω)) ds 1 . = Γ(β) a (t − s)1−β By (P2) 

D Yn (t, ω), Y (t, ω) ≤



D(Gn (s, ω, Yn (s, ω)), 0)ds (t − s)1−β  t D(F (s, ω, Yn (s, ω)), F (s, ω, Y (s, ω)))ds 1 + Γ(β) a (t − s)1−β 1 Γ(β)

t

a



and further we proceed as in the proof of Theorem 3.11.

Corollary 3.19 Let the assumptions of Theorem 3.18 be satisfied. Then for the global solution Y to (3.2) and the global solution Yn to (3.18) we have

 P.1 sup D Yn (t, ω), Y (t, ω) −→ 0

as n → ∞

t∈[a,b]

provided that Mn −→ 0 as n → ∞. Notice that results formulated in Theorems 3.14, 3.15, 3.16 and 3.18 can be repeated in terms of the local solution to (3.2). It is also worth mentioning that all the considerations presented in this paper for the random fuzzy fractional integral equations apply immediately to the random set-valued fractional integral equations. This is immediate since every ordinary set is also a fuzzy set by means of its characteristic function. Finally, we consider some examples of equations with solutions of nonincreasing diameter. Our aim is to find their explicit solutions. We present some counterparts of Examples A and B.

17

Example C. Let β > 0. Consider F : [0, T ] × Ω × F(Rd ) → F(Rd ) defined as F (t, ω, u) = tx(ω), where x : Ω → F(R) is a given fuzzy random variable. Applying these data to Eq. (3.2) we consider the following equation  t 1 −s [0,T ] P.1 Y0 (ω) = Y (t, ω) ⊕ x(ω)ds, (3.19) Γ(β) 0 (t − s)1−β where Y0 : Ω → F(Rd ) is a fuzzy random variable. We can write the solution to Eq. (3.19) as the following fuzzy stochastic process Y : [0, T ∗ ] × Ω → F(Rd ) −tβ+1 [0,T ∗ ] P.1 Y (t, ω) x(ω), = Y0 (ω)  β(β + 1)Γ(β) β+1

−t where T ∗ is a positive constant for which the Hukuhara differences Y0 (ω)  β(β+1)Γ(β) x(ω) exist for every ∗ t ∈ [0, T ] and P -a.a. ω. Observe further that

diam[Y (t, ω)]α

[0,T ∗ ] P.1

=

diam[Y0 (ω)]α −

tβ+1 diam[(−1)x(ω)]α . β(β + 1)Γ(β)

Hence for P -a.a. ω the functions t → diam[Y (t, ω)]α are nonincreasing. Example D. Consider β ∈ (0, 1], F : [0, T ] × Ω × F(R) → F(R) defined as F (t, ω, u) = u and Y0 : Ω → F(R) being a fuzzy random variable. Then Eq. (3.2) reads  t 1 −1 [0,T ] P.1 Y0 (ω) = Y (t, ω) ⊕ Y (s, ω)ds. (3.20) Γ(β) 0 (t − s)1−β This equation can also be used in population growth models incorporating fuzziness and randomness. Like in Example B we introduce the stochastic processes Yα− , Yα+ : [0, T ]×Ω → R and the random variables − + , Y0,α : Ω → R such that Y0,α − + [Y (t, ω)]α = [Yα− (t, ω), Yα+ (t, ω)] and [Y0 (ω)]α = [Y0,α (ω), Y0,α (ω)].

By (3.20) we arrive at ⎧ ⎨ Y − (t, ω) α ⎩ Y + (t, ω)

[0,T ] P.1

=

[0,T ] P.1

=

α

t

Yα+ (s,ω) ds, 0 (t−s)1−β −  t (s,ω) Y 1 α Γ(β) 0 (t−s)1−β ds.

− Y0,α (ω) +

1 Γ(β)

+ Y0,α (ω) +

Rewritting this system in terms of the classical Caputo’s fractional derivative C Dβ of the single-valued functions Yα− (·, ω), Yα+ (·, ω) we get  −    −     −  − [0,T ] P.1 Y0,α (ω) Yα (t, ω) Yα (0, ω) 0 1 Yα (t, ω) P.1 C β , . D = = + 1 0 Yα+ (t, ω) Yα+ (t, ω) Yα+ (0, ω) Y0,α (ω) Hence we obtain that

where Eβ (A) =  Eβ



Yα− (t, ω) Yα+ (t, ω)

∞

Ak k=0 Γ(βk+1)

0 1

1 0

 t

= =

=

∞   0 1 k=0  ∞  1 0 k=0

we have Yα− (t, ω)

[0,T ] P.1

=

∞  k=0

and Yα+ (t, ω)

[0,T ] P.1

=

[0,T ] P.1

 Eβ

0 1

1 0



 tβ

− Y0,α (ω) + Y0,α (ω)

 ,

for the matrix A. Since

 β



∞  k=0

k tβk 1 0 Γ(βk + 1)   ∞   t2kβ t(2k+1)β 0 0 1 + , 1 1 0 Γ(2kβ + 1) Γ((2k + 1)β + 1) k=0

∞

 t2kβ t(2k+1)β − Y0,α Y + (ω) (ω) + Γ(2kβ + 1) Γ((2k + 1)β + 1) 0,α k=0 ∞

 t2kβ t(2k+1)β + Y0,α Y − (ω). (ω) + Γ(2kβ + 1) Γ((2k + 1)β + 1) 0,α k=0

18

Thus we infer that the solution Y : [0, T ] × Ω → F(R) to (3.20) is of the form     ∞ ∞   t2kβ t(2k+1)β [0,T ] P.1  (−1)Y0 (ω) . Y (t, ω) = Y0 (ω) Γ(2kβ + 1) Γ((2k + 1)β + 1) k=0

k=0

Accordingly to Theorem 3.4 (ii) we have that P -a.e. the functions t → diam[Y (t, ω)]α are nonincreasing. In fact for the solution Y and α ∈ [0, 1] we get diam[Y (t, ω)]α

[0,T ] P.1

=

diam[Y0 (ω)]α

∞  k=0

∞

 t2kβ t(2k+1)β − diam[(−1)Y0 (ω)]α . Γ(2kβ + 1) Γ((2k + 1)β + 1) k=0

Since diam[Y0 (ω)]α = diam[(−1)Y0 (ω)]α , we obtain further ∞  ∞   t2kβ t(2k+1)β [0,T ] P.1 − . diam[Y (t, ω)]α = diam[Y0 (ω)]α Γ(2kβ + 1) Γ((2k + 1)β + 1) k=0

4

k=0

Concluding remarks

In this paper, for the first time, the random fuzzy fractional integral equations are studied. They can be viewed as some extensions of the random fuzzy differential equations [36, 37, 39] and the deterministic fuzzy fractional integral equations investigated in [1, 10, 58]. They are appropriate to model dynamic systems operating in fuzzy and random environment. A main purpose of the paper is to present a framework within which such equations can be investigated and to justify the well-posedness of introduced theory. Our aim is to give the theoretical foundations which can be a starting point for further developments in this area. To make the paper almost self-contained, we collected needed informations concerning properties of Hukuhara difference of fuzzy sets, definition of fuzzy integral, properties of this integral, definitions of fuzzy random variable and fuzzy stochastic process. Then, using a notion of fuzzy fractional integral we formulate two kinds of random fuzzy fractional integral equations. These equations coincide if the data are single-valued. However, in the fuzzy-setvalued case they are completely different and their solutions exhibit different geometrical properties. Namely, solutions of equations of the first type have trajectories with nondecreasing diameter of their consecutive values, whereas for solutions of equations of the second type this diameter is nonincreasing. Under suitable conditions we prove, for the two types of equations parallelly, the existence and uniqueness of solutions of such fractional equations. We establish a boundedness of solutions and their insensitivity to small changes of parameters. This confirms the well-posedness of introduced theory. Finally, we solve explicitly some examples of the random fuzzy fractional integral equations. Although we have found some solutions in their explicit, closed forms, we are aware that in general the solutions to random fuzzy fractional integral equations cannot be found explicitly. Therefore in a future research one could investigate some methods of approximation of solutions. Some numerical schemes will be needed to approximate the solutions. Then it will be possible to simulate trajectories of solutions. With a help of numerical methods one can approximate the solutions of random fuzzy fractional integral equations appearing in many real-life applications, for instance in modelling number of radioactive nuclei in a radioactive substance, in dynamics of harmonic oscillator, in biology in predator–prey models. The paper studies integral equations. It is worth mentioning that these equations can be used in some future examinations of random fuzzy fractional differential equations involving either a fuzzy Caputo’s derivative or a fuzzy Riemann–Liouville’s devivative.

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