CUBO A Mathematical Journal
Vol.19, No¯ 03, (31–42). October 2017

On the solution set of a fractional integro-differential
inclusion involving Caputo-Katugampola derivative

Aurelian Cernea

Faculty of Mathematics and Computer Science

University of Bucharest

Academiei 14, 010014 Bucharest, Romania

Academy of Romanian Scientists

Splaiul Independenţei 54, 050094 Bucharest, Romania

acernea@fmi.unibuc.ro

ABSTRACT

We study an initial value problem associated to a fractional integro-differential inclusion

defined by Caputo-Katugampola derivative and by a set-valued map with nonconvex

values. We prove the arcwise connectedness of the solution set and that the set of

selections corresponding to the solutions of the problem considered is a retract of the

space of integrable functions on a given interval.

RESUMEN

Estudiamos un problema de valor inicial asociado a la inclusión ı́ntegro-diferencial frac-

cionaria definida por la derivada de Caputo-Katugampola y por una aplicación multi-

valuada con valores no-convexos. Demostramos la arco-conexidad del conjunto solución

y que el conjunto de selecciones correspondientes a las soluciones del problema consid-

erado es un retracto del espacio de funciones integrables en un intervalo dado.

Keywords and Phrases: Differential inclusion, fractional derivative, initial value problem.

2010 AMS Mathematics Subject Classification: 34A60, 26A33, 34B15.



32 Aurelian Cernea CUBO
19, 3 (2017)

1 Introduction

In the last years one may see a strong development of the theory of differential equations and in-

clusions of fractional order ([2,5,7-9] etc.). The main reason is that fractional differential equations

are very useful tools in order to model many physical phenomena.

Recently, a generalized Caputo-Katugampola fractional derivative was proposed in [6] by

Katugampola and afterwards he provided the existence of solutions for fractional differential equa-

tions defined by this derivative. This Caputo-Katugampola fractional derivative extends the well

known Caputo and Caputo-Hadamard fractional derivatives. Also, in some recent papers [1,12],

several qualitative properties of solutions of fractional differential equations defined by Caputo-

Katugampola derivative were obtained.

Differential inclusion is a generalization of the notion of ordinary differential equation that

provides powerful tools for various fields of mathematical analysis. At the same time there are

dynamics of economical, social and biological systems that are set-valued. Therefore, differential

inclusions serve as natural models in such systems. An outstanding example comes from control

theory. Namely, the equivalence between a control system and the corresponding differential inclu-

sion, established by Filippov, allowed to obtained necessary and sufficient conditions of optimality

using set-valued techniques.

In the present paper we study the following Cauchy problem

Dα,ρc x(t) ∈ F(t, x(t), V(x)(t)) a.e. ([0, T]), x(0) = x0, (1.1)

where α ∈ (0, 1], ρ > 0, D
α,ρ
c is the Caputo-Katugampola fractional derivative, F : [0, T]×R×R →

P(R) is a set-valued map, V : C([0, T], R) → C([0, T], R) is a nonlinear Volterra integral operator

defined by V(x)(t) =
∫t
0
k(t, s, x(s))ds with k(., ., .) : [0, T] × R × R → R a given function and

x0 ∈ R.

Our goal is twofold. On one hand, we prove the arcwise connectedness of the solution set of

problem (1.1) when the set-valued map is Lipschitz in the second and third variable. On the other

hand, under such type of hypotheses on the set-valued map we establish a more general topological

property of the solution set of problem (1.1). Namely, we prove that the set of selections of the

set-valued map F that correspond to the solutions of problem (1.1) is a retract of L1([0, T], R).

Both results are essentially based on Bressan and Colombo results ([3]) concerning the existence

of continuous selections of lower semicontinuous multifunctions with decomposable values.

In the theory of ordinary differential equations Kneser’s theorem states that the solution set

of an ordinary differential equation is connected, i.e., it cannot be represented as a union of two

closed sets without common points. In the case of differential inclusions, although the solution set

multifunction is not, in general, convex valued we are able to prove its arcwise connectedness and

therefore, our result may be regarded as an extension of the classical theorem of Kneser.

We note that similar results for fractional differential inclusions defined by classical Caputo

fractional derivative are obtained in our previous paper [4]. The results in [4] extend to the case of



CUBO
19, 3 (2017)

On the solution set of a fractional integro-differential inclusion . . . 33

fractional differential inclusions the results in [10,11] obtained for ordinary differential inclusions.

The present paper generalizes and unifies all these results in the case of the more general problem

(1.1).

The paper is organized as follows: in Section 2 we present the notations, definitions and the

preliminary results to be used in the sequel and in Section 3 we prove our main results.

2 Preliminaries

Let T > 0, I := [0, T] and denote by L(I) the σ-algebra of all Lebesgue measurable subsets of I. Let

X be a real separable Banach space with the norm |.|. Denote by P(X) the family of all nonempty

subsets of X and by B(X) the family of all Borel subsets of X. If A ⊂ I then χA(.) : I → {0, 1}

denotes the characteristic function of A. For any subset A ⊂ X we denote by cl(A) the closure of

A.

The distance between a point x ∈ X and a subset A ⊂ X is defined as usual by d(x, A) =

inf{|x−a|; a ∈ A}. We recall that Pompeiu-Hausdorff distance between the closed subsets A, B ⊂ X

is defined by dH(A, B) = max{d
∗
(A, B), d

∗
(B, A)}, d

∗
(A, B) = sup{d(a, B); a ∈ A}.

As usual, we denote by C(I, X) the Banach space of all continuous functions x : I → X endowed

with the norm |x|C = supt∈I|x(t)| and by L
1(I, X) the Banach space of all (Bochner) integrable

functions x : I → X endowed with the norm |x|1 =
∫T
0
|x(t)|dt.

We recall first several preliminary results we shall use in the sequel.

A subset D ⊂ L1(I, X) is said to be decomposable if for any u, v ∈ D and any subset A ∈ L(I)

one has uχA + vχB ∈ D, where B = I\A.

We denote by D(I, X) the family of all decomposable closed subsets of L1(I, X).

Next (S, d) is a separable metric space; we recall that a multifunction G : S → P(X) is said

to be lower semicontinuous (l.s.c.) if for any closed subset C ⊂ X, the subset {s ∈ S; G(s) ⊂ C} is

closed. The next lemmas may be found in [3].

Lemma 2.1. If F : I → D(I, X) is a lower semicontinuous multifunction with closed nonempty

and decomposable values then there exists f : I → L1(I, X) a continuous selection from F.

Lemma 2.2. Let G(., .) : I×S → P(X) be a closed-valued L(I)⊗B(S)-measurable multifunction

such that G(t, .) is l.s.c. for any t ∈ I.

Then the multifunction G∗(.) : S → D(I, X) defined by

G∗(s) = {f ∈ L1(I, X); f(t) ∈ G(t, s) a.e. (I)}

is l.s.c. with nonempty closed values if and only if there exists a continuous mapping q(.) : S →

L1(I, X) such that

d(0, G(t, s)) ≤ q(s)(t) a.e. (I), ∀s ∈ S.



34 Aurelian Cernea CUBO
19, 3 (2017)

Lemma 2.3. Let H(.) : S → D(I, X) be a l.s.c. multifunction with closed decomposable

values and let a(.) : S → L1(I, X), b(.) : S → L1(I, R) be continuous such that the multifunction

F(.) : S → D(I, X) defined by

F(s) = cl{f ∈ H(s); |f(t) − a(s)(t)| < b(s)(t) a.e. (I)}

has nonempty values.

Then F(.) has a continuous selection.

Let ρ > 0.

Definition 2.4. ([6]) a) The generalized left-sided fractional integral of order α > 0 of a

Lebesgue integrable function f : [0, ∞) → R is defined by

Iα,ρf(t) =
ρ1−α

Γ(α)

∫t

0

(tρ − sρ)α−1sρ−1f(s)ds, (2.1)

provided the right-hand side is pointwise defined on (0, ∞) and Γ(.) is the (Euler’s) Gamma function

defined by Γ(α) =
∫
∞

0
tα−1e−tdt.

b) The generalized fractional derivative, corresponding to the generalized left-sided fractional

integral in (2.1) of a function f : [0, ∞) → R is defined by

Dα,ρf(t) = (t1−ρ
d

dt
)
n
(In−α,ρ)(t) =

ρα−n+1

Γ(n − α)
(t1−ρ

d

dt
)
n

∫t

0

sρ−1f(s)

(tρ − sρ)α−n+1
ds

if the integral exists and n = [α].

c) The Caputo-Katugampola generalized fractional derivative is defined by

Dα,ρc f(t) = (D
α,ρ[f(s) −

n−1∑

k=0

f(k)(0)

k!
sk])(t)

We note that if ρ = 1, the Caputo-Katugampola fractional derivative becames the well known

Caputo fractional derivative. On the other hand, passing to the limit with ρ → 0+, the above

definition yields the Hadamard fractional derivative.

In what follows ρ > 0 and α ∈ [0, 1]

Lemma 2.5. For a given integrable function h(.) : [0, T] → R, the unique solution of the

initial value problem

Dα,ρc x(t) = h(t) a.e. ([0, T]), x(0) = x0,

is given by

x(t) = x0 +
ρ1−α

Γ(α)

∫t

0

(tρ − sρ)α−1sρ−1h(s)ds



CUBO
19, 3 (2017)

On the solution set of a fractional integro-differential inclusion . . . 35

For the proof of Lemma 2.2, see [6]; namely, Lemma 4.2.

A function x ∈ C(I, R) is called a solution of problem (1.1) if there exists a function f ∈ L1(I, R)

with f(t) ∈ F(t, x(t), V(x)(t)) a.e. (I) such that D
α,ρ
c x(t) = f(t) a.e. (I) and x(0) = x0.

In this case (x(.), f(.)) is called a trajectory-selection pair of problem (1.1).

We shall use the following notations for the solution sets and for the selection sets of problem

(1.1).

S(x0) = {x ∈ C(I, R); x is a solution of(1.1)},

f̃(t) = x0 +
ρ
1−α

Γ(α)

∫t
0
(tρ − sρ)α−1sρ−1f(s)ds,

T (x0) = {f ∈ L
1(I, R); f(t) ∈ F(t, f̃(t), V(f̃)(t)) a.e. I}.

3 The main results

In order to prove our topological properties of the solution set of problem (1.1) we need the following

hypotheses.

Hypothesis 3.1. i) F(., .) : I × R × R → P(R) has nonempty closed values and is L(I) ⊗

B(R × R) measurable.

ii) There exists L(.) ∈ L1(I, (0, ∞)) such that, for almost all t ∈ I, F(t, ., .) is L(t)-Lipschitz in

the sense that

dH(F(t, x1, y1), F(t, x2, y2)) ≤ L(t)(|x1 − x2| + |y1 − y2|) ∀ x1, x2, y1, y2 ∈ R.

iii) There exists p ∈ L1(I, R) such that

dH({0}, F(t, 0, V(0)(t))) ≤ p(t) a.e. I.

iv) k(., ., .) : I × R × R → R is a function such that ∀x ∈ R, (t, s) → k(t, s, x) is measurable.

v) |k(t, s, x) − k(t, s, y)| ≤ L(t)|x − y| a.e. (t, s) ∈ I × I, ∀ x, y ∈ R.

We use next the following notations

M(t) := L(t)(1 +

∫t

0

L(u)du), t ∈ I, Iα,ρM := sup
t∈I

|Iα,ρM(t)|.

Theorem 3.2. Assume that Hypothesis 3.1 is satisfied and Iα,ρM < 1.

Then for any ξ0 ∈ R the solution set S(ξ0) is arcwise connected in the space C(I, R).

Proof. Let ξ0 ∈ R and x0, x1 ∈ S(ξ0). Therefore there exist f0, f1 ∈ L
1(I, R) such that

x0(t) = ξ0 +
ρ
1−α

Γ(α)

∫t
0
(tρ −uρ)α−1uρ−1f0(u)du and x1(t) = ξ0 +

ρ
1−α

Γ(α)

∫t
0
(tρ −uρ)α−1uρ−1f1(u)du,

t ∈ I.



36 Aurelian Cernea CUBO
19, 3 (2017)

For λ ∈ [0, 1] define

x0(λ) = (1 − λ)x0 + λx1 and g
0(λ) = (1 − λ)f0 + λf1

Obviously, the mapping λ 7→ x0(λ) is continuous from [0, 1] into C(I, R) and since |g0(λ) −

g0(λ0)|1 = |λ − λ0|.|f0 − f1|1 it follows that λ 7→ g
0(λ) is continuous from [0, 1] into L1(I, R).

Define the set-valued maps

Ψ1(λ) = {v ∈ L1(I, R); v(t) ∈ F(t, x0(λ)(t), V(x0(λ))(t)) a.e. I},

Φ1(λ) =






{f0} if λ = 0,

Ψ1(λ) if 0 < λ < 1,

{f1} if λ = 1

and note that Φ1 : [0, 1] → D(I, R) is lower semicontinuous. Indeed, let C ⊂ L1(I, R) be a closed

subset, let {λm}m∈N converges to some λ0 and Φ
1(λm) ⊂ C for any m ∈ N. Let v0 ∈ Φ

1(λ0).

Since the multifunction t 7→ F(t, x0(λm)(t), V(x
0(λm))(t)) is measurable, it admits a measurable

selection vm(.) such that

|vm(t) − v0(t)| = d(v0(t), F(t, x
0(λm)(t), V(x

0(λm))(t)) a.e. I.

Taking into account Hypothesis 3.1 one may write

|vm(t) − v0(t)| ≤ dH(F(t, x
0(λm)(t), V(x

0(λm))(t)), F(t, x
0(λ0)(t),

V(x0(λ0))(t)) ≤ L(t)[|x
0(λm)(t) − x

0(λ0)(t)| +

∫t

0

L(s)|x0(λm)(s)−

x0(λ0)(s)|ds] = L(t)|λm − λ0|[|x0(t) − x1(t)| +

∫t

0

L(s)|x0(s) − x1(s)|ds]

hence

|vm − v0|1 ≤ |λm − λ0|

∫T

0

L(t)[|x0(t) − x1(t)| +

∫t

0

L(s)|x0(s) − x1(s)|ds]dt

which implies that the sequence vm converges to v0 in L
1(I, R). Since C is closed we infer that

v0 ∈ C; hence Φ
1(λ0) ⊂ C and Φ

1(.) is lower semicontinuous.

Next we use the following notation

p0(λ)(t) = |g
0
(λ)(t)| + p(t) + L(t)(|x0(λ)(t)| +

∫t

0

L(s)|x0(λ)(s)|ds),

t ∈ I, λ ∈ [0, 1].

Since
|p0(λ)(t) − p0(λ0)(t)| ≤ |λ − λ0|[|f1(t) − f0(t)|+

L(t)(|x0(t) − x1(t)| +
∫t
0
L(s)|x0(s) − x1(s)|ds)]



CUBO
19, 3 (2017)

On the solution set of a fractional integro-differential inclusion . . . 37

we deduce that p0(.) is continuous from [0, 1] to L
1(I, R).

At the same time, from Hypothesis 3.1 it follows

d(g0(λ)(t), F(t, x0(λ)(t), V(x0(λ))(t)) ≤ p0(λ)(t) a.e. I. (3.1)

Fix δ > 0 and for m ∈ N we set δm =
m+1
m+2

δ.

We shall prove next that there exists a continuous mapping g1 : [0, 1] → L1(I, R) with the

following properties

a) g1(λ)(t) ∈ F(t, x0(λ)(t), V(x0(λ))(t)) a.e. I,

b) g1(0) = f0, g
1(1) = f1,

c) |g1(λ)(t) − g0(λ)(t)| ≤ p0(λ)(t) + δ0
ρ
α
Γ(α+1)

Tρα
a.e. I.

Define

G1(λ) = cl{v ∈ Φ1(λ); |v(t) − g0(λ)(t)| < p0(λ)(t) + δ0
ραΓ(α + 1)

Tρα
, a.e. I}

and, by (3.1), we find that G1(λ) is nonempty for any λ ∈ [0, 1]. Moreover, since the mapping

λ 7→ p0(λ) is continuous, we apply Lemma 2.3 and we obtain the existence of a continuous mapping

g1 : [0, 1] → L1(I, R) such that g1(λ) ∈ G1(λ) ∀λ ∈ [0, 1], hence with properties a)-c).

Define now

x1(λ)(t) = ξ0 +
ρ1−α

Γ(α)

∫t

0

(tρ − uρ)α−1uρ−1g1(λ)(u)du, t ∈ I

and note that, since |x1(λ) − x1(λ0)|C ≤
T

ρα

ραΓ(α+1)
|g1(λ) − g1(λ0)|1, x

1(.) is continuous from [0, 1]

into C(I, R).

Set pm(λ) := (I
α,ρM)m−1( T

ρα

ραΓ(α+1)
|p0(λ)|1 + δm).

We shall prove that for all m ≥ 1 and λ ∈ [0, 1] there exist xm(λ) ∈ C(I, R) and gm(λ) ∈

L1(I, R) with the following properties

i) gm(0) = f0, g
m(1) = f1,

ii) gm(λ)(t) ∈ F(t, xm−1(λ)(t), V(xm−1(λ))(t)) a.e. I,

iii) gm : [0, 1] → L1(I, R)is continuous,

iv) |g1(λ)(t) − g0(λ)(t)| ≤ p0(λ)(t) + δ0
ρ
α
Γ(α+1)

Tρα
,

v) |gm(λ)(t) − gm−1(λ)(t)| ≤ M(t)pm(λ), m ≥ 2,

vi) xm(λ)(t) = ξ0 +
ρ
1−α

Γ(α)

∫t
0
(tρ − uρ)α−1uρ−1gm(λ)(u)du, t ∈ I.

Assume that we have already constructed gm(.) and xm(.) with i)-vi) and define

Ψm+1(λ) = {v ∈ L1(I, R); v(t) ∈ F(t, xm(λ)(t), V(xm(λ))(t)) a.e. I},



38 Aurelian Cernea CUBO
19, 3 (2017)

Φm+1(λ) =






{f0} if λ = 0,

Ψm+1(λ) if 0 < λ < 1,

{f1} if λ = 1.

As in the case m = 1 we obtain that Φm+1 : [0, 1] → D(I, R) is lower semicontinuous.

From ii), v) and Hypothesis 3.1, for almost all t ∈ I, we have

|xm(λ)(t) − xm−1(λ)(t)| ≤
ρ1−α

Γ(α)

∫t

0

(tρ − uρ)α−1uρ−1|gm(λ)(u) − gm−1(λ)(u)|du ≤

ρ1−α

Γ(α)

∫t

0

(tρ − uρ)α−1uρ−1M(u)pm(λ)du = I
α,ρM(t)pm(λ) ≤ I

α,ρMpm(λ) < pm+1(λ).

For λ ∈ [0, 1] consider the set

Gm+1(λ) = cl{v ∈ Φm+1(λ); |v(t) − gm(λ)(t)| < M(t)pm+1(λ) a.e. I}.

To prove that Gm+1(λ) is not empty we note first that rm := (I
α,ρM)m(δm+1 − δm) > 0 and by

Hypothesis 3.1 and ii) one has

d(gm(t), F(t, xm(λ)(t), V(xm(λ))(t)) ≤ L(t)(|xm(λ)(t) − xm−1(λ)(t)|+

∫t

0

L(s)|xm(λ)(s) − xm−1(λ)(s)|ds) ≤ L(t)(1 +

∫t

0

L(s)ds)|Iα,ρM(t)|pm(λ)

= M(t)(pm+1(λ) − rm) < M(t)pm+1(λ).

Moreover, since Φm+1 : [0, 1] → D(I, R) is lower semicontinuous and the maps λ → pm+1(λ),

λ → hm(λ) are continuous we apply Lemma 2.3 and we obtain the existence of a continuous

selection gm+1 of Gm+1.

Therefore,

|xm(λ) − xm−1(λ)|C ≤ I
α,ρMpm(λ) ≤ (I

α,ρM)m(
Tρα

ραΓ(α + 1)
|p0(λ)|1 + δ)

and thus {xm(λ)}m∈N is a Cauchy sequence in the Banach space C(I, R), hence it converges to

some function x(λ) ∈ C(I, R).

Let g(λ) ∈ L1(I, R) be such that

x(λ)(t) = ξ0 +
ρ1−α

Γ(α)

∫t

0

(tρ − uρ)α−1uρ−1g(λ)(u)du, t ∈ I.

The function λ 7→ T
ρα

ραΓ(α+1)
|p0(λ)|1 + δ is continuous, so it is locally bounded. Therefore the

Cauchy condition is satisfied by {xm(λ)}m∈N locally uniformly with respect to λ and this implies

that the mapping λ → x(λ) is continuous from [0, 1] into C(I, R). Obviously, the convergence of

the sequence {xm(λ)} to x(λ) in C(I, R) implies that gm(λ) converges to g(λ) in L1(I, R).



CUBO
19, 3 (2017)

On the solution set of a fractional integro-differential inclusion . . . 39

Finally, from ii), Hypothesis 3.1 and from the fact that the values of F are closed we obtain

that x(λ) ∈ S(ξ0). From i) and v) we have x(0) = x0, x(1) = x1 and the proof is complete.

In what follows we use the notations

ũ(t) = x0 +
ρ1−α

Γ(α)

∫t

0

(tρ − sρ)α−1sρ−1u(s)ds, u ∈ L1(I, R) (3.2)

and

p0(u)(t) = |u(t)| + p(t) + L(t)(|ũ(t)| +

∫t

0

L(s)|ũ(s)|ds), t ∈ I (3.3)

Let us note that

d(u(t), F(t, ũ(t), V(ũ)(t)) ≤ p0(u)(t) a.e. I (3.4)

and, since for any u1, u2 ∈ L
1(I, R)

|p0(u1) − p0(u2)|1 ≤ (1 + |I
α,ρM(T)|)|u1 − u2|1

the mapping p0 : L
1(I, R) → L1(I, R) is continuous.

Proposition 3.3. Assume that Hypothesis 3.1 is satisfied and let φ : L1(I, R) → L1(I, R) be

a continuous map such that φ(u) = u for all u ∈ T (x0). For u ∈ L
1(I, R), we define

Ψ(u) = {u ∈ L1(I, R); u(t) ∈ F(t, φ̃(u)(t), V(φ̃(u))(t)) a.e. I},

Φ(u) =

{
{u} if u ∈ T (x0),

Ψ(u) otherwise.

Then the multifunction Φ : L1(I, R) → P(L1(I, R)) is lower semicontinuous with closed de-

composable and nonempty values.

The proof of Proposition 3.3 is similar to the proof of Proposition 3.2 in [4].

Theorem 3.4. Assume that Hypothesis 3.1 is satisfied, consider x0 ∈ R and assume I
α,ρM <

1.

Then there exists a continuous mapping g : L1(I, R) → L1(I, R) such that

i) g(u) ∈ T (x0), ∀u ∈ L
1(I, R),

ii) g(u) = u, ∀u ∈ T (x0).

Proof. Fix δ > 0 and for m ≥ 0 set δm =
m+1
m+2

δ and define pm(u) := (I
α,ρM)m−1( T

ρα

ραΓ(α+1)
|p0(u)|1+

δm), where ũ and p0(.) are defined in (3.2) and (3.3). By the continuity of the map p0(.), already

proved, we obtain that pm : L
1(I, R) → L1(I, R) is continuous.

We define g0(u) = u and we shall prove that for any m ≥ 1 there exists a continuous map

gm : L
1(I, R) → L1(I, R) that satisfies

a) gm(u) = u, ∀u ∈ T (x0),



40 Aurelian Cernea CUBO
19, 3 (2017)

b) gm(u)(t) ∈ F(t,
˜gm−1(u)(t), V(

˜gm−1(u))(t)) a.e. I,

c) |g1(u)(t) − g0(u)(t)| ≤ p0(u)(t) + δ0
ραΓ(α + 1)

Tρα
a.e. I,

d) |gm(u)(t) − gm−1(t)| ≤ M(t)pm(u) a.e. I, m ≥ 2.

For u ∈ L1(I, R), we define

Ψ1(u) = {v ∈ L
1(I, R); v(t) ∈ F(t, ũ(t), V(ũ)(t)) a.e. I},

Φ1(u) =

{
{u} if u ∈ T (x0),

Ψ1(u) otherwise

and by Proposition 3.3 (with φ(u) = u) we obtain that Φ1 : L
1(I, R) → D(I, R) is lower semicon-

tinuous. Moreover, due to (3.4), the set

G1(u) = cl{v ∈ Φ1(u); |v(t) − u(t)| < p0(u)(t) + δ0
ραΓ(α + 1)

Tρα
a.e. I}

is not empty for any u ∈ L1(I, R). So applying Lemma 2.3, we find a continuous selection g1(.) of

G1(.) that satisfies a)-c).

Suppose we have already constructed gi(.), i = 1, . . . , m satisfying a)-d). For u ∈ L
1(I, R) we

define

Ψm+1(u) = {v ∈ L
1(I, R); v(t) ∈ F(t, g̃m(u)(t), V(g̃m(u))(t)) a.e. I},

Φm+1(u) =

{
{u} if u ∈ T (x0),

Ψm+1(u) otherwise.

We apply Proposition 3.3 (with φ(u) = gm(u)) and obtain that Φm+1(.) is a lower semicontinuous

multifunction with closed decomposable and nonempty values. Define the set

Gm+1(u) = cl{v ∈ Φm+1(u); |v(t) − gm+1(u)(t)| < M(t)pm+1(u) a.e. I}.

To prove that Gm+1(u) is not empty we note first that rm := (I
α,ρM)m(δm+1 − δm) > 0 and by

Hypothesis 3.1 and b) one has

d(gm(t), F(t, g̃m(u)(t), V(g̃m(u))(t)) ≤ L(t)(|g̃m(u)(t) − ˜gm−1(u)(t)|+

∫t

0

L(s)|g̃m(u)(s) −
˜gm−1(u)(s)|ds ≤ M(t)(I

α,ρM)pm(u) = M(t)(pm+1(u) − rm)

< M(t)pm+1(u).

Thus Gm+1(u) is not empty for any u ∈ L
1(I, R). With Lemma 2.3, we find a continuous selection

gm+1 of Gm+1, satisfying a)-d).



CUBO
19, 3 (2017)

On the solution set of a fractional integro-differential inclusion . . . 41

Therefore, we obtain that

|gm+1(u) − gm(u)|1 ≤ (I
α,ρM)m(

Tρα−1

ραΓ(α + 1)
|p0(u)|1 + δ)

and this implies that the sequence {gm(u)}m∈N is a Cauchy sequence in the Banach space L
1(I, R).

Let g(u) ∈ L1(I, R) be its limit. The function u → |p0(u)|1 is continuous, hence it is locally

bounded and the Cauchy condition is satisfied by {gm(u)}m∈N locally uniformly with respect to

u. Hence the mapping g(.) : L1(I, R) → L1(I, R) is continuous.

From a) it follows that g(u) = u, ∀u ∈ T (x0) and from b) and the fact that F has closed

values we obtain that

g(u)(t) ∈ F(t, g̃(u)(t), V(g̃(u))(t)) a.e. I ∀u ∈ L1(I, R).

and the proof is complete.

Remark 3.5. We recall that if Y is a Hausdorff topological space, a subspace X of Y is called

retract of Y if there is a continuous map h : Y → X such that h(x) = x, ∀x ∈ X.

Therefore, by Theorem 3.4, for any x0 ∈ R, the set T (x0) of selections of solutions of (1.1) is

a retract of the Banach space L1(I, R).

Example 3.6. Consider α = 1
2
, ρ = 1, T = 1, x0 = 1 and c < min{1,

3

6+2Γ( 1
2
)
}. Define

F(., .) : I × R × R → P(R) by

F(t, x, y) = [−a
|x|

1 + |x|
, 0] ∪ [0, a

|y|

1 + |y|
], a = cΓ(

1

2
)

and k(., ., .) : I × R × R → R by k(t, s, x) = ax.

Since

sup{|u|; u ∈ F(t, x, y)} ≤ a ∀ t ∈ [0, 1], x, y ∈ R,

dH(F(t, x1, y1), F(t, x2, y2)) ≤ a|x1 − x2| + a|y1 − y2| ∀ x1, x2, y1, y2 ∈ R,

in this case p(t) ≡ a, L(t) ≡ a, M(t) = a(1 + at) and, taking into account the choice of c,

I
1
2
,1M(t) = 2ct1/2[1 +

2cΓ(1
2
)

3
t] ≤ 2c[1 +

2cΓ(1
2
)

3
] < 1 ∀t ∈ [0, 1].

Therefore, applying Theorems 3.2 and 3.4 to the problem

D
1
2
,1

c x(t) ∈ [−a
|x(t)|

1 + |x(t)|
, 0] ∪ [0, a2

|
∫t
0
x(s)ds|

1 + a|
∫t
0
x(s)ds|

], x(0) = 1,

we deduce that its solution set S(1) is arcwise connected in the space C([0, 1],

R) and its set of selections of solutions T (1) is a retract of the Banach space L1([0, 1], R).



42 Aurelian Cernea CUBO
19, 3 (2017)

References

[1] R. Almeida, A. B. Malinowski and T. Odzijewicz, Fractional differential equations with de-

pendence on the Caputo-Katugampola derivative, J. Comput. Nonlin. Dyn., 11 (2016), ID

061017, 11 pp.

[2] D. Băleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus Models and Numer-

ical Methods, World Scientific, Singapore, 2012.

[3] A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values,

Studia Math., 90 (1988), 69-86.

[4] A. Cernea, On a fractional integrodifferential inclusion, Electronic J. Qual. Theory Differ.

Equ., 2014 (2014), no. 25, 1-11.

[5] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, 2010.

[6] U. N. Katugampola, A new approach to generalized fractional derivative, Bull. Math. Anal.

Appl., 6 (2014) 1-15.

[7] A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential

Equations, Elsevier, Amsterdam, 2006.

[8] K. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations,

John Wiley, New York, 1993.

[9] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego,1999.

[10] V. Staicu, On the solution sets to differential inclusions on unbounded interval, Proc. Edin-

burgh Math. Soc., 43 (2000), 475-484.

[11] V. Staicu, Arcwise conectedness of solution sets to differential inclusions, J. Math. Sciences

120 (2004), 1006-1015.

[12] S. Zeng, D. Băleanu, Y. Bai, G. Wu, Fractional differential equations of Caputo-Katugampola

type and numerical solutions, Appl. Math. Comput., 315 (2017), 549-554.


	Introduction
	Preliminaries
	The main results