CUBO A Mathematical Journal

Vol.19, No¯ 01, (17–38). March 2017

Hilfer and Hadamard Functional Random Fractional

Differential Inclusions
Säıd Abbas1, Mouffak Benchohra2,

Jamal-Eddine Lazreg2 and Gaston M. N’Guérékata3

1 Laboratory of Mathematics, University of Säıda,

P.O. Box 138, Säıda 20000, Algeria
2 Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes,

P.O. Box 89, Sidi Bel-Abbès 22000, Algeria.
3 Department of Mathematics, Morgan State University,

1700 E. Cold Spring Lane, Baltimore M.D. 21252, USA.

benchohra@univ-sba.dz, lagregjamal@yahoo.fr, gaston.n’guerekata@morgan.edu

ABSTRACT

This paper deals with some existence and Ulam stability results for some functional

differential inclusions of Hilfer and Hilfer-Hadamard type with convex and non-convex

right hand side. We employ some multi-valued random fixed point theorems for the

existence of random solutions. Next we prove that our problems are generalized Ulam-

Hyers-Rassias stable.

RESUMEN

Este art́ıculo estudia algunos resultados de existencia y estabilidad de Ulam para al-

gunas inclusiones funcionales diferenciales de tipos Hilfer y Hilfer-Hadamard con lado

derecho convexo y no-convexo. Empleamos algunos teoremas aleatorios de punto fijo

multi-valuados para la existencia de soluciones aleatorias. A continuación demostramos

que nuestros problemas son Ulam-Hyers-Rassias estables generalizados.

Keywords and Phrases: Functional Random differential inclusion; left-sided Riemann-Liouville

integral of fractional order; left-sided Hadamard integral of fractional order; Hilfer fractional deriva-

tive; convex; non-convex; Hilfer-Hadamard fractional derivative; existence; Ulam stability; random

solution; fixed point.

2010 AMS Mathematics Subject Classification: 26A33. 34A60.



18 S. Abbas, M. Benchohra J.-e. Lazreg, G. M. N’Guérékata CUBO
19, 1 (2017)

1 Introduction

Fractional calculus is relative to the traditional integer order calculus put forward, which is the

order of calculus from integer orders extended to any order of the mathematical promotion. From

the theoretical point of view, the fractional differential calculus signal processing order extended

to any number from an integer, the ways and means of information processing were extended.

Fractional order differential equations have recently been applied in various areas of engineering,

mathematics, physics and bio-engineering, and other applied sciences [19, 35]. For some funda-

mental results in the theory of fractional calculus and fractional differential equations we refer the

reader to the monographs of Abbas et al. [7, 8], Kilbas et al. [26] and Zhou [41, 42], the papers

by Abbas et al. [1, 4, 5, 9, 10], Benchohra et al. [11], and the references therein.

The nature of a dynamic system in engineering or natural sciences depends on the accuracy

of the information we have concerning the parameters that describe that system. If the knowledge

about a dynamic system is precise then a deterministic dynamical system arises. Unfortunately in

most cases the available data for the description and evaluation of parameters of a dynamic system

are inaccurate, imprecise or confusing. In other words, evaluation of parameters of a dynamical

system is not without uncertainties. When our knowledge about the parameters of a dynamic

system are of statistical nature, that is, the information is probabilistic, the common approach in

mathematical modeling of such systems is the use of random differential equations or stochastic

differential equations. Random differential equations, as natural extensions of deterministic ones,

arise in many applications and have been investigated by many mathematicians. We refer the

reader to the monographs [12, 27, 37].

The stability of functional equations was originally raised by Ulam [38]). next by Hyers [21].

Thereafter, this type of stability is called the Ulam-Hyers stability. In 1978, Rassias [32] provided

a remarkable generalization of the Ulam-Hyers stability of mappings by considering variables. The

concept of stability for a functional equation arises when we replace the functional equation by an

inequality which acts as a perturbation of the equation. Considerable attention has been given to

the study of the Ulam-Hyers and Ulam-Hyers-Rassias stability of all kinds of functional equations;

one can see the monographs of [8, 22], and the papers of Abbas et al. [1, 2, 3, 4, 6, 9, 10], Petru et al.

[29], and Rus [33, 34] discussed the Ulam-Hyers stability for operatorial equations and inclusions.

More details from historical point of view, and recent developments of such stabilities are reported

in [23, 33].

Recently, considerable attention has been given to the existence of solutions of initial and

boundary value problems for fractional differential equations with Hilfer fractional derivative; see

[15, 16, 19, 20, 24, 36, 39]. Motivated by the above papers, in this article we discuss the existence

and the Ulam stability of solutions for the following problem of Random Hilfer fractional differential



CUBO
19, 1 (2017)

Hilfer and Hadamard functional random fractional differential . . . 19

inclusions of the form
⎧
⎪⎨

⎪⎩

(D
α,β

0
u)(t, w) ∈ F(t, u(t, w), w); t ∈ I := [0, T],

(I
1−γ

0
u)(t, w)|t=0 = φ(w),

w ∈ Ω, (1.1)

where α ∈ (0, 1), β ∈ [0, 1], γ = α + β − αβ, T > 0, (Ω, A) is a measurable space, φ : Ω → R

is a measurable function, F : I × R → P(R) is a given multivalued map, P(R) is the family of all

nonempty subsets of R, I
1−γ

0
is the left-sided Riemann-Liouville integral of order 1 − γ, and D

α,β

0

is the Hilfer fractional derivative of order α and type β.

Next, we consider the following problem of random Hilfer-Hadamard fractional differential

inclusions of the form
⎧
⎪⎨

⎪⎩

(HD
α,β

1
u)(t, w) ∈ G(t, u(t, w), w); t ∈ [1, T],

(HI
1−γ

1
u)(1, w) = φ0(w),

w ∈ Ω, (1.2)

where α ∈ (0, 1), β ∈ [0, 1], γ = α + β − αβ, T > 1, φ0 : Ω → R is a measurable function,

G : [1, T] × R → P(R) is a given multivalued map, HI1−γ
1

is the left-sided Hadamard integral of

order 1 − γ, and HD
α,β

1
is the Hilfer-Hadamard fractional derivative of order α and type β.

2 Preliminaries

Let C be the Banach space of all continuous functions v from I into R with the supremum (uniform)

norm

∥v∥∞ := sup
t∈I

|v(t)|.

As usual, AC(I) denotes the space of absolutely continuous functions from I into R. We denote by

AC1(I) the space defined by

AC1(I) := {w : I → R :
d

dt
w(t) ∈ AC(I)}.

By L1(I), we denote the space of Lebesgue-integrable functions v : I → R with the norm

∥v∥1 =

∫
T

0

|v(t)|dt.

Let L∞(I) be the Banach space of measurable functions u : I → R which are essentially bounded,

equipped with the norm

∥u∥L∞ = inf{c > 0 : |u(t)| ≤ c, a.e. t ∈ I}.

By Cγ(I) and C
1
γ
(I), we denote the weighted spaces of continuous functions defined by

Cγ(I) = {w : (0, T] → R : t
1−γw(t) ∈ C},



20 S. Abbas, M. Benchohra J.-e. Lazreg, G. M. N’Guérékata CUBO
19, 1 (2017)

with the norm

∥w∥Cγ := sup
t∈I

|t1−γw(t)|,

and

C1
γ
(I) = {w ∈ C :

dw

dt
∈ Cγ},

with the norm

∥w∥C1
γ
:= ∥w∥∞ + ∥w

′∥Cγ.

Throughout this paper, we denote ∥w∥Cγ by ∥w∥C.

For each u ∈ Cγ and w ∈ Ω, define the set of selections of F by

SF◦u(w) = {v : Ω → L
1(I) : v(t, w) ∈ F(t, u(t, w), w); t ∈ I}.

Let E be a Banach space, and denote Pcl(E) = {A ∈ P(E) : A closed},

Pcp,c(E) = {A ∈ P(E) : A compact and convex}.

Consider Hd : P(E) × P(E) −→ [0, ∞) ∪ {∞} given by

Hd(A, B) = max

{

sup
a∈A

d(a, B), sup
b∈B

d(A, b)

}

,

where d(A, b) = inf
a∈A

d(a, b), d(a, B) = inf
b∈B

d(a, b). Then (Pbd,cl(E), Hd) is a Hausdorff metric

space.

Definition 1. A multifunction F : Ω → E is called A- measurable if, for any open subset B of E,

the set F−1(B) = {w ∈ Ω : F(w) ∩ B ̸= ∅} ∈ A. Note that if F(w) ∈ Pcl(E) for all w ∈ Ω, then

F is measurable if and only if F−1(D) ∈ A for all D ∈ Pcl(E). A measurable operator u : Ω → E

is called a measurable selector for a measurable multifunction F : Ω → E, if u(w) ∈ F(w). Let

M ∈ Pcl(E), then a mapping f : Ω × M → E is called a random operator if, for each u ∈ M, the

mapping f(., u) : Ω → E is measurable. An operator u : Ω → E is said to be a random fixed point

of F if u is measurable and u(w) ∈ F(w, u(w)) for all w ∈ Ω.

Definition 2. A multifunction F : Ω × E → P(E) is called Carathéodory if F(·, u) is measurable

for all u ∈ E and F(w, ·) is continuous for all w ∈ Ω.

Definition 3. A multivalued map F : I × E × Ω → Pcp(E) is said to be random Carathéodory if

(i) (t, w) *−→ F(t, u, w) is jointly measurable for each u ∈ E; and

(ii) u *−→ F(t, u, w) is Hausdorff continuous for almost each t ∈ I, w ∈ Ω.

Definition 4. [17] Let E be a separable Banach space. If F : I× E → Pcp(E) is Carathéodory, then

the multivalued mapping (t, u(t)) → F(t, u(t)), is jointly measurable for any measurable E-valued

function u on I.



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19, 1 (2017)

Hilfer and Hadamard functional random fractional differential . . . 21

Definition 5. A multivalued random operator N : Ω × E → Pcl(E) is called multivalued random

contraction if there is a measurable function k : Ω → [0, ∞) such that

Hd(N(w)u, N(w)v) ≤ k(w)∥u − v∥E,

for all u, v ∈ E and w ∈ Ω, where k(w) ∈ [0, 1) on Ω.

Now, we give some results and properties of fractional calculus.

Definition 6. [7, 26] The Riemann-Liouville integral of order r > 0 of a function w ∈ L1(I) is

defined by

(Ir0w)(t) =
1

Γ(r)

∫t

0

(t − s)r−1w(s)ds; for a.e. t ∈ I,

where Γ(·) is the (Euler’s) Gamma function defined by

Γ(ξ) =

∫
∞

0

tξ−1e−tdt; ξ > 0.

Notice that for all r, r1, r2 > 0 and each w ∈ C, we have I
r
0
w ∈ C, and

(Ir1
0
Ir2
0
w)(t) = (Ir1+r2

0
w)(t); for a.e. t ∈ I.

Definition 7. [7, 26] The Riemann-Liouville fractional derivative of order r ∈ (0, 1] of a function

w ∈ L1(I) is defined by

(Dr
0
w)(t) =

!
d

dt
I1−r
0

w

"

(t)

=
1

Γ(1 − r)

d

dt

∫t

0

(t − s)−rw(s)ds; for a.e. t ∈ I.

Let r ∈ (0, 1], γ ∈ [0, 1) and w ∈ C1−γ(I). Then the following expression leads to the left

inverse operator as follows.

(Dr
0
Ir
0
w)(t) = w(t); for all t ∈ (0, T].

Moreover, if I1−r
0

w ∈ C1
1−γ

(I), then

(Ir0D
r
0w)(t) = w(t) −

(I1−r
0

w)(0+)

Γ(r)
tr−1; for all t ∈ (0, T].

Definition 8. [7, 26] The Caputo fractional derivative of order r ∈ (0, 1] of a function w ∈ AC(I)

is defined by

(cDr0w)(t) =

!

I1−r
0

d

dt
w

"

(t)

=
1

Γ(1 − r)

∫t

0

(t − s)−r
d

ds
w(s)ds; for a.e. t ∈ I.



22 S. Abbas, M. Benchohra J.-e. Lazreg, G. M. N’Guérékata CUBO
19, 1 (2017)

In [19], R. Hilfer studied applications of a generalized fractional operator having the Riemann-

Liouville and the Caputo derivatives as specific cases (see also [20, 24, 36].

Definition 9. (Hilfer derivative). Let α ∈ (0, 1), β ∈ [0, 1], w ∈ L1(I), I
(1−α)(1−β)
0

∈ AC1(I).

The Hilfer fractional derivative of order α and type β of w is defined as

(D
α,β

0
w)(t) =

!

I
β(1−α)
0

d

dt
I
(1−α)(1−β)
0

w

"

(t); for a.e. t ∈ I. (2.1)

Properties. Let α ∈ (0, 1), β ∈ [0, 1], γ = α + β − αβ, and w ∈ L1(I).

1. The operator (D
α,β

0
w)(t) can be written as

(D
α,β

0
w)(t) =

!

I
β(1−α)
0

d

dt
I
1−γ

0
w

"

(t) =
#
I
β(1−α)
0

D
γ

0
w
$
(t); for a.e. t ∈ I.

Moreover, the parameter γ satisfies

γ ∈ (0, 1], γ ≥ α, γ > β, 1 − γ < 1 − β(1 − α).

2. The generalization (2.1) for β = 0, coincides with the Riemann-Liouville derivative and for

β = 1 with the Caputo derivative.

Dα,00 = D
α
0 , and D

α,1
0 =

cDα0 .

3. If D
β(1−α)
0

w exists and in L1(I), then

(D
α,β

0
Iα0 w)(t) = (I

β(1−α)
0

D
β(1−α)
0

w)(t); for a.e. t ∈ I.

Furthermore, if w ∈ Cγ(I) and I
1−β(1−α)
0

w ∈ C1
γ
(I), then

(D
α,β

0
Iα0 w)(t) = w(t); for a.e. t ∈ I.

4. If D
γ

0
w exists and in L1(I), then

(Iα0 D
α,β

0
w)(t) = (I

γ

0
D

γ

0
w)(t) = w(t) −

I
1−γ

0
(0+)

Γ(γ)
tγ−1; for a.e. t ∈ I.

Corolary 1. Let h ∈ Cγ(I). Then the Cauchy problem

⎧
⎪⎨

⎪⎩

(D
α,β

0
u)(t) = h(t); t ∈ I,

(I
1−γ

0
u)(t)|t=0 = φ,

has a unique solution u ∈ L1(I) given by

u(t) =
φ

Γ(γ)
tγ−1 + (Iα0 h)(t).



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Hilfer and Hadamard functional random fractional differential . . . 23

From the above corollary, we conclude the following lemma.

Lemma 2.1. Let F : I× R×Ω → P(R) be such that SF◦u(w) ⊂ Cγ for any u ∈ Cγ. Then problem

(1.1) is equivalent to the problem of the solutions of the integral equation

u(t, w) =
φ(w)

Γ(γ)
tγ−1 + (Iα0 v)(t, w),

where v ∈ SF◦u(w).

Now, we consider the Ulam stability for the problem (1.1). Let ϵ > 0 and Φ : I × Ω → [0, ∞)

be a continuous function. We consider the following inequalities

Hd((D
α,β

0
u)(t, w), F(t, u(t, w), w)) ≤ ϵ; t ∈ I, w ∈ Ω. (2.2)

Hd((D
α,β

0
u)(t, w), F(t, u(t, w), w)) ≤ Φ(t, w); t ∈ I, w ∈ Ω. (2.3)

Hd((D
α,β

0
u)(t, w), F(t, u(t, w), w)) ≤ ϵΦ(t, w); t ∈ I, w ∈ Ω. (2.4)

Definition 10. [7, 33] The problem (1.1) is Ulam-Hyers stable if there exists a real number cF > 0

such that for each ϵ > 0 and for each random solution u : Ω → Cγ of the inequality (2.2) there

exists a random solution v : Ω → Cγ of (1.1) with

|u(t, w) − v(t, w)| ≤ ϵcF; t ∈ I, w ∈ Ω.

Definition 11. [7, 33] The problem (1.1) is generalized Ulam-Hyers stable if there exists cF :

C([0, ∞), [0, ∞)) with cF(0) = 0 such that for each ϵ > 0 and for each random solution u : Ω → Cγ
of the inequality (2.2) there exists a random solution v : Ω → Cγ of (1.1) with

|u(t, w) − v(t, w)| ≤ cF(ϵ); t ∈ I, w ∈ Ω.

Definition 12. [7, 33] The problem (1.1) is Ulam-Hyers-Rassias stable with respect to Φ if there

exists a real number cF,Φ > 0 such that for each ϵ > 0 and for each random solution u : Ω → Cγ
of the inequality (2.4) there exists a random solution v : Ω → Cγ of (1.1) with

|u(t, w) − v(t, w)| ≤ ϵcF,ΦΦ(t, w); t ∈ I, w ∈ Ω.

Definition 13. [7, 33] The problem (1.1) is generalized Ulam-Hyers-Rassias stable with respect to

Φ if there exists a real number cF,Φ > 0 such that for each random solution u : Ω → Cγ of the

inequality (2.3), there exists a random solution v : Ω → Cγ of (1.1) with

|u(t, w) − v(t, w)| ≤ cF,ΦΦ(t, w); t ∈ I, w ∈ Ω.

Remark 1. It is clear that

(i) Definition 10 ⇒ Definition 11,



24 S. Abbas, M. Benchohra J.-e. Lazreg, G. M. N’Guérékata CUBO
19, 1 (2017)

(ii) Definition 12 ⇒ Definition 13,

(iii) Definition 12 for Φ(., .) = 1 ⇒ Definition 10.

One can have similar remarks for the inequalities (2.2) and (2.4).

In the sequel, we employ the following random multi-valued fixed point theorems:

Theorem 2.1. [14]] Let (Ω, A) be a complete σ-finite measure space, X be a separable Banach

space, M(Ω, X) be the space of all measurable X-valued functions defined on Ω, and let N : Ω×X →

Pcp,cv(X) be a continuous and condensing multi-valued random operator. If the set {u ∈ M(Ω, X) :

λu ∈ N(w)u} is bounded for each w ∈ Ω and all λ > 1, then N(w) has a random fixed point.

Theorem 2.2. [28] Let (Ω, A) be a complete σ-finite measure space, E a separable Banach space,

and let N : Ω× E → Pcl(E) be a random multi-valued contraction. Then N(w) has a random fixed

point.

We recall an integral inequality which based on an iteration argument.

Lemma 2.2. [40] Suppose β > 0, a(t) is a nonnegative function locally integrable on 0 ≤ t < T

(some T ≤ +∞) and g(t) is a nonnegative, nondecreasing continuous function defined on 0 ≤ t <

T, g(t) ≤ M (constant), and suppose u(t) is nonnegative and locally integrable on 0 ≤ t < T with

u(t) ≤ a(t) + g(t)

∫t

0

(t − s)β−1u(s)ds

on this interval. Then

u(t) ≤ a(t) +

∫
t

0

%
∞∑

n=1

(g(t)Γ(β))n

Γ(nβ)
(t − s)nβ−1a(s)

&

ds, 0 ≤ t < T.

From the above lemma, we concluded with the following lemma.

Lemma 2.3. Suppose β > 0, a(t, w) is a nonnegative function locally integrable on [0, T) × Ω

(some T ≤ +∞) and g(t, w is a nonnegative, nondecreasing continuous function with respect to

t defined on [0, T) × Ω, g(t, w) ≤ M (constant), and suppose u(t, w) is nonnegative and locally

integrable with respect to t on [0, T) × Ω with

u(t, w) ≤ a(t, w) + g(t, w)

∫t

0

(t − s)β−1u(s, w)ds

on [0, T) × Ω. Then

u(t, w) ≤ a(t, w) +

∫t

0

%
∞∑

n=1

(g(t, w)Γ(β))n

Γ(nβ)
(t − s)nβ−1a(s, w)

&

ds, (t, w) ∈ [0, T) × Ω.



CUBO
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Hilfer and Hadamard functional random fractional differential . . . 25

3 Hilfer random fractional differential inclusions

In this section, we are concerned with the existence and the Ulam-Hyers-Rassias stability for

problem (1.1). Let us start by defining what we mean by a random solution of the problem (1.1).

Definition 14. By a random solution of the problem (1.1) we mean a measurable function u : Ω →

Cγ that satisfies the condition (I
1−γ

0
u)(0+, w) = φ(w), and the equation (D

α,β

0
u)(t, w) = v(t, w)

on I × Ω, where v ∈ SF◦u(w).

3.1 The convex case

We present now some existence and Ulam stabilities results for the problem (1.1) with convex

valued right hand side.

The following hypotheses will be used in the sequel.

(H1) The multifunction F : I × R × Ω → Pcp,cv(R) is random Carathéodory on I × R × Ω,

(H2) There exists a measurable and bounded function l : Ω → L
∞(I, [0, ∞)) satisfying for each

w ∈ Ω,

Hd(F(t, u, w), F(t, u, w)) ≤ t
1−γl(t, w)|u − u|; for every t ∈ I and u, u ∈ R.

and

d(0, F(t, 0, w)) ≤ t1−γl(t, w); for t ∈ I,

(H3) There exists λΦ > 0 such that for each t ∈ I, and w ∈ Ω, we have

∫
t

0

%
∞∑

n=1

(l∗)n

Γ(nα)
(t − s)nα−1Φ(s, w)

&

ds ≤ λΦΦ(t, w).

Remark 2. For each u : Ω → C, the set SF,u(w) is nonempty since by (H1), F has a measurable

selection (see [13], Theorem III.6).

Remark 3. The hypothesis (H2) implies that, for every t ∈ I, u ∈ R and w ∈ Ω, we get

Hd(F(t, u, w), F(t, 0, w)) ≤ l(t, w)|u|,

and
Hd(0, F(t, u, w)) ≤ Hd(0, F(t, 0, w)) + Hd(F(t, u, w), F(t, 0, w))

≤ l(t, w)(1 + |u|).

Set

l∗ = sup
w∈Ω

∥l(w)∥L∞ and φ
∗ = sup

w∈Ω

|φ(w)|.



26 S. Abbas, M. Benchohra J.-e. Lazreg, G. M. N’Guérékata CUBO
19, 1 (2017)

Theorem 3.1. Assume that the hypotheses (H1) and (H2) hold. Then, the problem (1.1) has a

random solution defined on I × Ω.

Proof. Define a multivalued operator N : Ω × Cγ → P(Cγ) by:

(N(w)u)(t) =

{

h : Ω → Cγ : h(t, w) =
φ(w)

Γ(γ)
tγ−1 + (Iα0 v)(t, w); t ∈ I, v ∈ SF◦u(w)

}

. (3.1)

The map φ is measurable for all w ∈ Ω. Again, as the indefinite integral is continuous on I, for

each v ∈ SF◦u(w), then N(w) defines a multivalued mapping N : Ω × Cγ → P(Cγ). Thus u is

a random solution for the problem (1.1) if and only if u ∈ N(w)u. We shall show that the mul-

tivalued operator N satisfies all conditions of Theorem 2.1. The proof will be given in several steps.

Step 1. N(w) is a multi-valued random operator on C.

Since F(t, u, w) is random Carathéodory, the map w → F(ty, u, w) is measurable in view of Def-

inition 4. Similarly, the product (t − s)α−1v(s, w) of a continuous function and a measurable

multifunction is again measurable for each v ∈ SF◦u(w). Further, the integral is a limit of a finite

sum of measurable functions, therefore, the map

w *→
φ(w)

Γ(γ)
tγ−1 +

∫
t

0

(t − s)α−1

Γ(α)
v(t, w)ds,

is measurable. As a result, N(w) is a multi-valued random operator on Cγ.

Step 2. N(w)u ∈ Pcv(Cγ) for each u ∈ Cγ.

Indeed, if h1, h2 belong to N(w)u, then there exist v1, v2 ∈ SF◦u(w) such that for each t ∈ I and

w ∈ Ω, we have

hi(t, w) =
φ(w)

Γ(γ)
tγ−1 + (Iα0 vi)(t, w); i = 1, 2.

Let 0 ≤ d ≤ 1. Then, for each t ∈ I and w ∈ Ω, we get

(dh1 + (1 − d)h2)(t, w) =
φ(w)

Γ(γ)
tγ−1 + (Iα0 [dv1 + (1 − d)v2])(t, w).

Since SF◦u(w) is convex (because F has convex values), we get

dh1 + (1 − d)h2 ∈ N(u).

Step 3. N(w) is continuous and N(w)u ∈ Pcp(Cγ) for each u ∈ Cγ.

The proof of this step will be given in several claims.

Claim 1: N(w) is continuous.

Let {un} be a sequence such that un → u in Cγ. Then from (H2), for each t ∈ I and w ∈ Ω, we



CUBO
19, 1 (2017)

Hilfer and Hadamard functional random fractional differential . . . 27

have

Hd(F(t, un(t, w), w), F(t, u(t, w), w))

≤ t1−γl(t, w)|un(t, w) − u(t, w)|

≤ l∗∥un − u∥C → 0 as n → ∞.

Thus, we obtain

Hd(F(t, un(t, w), w), F(t, u(t, w), w)) → 0 as n → ∞.

Claim 2: N(w) maps bounded sets into bounded sets in Cγ.

Let Bη∗ = {u ∈ Cγ : ∥u∥C ≤ η
∗} be bounded set in Cγ, and u ∈ Bη∗ . Then for each h ∈ N(w)u,

there exists v ∈ SF◦u(w) such that

h(t, w) =
φ(w)

Γ(γ)
tγ−1 + (Iα

0
v)(t, w).

By (H2), for each t ∈ I and w ∈ Ω, we obtain

|t1−γh(t, w)| ≤
|φ(w)|

Γ(γ)
+ T1−γ

∫t

0

(t − s)α−1

Γ(α)
|v(s, w)|ds

≤
|φ(w)|

Γ(γ)
+ T1−γ

∫t

0

(t − s)α−1

Γ(α)
|s1−γl(s, w)(1 + v(s, w))|ds

≤
φ∗

Γ(γ)
+ l∗T1−γ

∫t

0

(t − s)α−1

Γ(α)
(T1−γ + ∥v(s, w)∥C)ds

≤
φ∗

Γ(γ)
+

l∗T1+α−γ

Γ(1 + α)
(T1−γ + η∗) := ℓ.

Claim 3: N(w) maps bounded sets into equicontinuous sets in Cγ.

Let t1), t2 ∈ I, t1 < t2, and let Bη∗ be a bounded set of Cγ as in claim 2, and let u ∈ Bη∗ and

h ∈ N(w)u. Then, there exists v ∈ SF◦u(w) such that for each w ∈ Ω, we get

|t
1−γ

2
h(t2, w) − t

1−γ

1
h(t1, w)| ≤

l∗T1−γ+α

Γ(1 + α)
(t2 − t1)

α

+
l∗(T1−γ + η∗)

Γ(α)

∫t1

0

|t
1−γ

2
(t2 − s)

α−1 − t
1−γ

1
(t1 − s)

α−1ds.

As t1 → t2, the right-hand side of the above inequality tends to zero. As a consequence of claims

1 to 3, together with the Arzela-Ascoli theorem, we can conclude that N(w) is continuous and

completely continuous multi-valued random operator.

Step 4: The set E := {u ∈ Cγ : λu ∈ N(w)u} is bounded for some λ > 1.

Let u ∈ Cγ be arbitrary and let w ∈ Ω be fixed such that λu ∈ N(w)u for all λ > 1. Then, there

exists v ∈ SF◦u(w) such that for each t ∈ I, we have

u(t, w) =
φ(w)

λΓ(γ)
tγ−1 + λ−1(Iα

0
v)(t, w).



28 S. Abbas, M. Benchohra J.-e. Lazreg, G. M. N’Guérékata CUBO
19, 1 (2017)

This implies by (H2) that, for each t ∈ I, we get

|t1−γu(t, w)| ≤
φ∗

Γ(γ)
+

∫
t

0

(t − s)α−1

Γ(α)
l(s, w)(T1−γ + |s1−γv(s, w)|)ds

≤
φ∗

Γ(γ)
+

l∗T1−γ+α

Γ(1 + α)
+ l∗

∫
t

0

(t − s)α−1

Γ(α)
|s1−γv(s, w)|ds.

From Lemma 2.3, for each (t, w) ∈ [0, T) × Ω, we have

|t1−γu(t, w)| ≤

'
φ∗

Γ(γ)
+

l∗T1−γ+α

Γ(1 + α)

(%

1 +

∫
t

0

%
∞∑

n=1

(l∗)n

Γ(nα)
(t − s)nα−1

&

ds

&

≤

'
φ∗

Γ(γ)
+

l∗T1−γ+α

Γ(1 + α)

(%

1 +
∞∑

n=1

Tnα

Γ(1 + nα)

&

:= M.

Thus, for all t ∈ I and w ∈ Ω, we obtain ∥u∥∞ ≤ M.

As a consequence of steps 1 to 4, together with the Theorem 2.1, N has a random fixed point

u which is a random solution to problem (1.1).

Now, we are concerned with the generalized Ulam-Hyers-Rassias stability of our problem (1.1).

Theorem 3.2. Assume that the hypotheses (H1)−(H3) hold. Then the problem (1.1) is generalized

Ulam-Hyers-Rassias stable.

Proof. Let u be a random solution of the inequality (2.3), and let us assume that v is a

random solution of problem (1.1). Thus, we have

v(t, w) =
φ(w)

Γ(γ)
tγ−1 +

∫
t

0

(t − s)α−1
fv(s, w)

Γ(α)
ds,

where fv ∈ SF◦v(w). From the inequality (2.3) for each t ∈ I, and w ∈ Ω, we have

)
)
)
)u(t, w) −

φ(w)

Γ(γ)
tγ−1 −

∫
t

0

(t − s)α−1
f(s, w)

Γ(α)
ds

)
)
)
) ≤ (I

α
0
Φ)(t, w),

where f ∈ SF◦u(w). From hypotheses (H2) and (H3), for each t ∈ I, and w ∈ Ω, we get

|u(t, w) − v(t, w)| ≤

)
)
)
)u(t, w) −

φ(w)

Γ(γ)
tγ−1 −

∫
t

0

(t − s)α−1
f(s, w)

Γ(α)
ds

)
)
)
)

+

∫
t

0

(t − s)α−1
|f(s, w) − fv(s, w)|

Γ(α)
ds

≤ (Iα
0
Φ)(t, w) +

l∗T1−γ

Γ(α)

∫
t

0

(t − s)α−1|u(s, w) − v(s, w)|ds.



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Hilfer and Hadamard functional random fractional differential . . . 29

From Lemma 2.3, we obtain

∥u(t, w) − v(t, w)∥ ≤
λφ
l∗

%

Φ(t, w) +

∫
t

0

%
∞∑

n=1

(l∗)n

Γ(nα)
(t − s)nα−1Φ(s, w)

&

ds

&

≤
λφ
l∗

[1 + λΦ]Φ(t, w)

:= cF,ΦΦ(t, w).

Finally, the problem (1.1) is generalized Ulam-Hyers-Rassias stable.

3.2 The Non-convex case

We present now some existence and Ulam stabilities results for the problem (1.1) with non-convex

valued right hand side.

The following hypotheses will be used in the sequel.

(H01) The multifunction F : I × R × Ω → Pcp(R) is random Carathéodory on I × R × Ω,

(H02) There exists a measurable and bounded function l : Ω → L
∞(I, [0, ∞)) satisfying for each

w ∈ Ω,

Hd(F(t, u, w), F(t, u, w)) ≤ t
1−γl(t, w)|u − u|; for every t ∈ I and u, u ∈ R.

Set

l∗ = sup
w∈Ω

∥l(w)∥L∞ .

Now, we shall prove the following theorem concerning the existence of random solutions of

problem (1.1).

Theorem 3.3. Assume that the hypotheses (H01) and (H02) hold. If

l∗T1+α−γ

Γ(1 + α)
< 1, (3.2)

Then the problem (1.1) has at least one random solution defined on I × Ω.

Proof. LetN : Ω × Cγ → P(Cγ) be the multivalued operator defined in (3.1). We know that

N(w) is a multi-valued random operator on Cγ. We shall show that the multivalued operator N

satisfies all conditions of Theorem 2.2. The proof will be given in two steps.



30 S. Abbas, M. Benchohra J.-e. Lazreg, G. M. N’Guérékata CUBO
19, 1 (2017)

Step 1. N(w)u ∈ Pcl(Cγ) for each u ∈ Cγ.

let {un}n≥0 ∈ N(w)u such that un −→ ũ in Cγ. Then, ũ ∈ Cγ and there exists fn(·, ·, ·) ∈ SF◦u(w)

be such that, for each t ∈ I and w ∈ Ω, we have

un(t, w) =
φ(w)

Γ(γ)
tγ−1 + (Iα

0
fn)(t, w).

Using the fact that F has compact values and from (H01), we may pass to a subsequence if necessary

to get that fn(·, ·, ·) converges to f in L
1(I), and hence f ∈ SF◦u(w). Then, for each t ∈ I and w ∈ Ω,

we get

un(t, w) −→ ũ(t, w) =
φ(w)

Γ(γ)
tγ−1 + (Iα

0
f)(t, w).

So, ũ ∈ N(w)u.

Step 2. There exists 0 ≤ λ < 1 such that, for each w ∈ Ω,

Hd(N(w)u, N(w)u) ≤ λ∥u − u∥C for each u, u ∈ Cγ.

Let u, u ∈ Cγ and h ∈ N(w)u. Then, there exists f(t, w) ∈ F(t, u(t, w), w) such that for each t ∈ I

and w ∈ Ω, we have

h(t, w) =
φ(w)

Γ(γ)
tγ−1 + (Iα

0
f)(t, w).

From (H02) it follows that

Hd(F(t, u(t, w), w), F(t, u(t, w), w)) ≤ t
1−γl(t, w)|u(t, w) − u(t, w)|.

Hence, there exists v ∈ SF◦u such that

|f(t, w) − v(t, w)| ≤ t1−γl(t, w)|u(t, w) − u(t, w)|.

Consider U : I × Ω → P(R) given by

U(t, w) = {v(t, w) ∈ R : |f(t, w) − v(t, w)| ≤ t1−γl(t, w)|u(t, w) − u(t, w)|}.

Since the multivalued operator u(t, w) = U(t, w) ∩ F(t, u(t, w), w) is measurable (see Proposition

III.4 in [13]), there exists a function f(t, w) which is a measurable selection for u. So, f(t, w) ∈

F(t, u(t, w), w), and for each t ∈ I and w ∈ Ω, we get

|f(t, w) − f(t, w)| ≤ t1−γl(t, w)|u(t, w) − u(t, w)|.

Let us define for each t ∈ I and w ∈ Ω,

h(t, w) =
φ(w)

Γ(γ)
tγ−1 + (Iα

0
f)(t, w).



CUBO
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Hilfer and Hadamard functional random fractional differential . . . 31

Then for each t ∈ I and w ∈ Ω, we obtain

|t1−γh(t, w) − t1−γh(t, w)| ≤ t1−γIα0 |f(t, w) − f(t, w)|

≤
T1−γ

Γ(α)

∫t

0

(t − s)α−1l(s, w)|s1−γu(s, w) − s1−γu(s, w)|ds

≤
l∗T1−γ∥u − u∥C

Γ(α)

∫t

0

(t − s)α−1ds.

Hence

∥h − h∥C ≤
l∗T1+α−γ

Γ(1 + α)
∥u − u∥C.

By an analogous relation, obtained by interchanging the roles of u and u, it follows that

Hd(N(w)u, N(w)u) ≤
l∗T1+α−γ

Γ(1 + α)
∥u − u∥C.

So by (3.2), N is random contraction and thus, by Theorem 2.2, N has a random fixed point u

which is a random solution to problem (1.1).

Now, we can show the following generalized Ulam-Hyers-Rassias stability result.

Theorem 3.4. Assume that the hypotheses (H01), (H02), (H3) and the condition (3.2) hold, then

the problem (1.1) is generalized Ulam-Hyers-Rassias stable.

4 Hilfer-Hadamard fractional random differential inclusions

Now, we are concerned with the existence and the Ulam-Hyers-Rassias stability for problem (1.2).

Set C := C([1, T]). Denote the weighted space of continuous functions defined by

Cγ,ln([1, T]) = {w(t) : (ln t)
1−γw(t) ∈ C},

with the norm

∥w∥Cγ,ln := sup
t∈[1,T]

|(ln t)1−rw(t)|.

Let us recall some definitions and properties of Hadamard fractional integration and differen-

tiation. We refer to [18, 26] for a more detailed analysis.

Definition 15. [18, 26] (Hadamard fractional integral). The Hadamard fractional integral of order

q > 0 for a function g ∈ L1([1, T]), is defined as

(HI
q

1
g)(x) =

1

Γ(q)

∫x

1

#
ln

x

s

$q−1 g(s)

s
ds,

provided the integral exists.



32 S. Abbas, M. Benchohra J.-e. Lazreg, G. M. N’Guérékata CUBO
19, 1 (2017)

Example 1. Let 0 < q < 1. Then

HI
q

1
ln t =

1

Γ(2 + q)
(ln t)1+q, for a.e. t ∈ [0, e].

Set

δ = x
d

dx
, q > 0, n = [q] + 1,

and

ACnδ := {u : [1, T] → E : δ
n−1[u(x)] ∈ AC(I)}.

Analogous to the Riemann-Liouville fractional calculus, the Hadamard fractional derivative is

defined in terms of the Hadamard fractional integral in the following way:

Definition 16. [18, 26] (Hadamard fractional derivative). The Hadamard fractional derivative of

order q > 0 applied to the function w ∈ ACn
δ
is defined as

(HD
q

1
w)(x) = δn(HI

n−q

1
w)(x).

In particular, if q ∈ (0, 1], then

(HD
q

1
w)(x) = δ(HI

1−q

1
w)(x).

Example 2. Let 0 < q < 1. Then

HD
q

1
ln t =

1

Γ(2 − q)
(ln t)1−q, for a.e. t ∈ [0, e].

It has been proved (see e.g. Kilbas [[25], Theorem 4.8]) that in the space L1(I, E), the

Hadamard fractional derivative is the left-inverse operator to the Hadamard fractional integral,

i.e.

(HD
q

1
)(HI

q

1
w)(x) = w(x).

From Theorem 2.3 of [26], we have

(HI
q

1
)(HD

q

1
w)(x) = w(x) −

(HI
1−q

1
w)(1)

Γ(q)
(ln x)q−1.

Analogous to the Hadamard fractional calculus, the Caputo-Hadamard fractional derivative

is defined in the following way:

Definition 17. (Caputo-Hadamard fractional derivative). The Caputo-Hadamard fractional deriva-

tive of order q > 0 applied to the function w ∈ ACn
δ
is defined as

(HcD
q

1
w)(x) = (HI

n−q

1
δnw)(x).



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Hilfer and Hadamard functional random fractional differential . . . 33

In particular, if q ∈ (0, 1], then

(HcD
q

1
w)(x) = (HI

1−q

1
δw)(x).

From the Hadamard fractional integral, the Hilfer-Hadamard fractional derivative (introduced

for the first time in [30]) is defined in the following way:

Definition 18. (Hilfer-Hadamard fractional derivative). Let α ∈ (0, 1), β ∈ [0, 1], γ = α + β −

αβ, w ∈ L1(I), and HI
(1−α)(1−β)
1

w ∈ AC1(I). The Hilfer-Hadamard fractional derivative of order

α and type β applied to the function w is defined as

(HD
α,β

1
w)(t) =

#
HI

β(1−α)
1

(HD
γ

1
w)
$
(t)

=
#
HI

β(1−α)
1

δ(HI
1−γ

1
w)
$
(t); for a.e. t ∈ [1, T].

(4.1)

This new fractional derivative (4.1) may be viewed as interpolating the Hadamard fractional

derivative and the Caputo-Hadamard fractional derivative. Indeed for β = 0 this derivative reduces

to the Hadamard fractional derivative and when β = 1, we recover the Caputo-Hadamard fractional

derivative.
HDα,0

1
= HDα

1
, and HDα,1

1
= HcDα

1
.

From Theorem 21 in [31], we concluded the following lemma

Lemma 4.1. Let G : [1, T]×R×Ω → P(R) be such that SG◦u(w) ∈ Cγ,ln([1, T]) for any u(·, w) ∈

Cγ,ln([1, T]). Then problem (1.2) is equivalent to the following volterra integral equation

u(t, w) =
φ0(w)

Γ(γ)
(ln t)γ−1 + (HIα

1
g(·, w))(t); w ∈ Ω,

where g ∈ SG◦u(w).

Definition 19. By a random solution of the problem (1.2) we mean a measurable function u ∈

Cγ,ln that satisfies the condition (
HI

1−γ

1
u)(1+, w) = φ0(w), and the equation (

HD
α,β

1
u)(t, w) =

g(t, w) on [1, T] × Ω, where g ∈ SG◦u(w).

Now we give (without proof) existence and Ulam slability results for problem (1.2). The

following hypotheses will be used in the sequel.

(H′
1
) The multifunction G : [1, T] × R × Ω → Pcp,cv(R) is random Carathéodory,

(H′
2
) There exists a measurable and bounded function l : Ω → L∞([1, T], [0, ∞)) satisfying for each

w ∈ Ω,

Hd(G(t, u, w), G(t, u, w)) ≤ t
1−γl(t, w)|u − u|; for every t ∈ [1, T] and u, u ∈ R.

and

d(0, G(t, 0, w)) ≤ (ln t)1−γl(t, w); for t ∈ [1, T],



34 S. Abbas, M. Benchohra J.-e. Lazreg, G. M. N’Guérékata CUBO
19, 1 (2017)

(H′
3
) There exists λΦ > 0 such that for each t ∈ [1, T], and w ∈ Ω, we have

∫t

1

%
∞∑

n=1

(l∗)n

Γ(nα)

!

ln
t

s

"nα−1
Φ(s, w)

&
ds

s
≤ λΦΦ(t, w).

Theorem 4.1. Assume that the hypotheses (H′
1
) and (H′

2
) hold. Then, the problem (1.1) has a

random solution defined on [1, T] × Ω. Moreover; if the hypothesis (H′
3
) holds, then the problem

(1.1) is generalized Ulam-Hyers-Rassias stable.

Finally, we give (without proof) existence and Ulam stability results for problem (1.2) with non-

convex valued right hand side. The following hypotheses will be used in the sequel.

(H′
01
) The multifunction G : [1, T] × R × Ω → Pcp(R) is random Carathéeodory on [1, T] × R × Ω,

(H′
02
) There exists a measurable and bounded function p : Ω → L∞([1, T], [0, ∞)) satisfying for

each w ∈ Ω,

Hd(G(t, u, w), G(t, u, w)) ≤ (ln t)
1−γp(t, w)|u − u|; for every t ∈ [1, T] and u, u ∈ R.

Set

l∗ = sup
w∈Ω

∥l(w)∥L∞ .

Theorem 4.2. Assume that the hypotheses (H′
01
) and (H′

02
) hold. If

l∗(ln T)1+α−γ

Γ(1 + α)
< 1, (4.2)

then the problem (1.2) has at least one random solution defined on [1, T] × Ω. Moreover, if the

hypothesis (H′
3
) holds, then the problem (1.2) is generalized Ulam-Hyers-Rassias stable.

5 Examples

Let Ω = (−∞, 0) be equipped with the usual σ-algebra consisting of Lebesgue measurable subsets

of (−∞, 0).

Example 1. Consider Hilfer fractional differential inclusion of the form
{
(D

1

2
,
1

2

0
u)(t, w) ∈ F(t, u(t, w), w); t ∈ [0, 1],

(I
1

4

0
u)(0, w) = 1,

w ∈ Ω, (5.1)

where

F(t, u(t, w), w) = {v : Ω → C([0, 1], R) : |f1(t, u(t, w), w)| ≤ |v(w)| ≤ |f2(t, u(t, w), w)|};



CUBO
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Hilfer and Hadamard functional random fractional differential . . . 35

t ∈ [0, 1], w ∈ Ω, with f1, f2 : [0, 1] × R × Ω → R, such that

f1(t, u(t, w), w) =
t2u

(1 + w2 + |u|)e10+t
,

and

f2(t, u(t, w), w) =
t2u

(1 + w2)e10+t
.

Set α = β = 1
2
, then γ = 3

4
. We assume that F is closed and convex valued. A simple computation

shows that conditions of Theorem 3.1 are satisfied. Hence, the problem (5.1) has at least one

random solution defined on [0, 1]. Also, the hypothesis (H3) is satisfied with

Φ(t, w) =
e3

1 + w2
, and λΦ =

∞∑

n=1

e−10n

Γ(1 + nα)
.

Φ(t, w) =
e3

1 + w2
, and λΦ =

1

Γ(1 + α)
.

Indeed, for each t ∈ [0, 1], and w ∈ Ω, we get

(Iα
0
Φ)(t, w) ≤

e3

(1 + w2)

∞∑

n=1

e−10n

Γ(1 + nα)
.

= λΦΦ(t, w).

Consequently, Theorem 3.2 implies that the problem (5.1) is generalized Ulam-Hyers-Rassias stable.

Example 2. Consider Hilfer fractional differential inclusion of the form

{
(D

1

2
,
1

2

0
u)(t, w) ∈ F(t, u(t, w), w); t ∈ [0, 1],

(I
1

4

0
u)(0, w) = 1,

w ∈ Ω, (5.2)

where

F(t, u(t, w), w) =
t2

(1 + w2 + |u|)e10+t
[u − 1, u]; t ∈ [0, 1], w ∈ Ω.

Set α = β = 1
2
, then γ = 3

4
. We assume that F is closed valued. A simple computation shows

that conditions of Theorem 3.3 are satisfied. Hence, the problem (5.2) has at least one random

solution defined on [0, 1]. Also, Theorem 3.4 implies that the problem (5.2) is generalized Ulam-

Hyers-Rassias stable.

References

[1] S. Abbas, W. A. Albarakati, M. Benchohra and J. Henderson, Existence and Ulam stabili-

ties for Hadamard fractional integral equations with random effects, Electron. J. Differential

Equations 2016 (2016), No. 25, pp 1-12.



36 S. Abbas, M. Benchohra J.-e. Lazreg, G. M. N’Guérékata CUBO
19, 1 (2017)

[2] S. Abbas, W. Albarakati, M. Benchohra and G. M. N’Guérékata, Existence and Ulam stabil-

ities for Hadamard fractional integral equations in Fréchet spaces, J. Frac. Calc. Appl. 7 (2)

(2016), 1-12.

[3] S. Abbas, W.A. Albarakati, M. Benchohra and S. Sivasundaram, Dynamics and stability of

Fredholm type fractional order Hadamard integral equations, J. Nonlinear Stud. 22 (4) (2015),

673-686.

[4] S. Abbas and M. Benchohra, Uniqueness and Ulam stabilities results for partial fractional

differential equations with not instantaneous impulses, Appl. Math. Comput. 257 (2015),

190-198.

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