Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
A
NOTE
ON
THE
EXISTENCE
OF
SOLUTIONS
OF
A
FRACTIONAL
DIFFERENTIAL
INCLUSION
OF
HILFER-HADAMARD
TYPE
∗
Aurelian
Cernea
†
Dedicated
to
the
memory
of
Professor
Mihail
Megan
DOI
10.56082/annalsarscimath.2026.3.29
Abstract
A
fractional
differential
inclusion
with
multi-point
boundary
con-
ditions
involving
Hilfer-Hadamard
derivatives
is
considered.
An
exis-
tence
result
is
established
when
the
set-valued
map
is
lower
semicon-
tinuous.
Keywords:
differential
inclusion,
fractional
derivative,
boundary
value
problem.
MSC:
34A60,
34A12,
34A08.
1
Introduction
In
this
note
we
consider
the
following
differential
inclusion
D
α,β
HH
x
(
t
)
∈
F
(
t,
x
(
t
))
a.e.
([1
,
e
])
(1)
with
boundary
conditions
of
the
form
x
(1
+
ε
)
=
0
,
D
1
,
1
HH
x
(
e
)
=
n
i
=1
ν
i
D
1
,
1
HH
x
(
η
i
)
,
(2)
∗
Accepted
for
publication
on
February
05,
2026
†
acernea@fmi.unibuc.ro
,
Faculty
of
Mathematics
and
Computer
Science,
University
of
Bucharest,
Academiei
14,
010014
Bucharest
and
Academy
of
Romanian
Scientists,
Ilfov
3,
050044
Bucharest,
Romania
29
A
Hilfer-Hadamard
fractional
integro-differential
inclusion
30
where
D
α,β
HH
is
the
Hilfer-Hadamard
fractional
derivative
of
order
α
∈
(1
,
2]
and
type
β
∈
[0
,
1],
η
i
∈
(1
,
e
),
ν
i
∈
R
,
i
=
1
,
n
,
ε
∈
(0
,
1)
and
F
:
[1
,
e
]
×
R
→
P
(
R
)
is
a
set-valued
map.
We
recall
that
Hilfer
fractional
derivative
introduced
in
[
14
]
is
an
interpo-
lation
between
Riemann-Liouville
and
Caputo
derivatives.
The
modification
of
Hilfer
fractional
derivative
resulted
in
the
concept
of
Hilfer-Hadamard
fractional
derivative.
Similarly,
the
Hilfer-Hadamard
fractional
derivative
covers
the
cases
of
Riemann-Liouville-Hadamard
and
Caputo-Hadamard
fractional
derivatives.
Several
recent
papers
are
devoted
to
the
study
to
boundary
value
prob-
lems
associated
to
fractional
differential
equations
and
inclusions
defined
by
Hilfer-Hadamard
fractional
derivative
[
1
,
9
,
10
,
18
]
etc.
For
background
facts
concerning
fractional
calculus
and
differential
equations
of
fractional
order
we
refer
to
[
2
,
12
,
15
,
17
]
etc.
The
present
study
is
motivated
by
a
recent
paper
[
3
],
where
two
existence
results
for
problem
(1)-(2)
are
provided.
A
result
is
obtained
in
the
case
when
the
values
of
the
set-valued
map
are
convex
and
the
set-valued
map
is
upper
semicontinuous.
The
other
result
is
proved
in
the
situation
when
the
set-valued
map
is
Lipschitz
in
the
state
variable
with
non
convex
values.
Our
aim
is
to
complete
the
study
in
[
3
]
by
taking
into
account
the
case
when
the
set-valued
map
has
non
convex
values,
but
instead
the
Lipschitz
hypothesis
in
[
3
]
we
assume
that
F
(
.,
.
)
is
lower
semicontinuous.
The
proof
of
our
existence
result
relies
on
a
nonlinear
alternative
of
Leray-Schauder
type
and
on
Bressan-Colombo-Fryszkowski
selection
theorem
for
lower
semi-
continuous
set-valued
maps
with
decomposable
values.
We
emphasize
that
such
kind
of
approach
is
usual
in
the
study
of
differ-
ential
inclusions
(e.g.,
[
5
–
8
,
11
]),
however
its
presentation
in
the
framework
of
problem
(1)-(2)
is
new.
The
paper
is
organized
as
follows:
in
Section
2
we
recall
some
preliminary
facts
that
we
need
in
the
sequel,
in
Section
3
we
prove
our
result.
2
Preliminaries
Let
(
X,
d
)
be
a
metric
space
with
the
corresponding
norm
|
.
|
and
let
I
⊂
R
be
a
compact
interval.
Denote
by
L
(
I
)
the
σ
-algebra
of
all
Lebesgue
measurable
subsets
of
I
,
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
χ
A
(
.
)
:
I
→
{
0
,
1
}
denotes
the
characteristic
function
of
A
.
For
any
subset
A
⊂
X
we
denote
by
A
the
closure
of
A
.
B
r
(0)
and
B
r
(0)
will
be
the
open
and
closed
balls
centered
at