Boundary value problems of differential inclusions with Riemann-Liouville fractional derivative
In this paper, we establish sufficient conditions for the existence of solutions for a class of boundary value problem for fractional differential inclusions involving the Riemann-Liouville fractional derivative. The cases of both convex and nonconvex valued right-hand side are considered. The top...
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nasplib_isofts_kiev_ua-123456789-1753112025-02-23T17:21:02Z Boundary value problems of differential inclusions with Riemann-Liouville fractional derivative Граничні задачі для диференціальних включень з дробовою похідною Рімана-Ліувілля Краевые задачи для дифференциальных включений с дробной производной Римана-Лиувилля Benchohra, M. Djebali, S. Hamani, S. In this paper, we establish sufficient conditions for the existence of solutions for a class of boundary value problem for fractional differential inclusions involving the Riemann-Liouville fractional derivative. The cases of both convex and nonconvex valued right-hand side are considered. The topological structure of the set of solutions is also examined. Отримано достатнi умови iснування розв’язкiв для класу граничних задач для диференцiальних включень з похiдною дробового порядку, включаючи дробову похiдну Рiмана – Лiувiлля. Розглянуто випадки опуклої та неопуклої правої частини. Також вивчено топологiчну структуру множини розв’язкiв. The authors are grateful to the referee for his remarks which led to an improvement of the first mansucript. 2011 Article Boundary value problems of differential inclusions with Riemann-Liouville fractional derivative / M. Benchohra, S. Djebali, S. Hamani // Нелінійні коливання. — 2011. — Т. 14, № 1. — С. 7-20. — Бібліогр.: 20 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/175311 517.9 en Нелінійні коливання application/pdf Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
In this paper, we establish sufficient conditions for the existence of solutions for a class of boundary value
problem for fractional differential inclusions involving the Riemann-Liouville fractional derivative. The
cases of both convex and nonconvex valued right-hand side are considered. The topological structure of
the set of solutions is also examined. |
| format |
Article |
| author |
Benchohra, M. Djebali, S. Hamani, S. |
| spellingShingle |
Benchohra, M. Djebali, S. Hamani, S. Boundary value problems of differential inclusions with Riemann-Liouville fractional derivative Нелінійні коливання |
| author_facet |
Benchohra, M. Djebali, S. Hamani, S. |
| author_sort |
Benchohra, M. |
| title |
Boundary value problems of differential inclusions with Riemann-Liouville fractional derivative |
| title_short |
Boundary value problems of differential inclusions with Riemann-Liouville fractional derivative |
| title_full |
Boundary value problems of differential inclusions with Riemann-Liouville fractional derivative |
| title_fullStr |
Boundary value problems of differential inclusions with Riemann-Liouville fractional derivative |
| title_full_unstemmed |
Boundary value problems of differential inclusions with Riemann-Liouville fractional derivative |
| title_sort |
boundary value problems of differential inclusions with riemann-liouville fractional derivative |
| publisher |
Інститут математики НАН України |
| publishDate |
2011 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/175311 |
| citation_txt |
Boundary value problems of differential inclusions with Riemann-Liouville fractional derivative / M. Benchohra, S. Djebali, S. Hamani // Нелінійні коливання. — 2011. — Т. 14, № 1. — С. 7-20. — Бібліогр.: 20 назв. — англ. |
| series |
Нелінійні коливання |
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2025-11-24T02:52:44Z |
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2025-11-24T02:52:44Z |
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| fulltext |
UDC 517 . 9
BOUNDARY-VALUE PROBLEMS
OF DIFFERENTIAL INCLUSIONS
WITH RIEMANN – LIOUVILLE FRACTIONAL DERIVATIVE
ГРАНИЧНI ЗАДАЧI ДЛЯ ДИФЕРЕНЦIАЛЬНИХ ВКЛЮЧЕНЬ
З ДРОБОВОЮ ПОХIДНОЮ РIМАНА – ЛIУВIЛЛЯ
M. Benchohra
Laboratoire de Mathématiques, Université de Sidi Bel-Abbès
B.P. 89, 22000, Sidi Bel-Abbès, Algérie
e-mail: benchohra@yahoo.com
S. Djebali
Department Math., E.N.S., B.P. 92
Kouba 16050, Algiers, Algeria
e-mail: djebali@ens-kouba.dz
S. Hamani
Laboratoire de Mathématiques, Université de Sidi Bel-Abbès
B.P. 89, 22000, Sidi Bel-Abbès, Algérie
e-mail: hamani_samira@yahoo.fr
In this paper, we establish sufficient conditions for the existence of solutions for a class of boundary value
problem for fractional differential inclusions involving the Riemann-Liouville fractional derivative. The
cases of both convex and nonconvex valued right-hand side are considered. The topological structure of
the set of solutions is also examined.
Отримано достатнi умови iснування розв’язкiв для класу граничних задач для диференцiаль-
них включень з похiдною дробового порядку, включаючи дробову похiдну Рiмана – Лiувiлля. Роз-
глянуто випадки опуклої та неопуклої правої частини. Також вивчено топологiчну структуру
множини розв’язкiв.
1. Introduction. This paper deals with the existence of solutions for the boundary-value problem
(BVP for short) for fractional order differential inclusions of the form
Dαy(t) ∈ F (t, y(t)), for a.e. t ∈ J, 1 < α ≤ 2, (1)
y(0) = 0, y′(T ) = 0, (2)
where Dα is the Riemann – Liouville fractional derivative, F : J × R → P(R) is a multivalued
map, P(R) is the family of all nonempty subsets of R, and J = [0, T ]. Differential equations
of fractional order have recently been proved to be valuable tools in the modeling of many
phenomena in various fields of science and engineering. Indeed, we can find numerous appli-
cations in viscoelasticity, electrochemistry, control, porous media, electromagnetic. A significant
c© M. Benchohra, S. Djebali, S. Hamani, 2011
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1 7
8 M. BENCHOHRA, S. DJEBALI, S. HAMANI
development in fractional differential and partial differential equations has appeared in recent
years; we refer to the monographs by Kilbas et al. [15], Podlubny [17], Samko et al. [18] and the
papers by Agarwal et al. [1], Benchohra and Hamani [2], Furati and Tatar [9], and Ouahab [16],
and the references therein. In [3], the authors studied the existence and uniqueness of solutions
of classes of initial value problems for functional differential equations with infinite delay and
fractional order. The aim of this work is the study of a BVP for differential inclusion with a
Riemann – Liouville fractional derivative.
This paper is organized as follows. In Section 2, we introduce some preliminary results
needed in the subsequent sections. In Section 3, using the nonlinear alternative of Leray and
Schauder, we present an existence result for problem (1), (2) when the right-hand side is convex-
valued. In Section 4, two results, for the case of nonconvex-valued right-hand side, are given.
The first one is based upon a fixed point theorem for contraction multivalued maps due to Co-
vitz and Nadler while the second one employs the nonlinear alternative of Leray and Schauder
for single-valued maps [11], combined with a selection theorem due to Bressan – Colombo
[4] for lower semicontinuous multivalued maps with decomposable values. The topological
structure of the solutions set is also considered in Section 5. These results extend to the multi-
valued case some results from the above cited literature, and constitute a contribution to this
emerging field of research.
2. Preliminaries. In this section, we introduce notations, definitions, and preliminary facts
that will be used in the remainder of this paper. Let C(J,R) be the Banach space of all conti-
nuous functions from J into R with the norm
‖y‖∞ = sup{|y(t)| : 0 ≤ t ≤ T},
and let L1(J,R) denote the Banach space of functions y : J −→ R that are Lebesgue integrable
with norm
‖y‖L1 =
T∫
0
|y(t)|dt.
AC(J,R) is the space of functions y : J → R, which are absolutely continuous. Given a
topological vector space X, let P(X) be the set of all nonempty subsets of X. Denote by
Pcl(X) = {Y ∈ P(X) : Y closed}, Pb(X) = {Y ∈ P(X) : Y bounded}, Pcp(X) =
= {Y ∈ P(X) : Y compact}, Pcp,cv(X) = {Y ∈ P(X) : Y compact and convex}, Pcl,cv(X) =
= {Y ∈ P(X) : Y closed and convex} and so on. A multivalued map G : X → P (X) is
convex (closed) valued if G(x) is convex (closed) for all x ∈ X. G is bounded on bounded
sets if G(A) = dx∈AG(x) is bounded in X for all A ∈ Pb(X), i.e. supx∈A{sup{|y| : y ∈
∈ G(x)}} < ∞. G is called upper semicontinuous (u.s.c.) on X if for each x0 ∈ X, the set
G(x0) is a nonempty closed subset of X, and for each open set N of X containing G(x0), there
exists an open neighborhood N0 of x0 such that G(N0) ⊆ N, Equivalently, G is u.s.c. if the set
{x ∈ X : G(x) ⊂ B} is open for any open set B in X. G is lower semicontinuous (l.s.c.) if
the set {x ∈ X : G(x) ∩ B 6= ∅} is open for any open set B in X . G is said to be completely
continuous if G(A) is relatively compact for every A ∈ Pb(X). If the multivalued map G is
completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a
closed graph, i.e., (xn −→ x∗, yn −→ y∗, yn ∈ G(xn) imply y∗ ∈ G(x∗)). G has a fixed point
if there is x ∈ X such that x ∈ G(x). The fixed point set of the multivalued operator G will
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
BOUNDARY-VALUE PROBLEMS OF DIFFERENTIAL INCLUSIONS . . . 9
be denoted by FixG. A multivalued map G : J → Pcl(R) is said to be measurable provided
for every open U ⊂ R, the set {t ∈ J, G(t) ⊂ U} is Lebesgue measurable in J. We have the
following lemma.
Lemma 2.1 (see [5, 10]). G is measurable if and only if for each x ∈ R, the function ζ :
J → [0,+∞) defined by ζ(t) = dist(x,G(t)) = inf{‖x − y‖, y ∈ G(t)}, t ∈ J is Lebesgue
measurable.
The following lemma is known as the Kuratowski – Ryll – Nardzewski selection theorem.
Lemma 2.2 (see [10], Theorem 19.7 or [5], Theorem III.6). Let E be a separable metric
space and G : [a, b] → P(E) a measurable multivalued map with closed values. Then G has a
measurable selection.
The following one is taken from [20], Lemma 3.2.
Lemma 2.3. Let G : [0, b] → P(E) be a measurable multifunction and u : [0, b] → E a
measurable function. Then, for any measurable v : [0, b] → (0,+∞), there exists a measurable
selection gv of G such that, for a.e. t ∈ [0, b],
|u(t)− gv(t)| ≤ d(u(t), G(t)) + v(t).
Definition 2.1. A multivalued map F : J × R → P(R) is said to be Carathéodory if
(a) the map t 7−→ F (t, u) is measurable for each u ∈ R;
(b) the map u 7−→ F (t, u) is upper semicontinuous for a.e. t ∈ J ;
(c) it is L1−Carathéodory if it is further integrably bounded, i.e., there exists h ∈ L1(J,R+)
such that
‖F (t, x)‖P ≤ h(t) for a.e. t ∈ J and all x ∈ R,
where ‖F (t, u)‖P = sup{|v| : v ∈ F (t, u)}.
For each y ∈ C(J,R), define the set of selections of F by
SF,y = {v ∈ L1(J,R) : v(t) ∈ F (t, y(t)), a.e. t ∈ J}.
Remark 2.1. From ([19], Theorem 5.10), we know that SF,y is nonempty if and only if the
mapping t 7→ {inf ‖v‖ : v ∈ F (t, y(t)} belongs to L1(J). It is bounded if and only if the
mapping t 7→ ‖F (t, y(t)‖P = {sup ‖v‖ : v ∈ F (t, y(t)} belongs to L1(J); this particularly
holds true when F is integrably bounded.
Let (X, d) be a metric space. Define the Hausdorff pseudometric distance Hd : P(X) ×
×P(X) −→ R+ ∪ {∞} by
Hd(A,B) = max{sup
a∈A
d(a,B), sup
b∈B
d(A, b)},
where d(A, b) = infa∈A d(a, b) and d(a,B) = infb∈B d(a, b). Then (Pb,cl(X), Hd) is a metric
space and (Pcl(X), Hd) is a generalized metric space (see [14]).
Definition 2.2. A multivalued operator N : X → Pcl(X) is called
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
10 M. BENCHOHRA, S. DJEBALI, S. HAMANI
(a) γ-Lipschitz if there exists γ > 0 such that
Hd(N(x), N(y)) ≤ γd(x, y), for each x, y ∈ X,
(b) a contraction if it is γ-Lipschitz with 0 < γ < 1.
The following result is known as the Covitz – Nadler fixed point theorem.
Lemma 2.4 [6]. Let (X, d) be a complete metric space. If N : X → Pcl(X) is a contraction,
then FixN 6= ∅.
For more details about multivalued maps, we refer to the books by Deimling [7], Górnie-
wicz [10], and Kisielewicz [14]. We end this section with the definitions of fractional order
integral and derivative (see [15, 17]).
Definition 2.3. The fractional (arbitrary) order integral of a function h ∈ L1([a, b],R+) of
order α > 0 is defined by
Iαa h(t) =
1
γ(α)
t∫
a
(t− s)α−1h(s)ds,
where γ is the Gamma function. When a = 0, we write Iαh(t) = h(t) ∗ ϕα(t), where ϕα(t) =
=
tα−1
γ(α)
for t > 0 and ϕα(t) = 0 for t ≤ 0. Note ϕα behaves as the Delta function when
α → 0. Indeed, it is shown (see e.g. (2.89) in [17], p. 65) that limα→0 I
αh(t) = h(t) whenever h
is continuous.
Definition 2.4. For a function h defined on the interval [a, b], the α − th Riemann – Liouville
fractional-order derivative of h is given by
(Dα
a+h)(t) =
1
γ(n− α)
(
d
dt
)n t∫
a
(t− s)n−α−1h(s) ds,
where n = [α] + 1 and [α] denotes the integer part of α.
3. The convex case. In this section, we are concerned with the existence of solutions to
problem (1), (2) when the right-hand side takes convex, compact values. Let us start by defining
what we mean by a solution of problem (1), (2).
Definition 3.1. A function y ∈ AC(J,R) is said to be a solution of problem (1), (2), if there
exists a function v ∈ L1(J,R) with v(t) ∈ F (t, y(t)), for a.e. t ∈ J, such that
Dαy(t) = v(t), a.e. t ∈ J, 1 < α ≤ 2,
and the function y satisfies the conditions (2).
For the existence of solutions for problem (1), (2), we need an auxiliary lemma:
Lemma 3.1 [15]. Let α > 0. If we assume h ∈ C((0, T ),R)
⋂
L((0, T ),R), then the fractional
differential equation
Dαh(t) = 0
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
BOUNDARY-VALUE PROBLEMS OF DIFFERENTIAL INCLUSIONS . . . 11
has solutions
h(t) = c1t
α−1 + c2t
α−2 + . . .+ cnt
α−n, for ci ∈ R, i = 1, 2, . . . , n.
Lemma 3.2 [15]. Assume h ∈ C((0, T ),R)
⋂
L((0, T ),R) with a fractional derivative of order
α > 0. Then
IαDαh(t) = h(t) + c1t
α−1 + c2t
α−2 + . . .+ cnt
α−n
for some constants ci, i = 1, 2, . . . , n.
As a consequence of Lemmas 3.1 and 3.2, we have the following result which provides the
integral formulation for problem (1), (2).
Lemma 3.3. Let 1 < α ≤ 2 and let σ : J → R be continuous. A function y is a solution of
the fractional integral equation
y(t) =
T∫
0
G(t, s)σ(s) ds, (3)
where
G(t, s) =
(t− s)α−1
γ(α)
− tα−1(T − s)α−2
(α− 1)Tα−2γ(α− 1)
, 0 ≤ s ≤ t,
− tα−1(T − s)α−2
(α− 1)Tα−2γ(α− 1)
, t ≤ s < T,
(4)
if and only if y is a solution of the fractional BVP
Dαy(t) = σ(t), t ∈ J, (5)
y(0) = 0, y′(T ) = 0. (6)
Proof. Assume that y satisfies (5); then Lemma 3.2 implies that
y(t) = c1t
α−1 + c2t
α−2 +
1
γ(α)
t∫
0
(t− s)α−1σ(s) ds.
From (6), a simple calculation yields c2 = 0 and
c1 =
−1
(α− 1)Tα−2γ(α− 1)
T∫
0
(T − s)α−2σ(s) ds,
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
12 M. BENCHOHRA, S. DJEBALI, S. HAMANI
whence equation (3). Conversely, it is clear that if y satisfies equation (3), then by Lemmas 3.2,
3.3, equations (5), (6) hold.
Remark 3.1. The function t ∈ J 7→
∫ T
0 |G(t, s)|ds is continuous on [0, T ], hence bounded.
Let
G∗ = sup
T∫
0
|G(t, s)|ds, t ∈ J
.
Our first existence result is based on the nonlinear alternative of Leray – Schauder type for
multivalued maps [10, 11] which we recall for the reader’s convenience:
Lemma 3.4. Let (X, ‖ · ‖) be a Banach space and F : X → Pcl,cv(X) a compact, u.s.c.
multivalued map. Then either one of the following conditions hold:
(a) F has at least one fixed point,
(b) the setM := {x ∈ X, x ∈ λF (x), λ ∈ (0, 1)} is unbounded.
We have the following theorem.
Theorem 3.1. Assume the following hypotheses hold:
(H1) F : J × R −→ Pcp,cv(R) is a Carathéodory multivalued map.
(H2) There exist p ∈ L∞(J,R) and ψ : [0,∞) → (0,∞) continuous and nondecreasing such
that
‖F (t, u)‖P ≤ p(t)ψ(|u|) for t ∈ J and each u ∈ R.
(H3) There exists a constant M > 0 such that
M
p∗G∗ψ(M)
> 1, (7)
where
p∗ = ‖p‖L∞ .
Then problem (1), (2) has at least one solution on J.
Proof. In order to transform problem (1), (2) into a fixed point problem, consider the multi-
valued operator N : C(J,R) −→ P(C(J,R)) defined by
N(y) =
h ∈ C(J,R) : h(t) =
T∫
0
G(t, s)v(s)ds, v ∈ SF,y
where the Green function G(t, s) is given by (4). Clearly, from Lemma 3.3, the fixed points of
N are solutions of (1) – (3). We shall show that N satisfies the assumptions of Lemma 3.4. The
proof will be given in four steps. First, since SF,y is convex (because F has convex values), then
N(y) is convex for each y ∈ C(J,R).
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
BOUNDARY-VALUE PROBLEMS OF DIFFERENTIAL INCLUSIONS . . . 13
Step 1: N maps bounded sets into bounded sets in C(J,R). Let Bη∗ = {y ∈ C(J,R) :
‖y‖∞ ≤ η∗} be a bounded set in C(J,R) and y ∈ Bη∗ . Then, for each h ∈ N(y) and t ∈ J,
there exists v ∈ SF,y such that, by (H2) we have
|h(t)| ≤
T∫
0
G(t, s)|v(s))|ds ≤ p∗G∗ψ(‖y‖∞).
Thus
‖h‖∞ ≤ p∗G∗ψ(η∗).
Step 2: N maps bounded sets into equicontinuous sets of C(J,R). Let t1, t2 ∈ J, t1 < t2, Bη∗
be a bounded set of C(J,R), y ∈ Bη∗ and h ∈ N(y). As in Step 1, we have
|h(t2)− h(t1)| ≤
T∫
0
|G(t2, s)−G(t1, s)| |v(s)| ds+ p∗ψ(η∗) sup
s∈J
∣∣∣∣∣∣
T∫
0
G(t2, s)−G(t1, s)
∣∣∣∣∣∣ ds.
As t1 −→ t2, the right-hand side of the above inequality tends to zero. As a consequence of
Steps 1, 2 together with the Arzéla – Ascoli theorem, we conclude that N is completely conti-
nuous.
Step 3: N has a closed graph. Let yn → y∗, hn ∈ N(yn) and hn → h∗. We need to show that
h∗ ∈ N(y∗). hn ∈ N(yn) means that there exists vn ∈ SF,yn such that, for each t ∈ J,
hn(t) =
T∫
0
G(t, s) vn(s) ds.
We must show that there exists v∗ ∈ SF,y∗ such that, for each t ∈ J,
h∗(t) =
T∫
0
G(t, s) v∗(s) ds.
Since F (t, ·) is upper semicontinuous, for every ε > 0, there exists n0(ε) ≥ 0 such that for every
n ≥ n0, we have
vn(t) ∈ F (t, yn(t)) ⊂ F (t, y∗(t)) + εB(0, 1), a.e. t ∈ J.
Since F (·, ·) has compact values, there exists a subsequence vnm(·) such that
vnm(·) → v∗(·), as m → ∞
and then
v∗(t) ∈ F (t, y∗(t)), a.e. t ∈ J.
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
14 M. BENCHOHRA, S. DJEBALI, S. HAMANI
Since ∣∣∣∣∣∣h∗(t)−
T∫
0
G(t, s) v∗(s) ds
∣∣∣∣∣∣ ≤ |h∗(t)− hnm(t)|+
∣∣∣∣∣∣hnm(t)−
T∫
0
G(t, s)v∗(s)ds
∣∣∣∣∣∣ ≤
≤ |h∗(t)− hnm(t)|+
∣∣∣∣∣∣
T∫
0
G(t, s)|vnm(s)− v∗(s)| ds
∣∣∣∣∣∣ ,
our claim follows from the Lebesgue dominated convergence theorem.
Step 4: A priori bounds on solutions. Let y be such that y ∈ λN(y) for λ ∈ [0, 1]. Then,
there exists v ∈ SF,y such that, for each t ∈ J,
|y(t)| ≤
T∫
0
G(t, s)p(s)ψ(|y(s)|)ds ≤ p∗G∗ψ(‖y‖∞).
Thus
‖y‖∞
p∗G∗ψ(‖y‖∞)
≤ 1.
Condition (7) implies that ‖y‖∞ 6= M. Given
U = {y ∈ C(J,R) : ‖y‖∞ < M},
there is no y ∈ ∂U such that y ∈ λN(y) for some λ ∈ (0, 1). Moreover, the operator N : U →
→ P(C(J,R)) is upper semicontinuous and completely continuous. Therefore, with Lemma 3.4,
we deduce that N has a fixed point y in U, a solution of problem (1), (2).
Theorem 3.1 is proved.
4. The nonconvex case. In this section, two existence results for problem (1), (2) are given
when the right-hand side takes nonconvex values.
4.1. A first result.
Theorem 4.1. Assume that
(H4) F : J × R −→ Pcp(R) is integrably bounded and F (·, u) : J → Pcp(R) is measurable
for each u ∈ R.
(H5) There exists l ∈ L∞(J,R) such that d(0, F (t, 0)) ≤ l(t), a.e. t ∈ J and
Hd(F (t, u), F (t, u)) ≤ l(t)|u− u| for every u, u ∈ R.
If further
‖l‖L∞G∗ < 1, (8)
then problem (1), (2) has at least one solution on J .
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BOUNDARY-VALUE PROBLEMS OF DIFFERENTIAL INCLUSIONS . . . 15
Proof. For each y ∈ C(J,R), the set SF,y is nonempty by (H4) (see Remark 2.1). By
Lemma 2.2, F has a measurable selection. We shall show that N satisfies the assumptions of
Lemma 2.4. The proof will be given in two steps.
Step 1: N(y) ∈ Pcl(C(J,R)) for each y ∈ C(J,R). Indeed, let (yn)n≥0 ∈ N(y) be such that
yn −→ ỹ in C(J,R). Then, ỹ ∈ C(J,R) and there exists vn ∈ SF,y such that, for each t ∈ J,
yn(t) =
T∫
0
G(t, s) vn(s) ds. (9)
By Assumption (H4), the sequence vn is integrably bounded. Moreover F has compact values.
Then by the Dunford-Pettis theorem (see [13], Proposition 4.2.1), we may pass to a subsequence,
if necessary, to get that (vn)n∈N converges weakly to v in L1
w(J,R) (the space endowed with the
weak topology). Define the linear operator
γ : L1(J,R) −→ C(J,R)
by (γv)(t) =
∫ T
0 G(t, s)v(s)ds. The Ascoli – Arzéla lemma implies that γ is completely conti-
nuous. As a consequence, (vn)n∈N admits a subsequence (vnk
)k∈N such that γ(vnk
) converges
strongly to γ(v) in C(J,R). Passing to the limit in (9) with n = nk, as k → +∞, yields that, for
each t ∈ J
ỹ(t) =
T∫
0
G(t, s) v(s) ds,
hence ỹ ∈ N(y) and N(y) is closed.
Step 2: There exists γ < 1 such that
Hd(N(y), N(y)) ≤ γ‖y − y‖∞, for each y, y ∈ C(J,R).
Let y, y ∈ C(J,R) and h1 ∈ N(y). Then, there exists v1 ∈ SF,y such that for each t ∈ J ,
h1(t) =
T∫
0
G(t, s)v1(s)ds, a.e. t ∈ J.
From (H5), we deduce
Hd(F (t, y(t)), F (t, y(t))) ≤ l(t)|y(t)− y(t)|.
Hence, there exists w ∈ F (t, y(t)) such that
|v1(t)− w| ≤ l(t)|y(t)− y(t)|, t ∈ J.
Consider the multivalued map U : J → P(R) defined by
U(t) = {w ∈ R : |v1(t)− w| ≤ l(t)|y(t)− y(t)|} := B(v1(t), γ(t)),
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16 M. BENCHOHRA, S. DJEBALI, S. HAMANI
where γ(t) = l(t)|y(t) − y(t)|. Assumptions (H4) and (H5) imply that the multimap t 7→
7→ F (t, y(t)) is measurable. Since v1 and γ are measurable, Theorem III.4.1 in [5] tells us that
the closed ball B is measurable. Moreover, the set V (t) = U(t)∩F (t, y(t)) is nonempty. Indeed,
taking the measurable function v = 0 in Lemma 2.3, we obtain a measurable selection u of
F (t, y(t)) such that
|u(t)− v1(t)| ≤ d(v1(t), F (t, y(t)) ≤ γ(t).
Then u ∈ U(t), hence u ∈ V (t), proving our claim. Finally, since the multivalued operator V
defined by V (t) = U(t) ∩ F (t, y(t)) is measurable (see [5, 10]), there exists, by Lemma 2.2, a
function which is a measurable selection for V. So v2(t) ∈ F (t, y(t)), and for each t ∈ J,
|v1(t)− v2(t)| ≤ l(t)|y(t)− y(t)|.
Let us define for a.e. t ∈ J,
h2(t) =
T∫
0
G(t, s) v2(s) ds.
Then, for a.e. t ∈ J
|h1(t)− h2(t)| ≤ G∗‖l‖L∞‖y − y‖∞ ≤ ‖l‖L∞G∗‖y − y‖∞.
Hence
‖h1 − h2‖∞ ≤ ‖l‖L∞G∗‖y − y‖∞.
By an analogous relation, obtained by interchanging the roles of y and y, we get
Hd(N(y), N(y)) ≤ ‖l‖L∞G∗‖y − y‖∞.
Finally, condition (8) implies that N is a contraction and thus, by Lemma 2.4, N has a fixed
point y, solution to problem (1), (2).
Theorem 4.1 is proved.
4.2. A second result. Now, we present a result for problem (1), (2) in the spirit of the
nonlinear alternative of Leray – Schauder type [11] for single-valued maps, combined with a
selection theorem due to Bressan – Colombo [4] for lower semicontinuous multivalued maps
with decomposable values. Details on multivalued maps with decomposable values and their
properties can be found in the book by Fryszkowski [8]. Let A be a subset of J × R.
Definition 4.1. (a) A is called L⊗B measurable if A belongs to the σ-algebra generated by all
sets of the form I ×D where I is Lebesgue measurable in J and D is Borel measurable in R.
(b) A subset A ⊂ L1(J,R) is decomposable if for all u, v ∈ A and for every Lebesgue
measurable set I ⊂ J, uχI + vχJ\I ∈ A, where χ stands for the characteristic function.
Let F : J × R → P(R) be a multivalued map with nonempty closed values. Assign to F
the multivalued operator F : C(J,R) → P(L1(J,R)) defined by F(y) = SF,y and let F(t, y) =
= SF,y(t), t ∈ J, y ∈ C(J,R). The operatorF is called the Nemyts’kı̆i operator associated to F.
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BOUNDARY-VALUE PROBLEMS OF DIFFERENTIAL INCLUSIONS . . . 17
Definition 4.2. Let F : J × R → P(R) be a multivalued function with nonempty compact
values. We say that F is of lower semicontinuous type (l.s.c. type) if its associated Nemyts’kı̆i
operator F is lower semicontinuous and has nonempty closed and decomposable values.
Lemma 4.1 [4]. Let X be a separable metric space and let E be a Banach space. Then every
l.s.c. multivalued operator N : X → Pcl(L1([0, T ], E)) with nonempty closed decomposable
values has a continuous selection, i.e. there exists a continuous single-valued function f : X →
→ L1(J,E) such that f(x) ∈ N(x) for every x ∈ X.
Let us introduce the following hypotheses:
(H6) F : [0, T ]× R −→ P(R) is a nonempty compact-valued multivalued map such that:
(a) the map (t, u) 7→ F (t, u) is L ⊗ B measurable;
(b) the map y 7→ F (t, y) is lower semicontinuous for a.e. t ∈ [0, T ].
(H7) F is locally integrably bounded, i.e., for each q > 0, there exists a function hq ∈
∈ L1([0, T ],R+) such that
‖F (t, u)‖P ≤ hq(t), for a.e. t ∈ [0, T ] and for y ∈ R with |u| ≤ q.
The following lemma is crucial in the proof of our main existence theorem. The second one
is the classical Nonlinear Alternative of Leray and Schauder for single-valued mappings.
Lemma 4.2 [7]. Let F : [0, T ] × R → P(R) be a multivalued map with nonempty, compact
values. Assume that (H6) and (H7) hold. Then F is of l.s.c. type.
Lemma 4.3 [11]. Let X be a Banach space and C ⊂ X a nonempty bounded, closed, convex
subset. Assume U is an open subset of C with 0 ∈ U and let G : Ū → C be a a continuous
compact map. Then
(a) either there is a point u ∈ ∂U and λ ∈ (0, 1) with u = λG(u),
(b) or G has a fixed point in U.
Theorem 4.2. Suppose that Assumptions (H2), (H3), (H6), (H7) are satisfied. Then problem
(1), (2) has at least one solution.
Proof. (H6) and (H7) together with Lemma 4.2 imply that F is of lower semi-continuous
type. Then, from Theorem 4.1, there exists a continuous function f : C([0, T ],R) → L1([0, T ],R)
such that f(y) ∈ F(y) for all y ∈ C([0, T ],R). Consider the problem
cDαy(t) = f(y)(t), for a.e. t ∈ J, 1 < α ≤ 2, (10)
y(0) = 0, y′(T ) = 0. (11)
If y ∈ AC([0, T ],R) is a solution of problem (10), (11), then y is a solution to problem (1), (2).
Problem (10), (11) is then reformulated as a fixed point problem for the single-valued operator
N1 : C([0, T ],R) → C([0, T ],R) defined by
N1(y)(t) =
T∫
0
G(t, s) f(y)(s) ds,
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18 M. BENCHOHRA, S. DJEBALI, S. HAMANI
where the functions G is given by (4). Using (H2), (H3), we can easily show (using similar
argument as in Theorem 3.1) that the operator N1 satisfies all conditions of Lemma 4.3, which
completes the proof of Theorem 4.2.
5. Topological structure. In this section, we present a result on the topological structure of
the set of solutions of problem (1), (2).
Theorem 5.1. Assume that (H1) – (H3) hold. Then the solution set for problem (1), (2) is
nonempty and compact in C(J,R).
Proof. Let
S = {y ∈ C(J,R) : y is solution of problem (1), (2)}.
From Theorem 3.1, S 6= ∅; thus we only prove that S is a compact set. Let (yn)n∈N ∈ S, then
there exists vn ∈ SF,yn such that for t ∈ J
yn(t) =
T∫
0
G(t, s) vn(s) ds,
where the function G(t, s) is given by (4). From (H2), we can prove that there exists a constant
M1 > 0 such that
‖yn‖∞ ≤ M1, for every n ≥ 1.
As in Step 2 in Theorem 3.1, we can easily show that the set {yn : n ≥ 1} is equicontinuous
in C(J,R). By the Arzéla – Ascoli theorem we can conclude that there exists a subsequence of
{yn} and still denoted again by {yn}, such that yn converges to some limit y in C(J,R).We shall
show that there exists v(·) ∈ F (·, y(·)) such that
y(t) =
T∫
0
G(t, s) v(s) ds.
Since F (t, .) is upper semicontinuous, then for every ε > 0, there exists n0(ε) ≥ 0 such that for
every n ≥ n0, we have
vn(t) ∈ F (t, yn(t)) ⊂ F (t, y(t)) + εB(0, 1), a.e. t ∈ J.
Since F (., .) has compact values, there exists subsequence vnm(.) such that
vnm(.) → v(.), as m → ∞
and
v(t) ∈ F (t, y(t)), a.e. t ∈ J.
Moreover
|vnm(t)| ≤ p(t), a.e. t ∈ J.
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BOUNDARY-VALUE PROBLEMS OF DIFFERENTIAL INCLUSIONS . . . 19
By the Lebesgue dominated convergence theorem, we conclude that v ∈ L1(J,R) which impli-
es that v ∈ SF,y. Thus
y(t) =
T∫
0
G(t, s) v(s) ds, t ∈ J,
that is y ∈ S. Hence S ∈ Pcp(C(J,R)), as claimed.
6. An example. As an application of our results, consider the fractional differential inclusion
Dαy(t) ∈ F (t, y), a.e. t ∈ J = [0, 1], 1 < α ≤ 2, (12)
y(0) = 0, y′(1) = 0, (13)
where F (t, y) = {v ∈ R : f1(t, y) ≤ v ≤ f2(t, y)} and f1, f2 : J×R → R are two single-valued
functions. Assume that for each t ∈ J, the function f1(t, ·) is lower semicontinuous (i.e., the set
{y ∈ R : f1(t, y) > µ} is open for each µ ∈ R), and that for each t ∈ J, the function f2(t, ·)
is upper semicontinuous (i.e., the set {y ∈ R : f2(t, y) < µ} is open for each µ ∈ R). Assume
further that there are p ∈ L∞(J,R+) and ψ : [0,∞) → (0,∞) continuous and nondecreasing
such that
max(|f1(t, y)|, |f2(t, y)|) ≤ p(t)ψ(|y|), t ∈ J and y ∈ R.
From (4), the Green function G is given by
G(t, s) =
(t− s)α−1
γ(α)
− tα−1(1− s)α−2
(α− 1)γ(α− 1)
, 0 ≤ s ≤ t,
− tα−1(1− s)α−2
(α− 1)γ(α− 1)
, t ≤ s ≤ 1.
Then, simple computations show that
G∗ = sup
1∫
0
|G(t, s)| ds : t ∈ [0, 1]
=
1
αγ(α)
+
1
(α− 1)2γ(α− 1)
.
It is clear that F is compact and convex valued, and it is upper semicontinuous (see [7]). If there
exists a constant M > 0 such that
M
p∗G∗ψ(M)
> 1,
then all the conditions of Theorem 3.1 are met. As a consequence, BVP (12), (13) has at least
one solution y on J.
Acknowledgement. The authors are grateful to the referee for his remarks which led to an
improvement of the first mansucript.
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20 M. BENCHOHRA, S. DJEBALI, S. HAMANI
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Received 15.07.08,
after revision — 29.11.09
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