In this paper, a twopoint fractional boundary value problem at resonance is considered. By using the coincidence degree theory some existence results of solutions are established.
AMS Mathematics Subject Classification : 34B15.
1. Introduction
In this paper, we investigate the existence of solutions for the following twopoint fractional boundary value problem (BVP) at resonance
where
f
: [0, 1] × IR
^{3}
→ IR is continuous, 2 <
α
< 3 and
denotes the Caputo’s fractional derivative. The twopoint boundary value problem (1.1) happens to be at resonance in the sense that its associated linear homogeneous boundary value problem
has a nontrivial solution
u
(
t
) =
ct
^{2}
,
c
∈ IR. To solve this problem, we use the coincidence degree of Mawhin
[12
,
13]
. This method is based on an equivalent formulation in an abstract space and a theory of topological degree. This formulation generally leads to an abstract operator of the form
N
+
L
, where
L
is a Fredholm operator of index zero and
N
is generally a nonlinear operator having some compactness properties with respect to
L
. Fractional differential equations arise in different areas of sciences such as in rheology, fluid flows, viscoelasticity, chemical physics, and so on
[1
,
6
,
8
,
11
,
14

19]
. Recently, boundary value problems for fractional differential equations at nonresonance have been studied by many authors
[1
,
3
,
4
,
10]
by using fixed point theorems, lower and upper solution. Moreover, boundary value problems for differential equations at resonance have also been studied in some papers, see
[2
,
7]
. In
[20]
, the authors studied, by using the coincidence degree theory, the following BVP of fractional equation at resonance
where
denotes the Caputo’s fractional differential operator of order α, 1 <
α
≤ 2 and
f
: [0, 1] × IR
^{2}
→ IR is a continuous function.
A similar boundary value problem at resonance involving RiemannLiouville fractional derivative is considered in
[7]
. The author solved the problem
by applying degree theory theorem for coincidences.
By the same method, in
[5]
the authors established the existence of solutions for the following thirdorder differential equation
where
f
: [0, 1] × IR
^{2}
→ IR is a Caratheodory function, and
η
∈ (0, 1).
The organization of this paper is as follows. We present in Section 2 some notation and some basic results involved in the reformulation of the problem. In section 3, we give the main theorem and some lemmas, then we will see that the proof of the main theorem is an immediate consequence of these lemmas and the coincidence degree of Mawhin. At the end of this section, we give an example to illustrate the main result.
2. Preliminaries
We begin by introducing the fundamental tools of fractional calculus and the coincidence degree theory which will be used throughout this paper.
Definition 2.1.
The RiemannLiouville fractional integral is defined by
for
g
∈
C
([
a
,
b
]) and
α
> 0.
Definition 2.2.
Let
f
∈
C^{n}
([
a
,
b
]), the Caputo fractional derivative of order
α
≥ 0 of
f
is defined by
, where
n
= [
α
] + 1 ([
α
] is the integer part of
α
).
Lemma 2.3.
For
α
> 0,
g
∈
C
([0, 1], IR)
, the homogenous fractional differential equation
has a solution g
(
t
) =
c
_{0}
+
c
_{1}
t
+
c
_{2}
t
^{2}
+…+
c
_{n−1}
t
^{n−1}
, where c_{i}
∈ IR
, i
= 0,…,
n
−1,
here n is the smallest integer greater than or equal to α
.
Definition 2.4.
Let
X
and
Y
be real normed spaces. A linear mapping
L
:
domL
⊂
X
→
Y
is called a Fredholm mapping if the following two conditions hold:
(i)
kerL
has a finite dimension, and
(ii)
ImL
is closed and has a finite codimension.
If
L
is a Fredholm mapping, its index is the integer
IndL
=
dimkerL
−
codimImL
.
From here if
L
Fredholm mapping of index zero then there exist continuous projectors
P
:
X
→
X
and
Q
:
Y
→
Y
such that
ImP
=
KerL
,
KerQ
=
ImL
,
X
=
KerL
⊕
KerP
,
Y
=
ImL
⊕
ImQ
and that the mapping
L

_{domL∩KerP}
:
domL
∩
KerP
→
ImL
is onetoone and onto. We denote its inverse by
K_{P}
. Moreover, since
dimImQ
=
codimImL
, there exists an isomorphism
J
:
ImQ
→
KerL
.
If Ω is an open bounded subset of
X
, and
, the map
N
:
X
→
Y
will be called
L−compact
on
if
QN
(
) is bounded and
K_{P}
(
I
−
Q
)
N
:
→
X
is compact.
Lemma 2.5
(
[13]
).
Let L
:
domL
⊂
X
→
Y be a Fredholm operator of index zero and N
:
X
→
Y
,
L
−
compact on
.
Assume that the following conditions are satisfied
(1) Lu
≠
Nu for every
(
u
,
λ
) ∈ [(
domL KerL
)] ∩ ∂Ω × (0, 1);
(2) Nx
∉
ImL for every x
∈
KerL
∩ ∂Ω;
(3) deg
(
JQN

_{KerL}
,
KerL
∩ Ω, 0) ≠ 0, where
Q
:
Y
→
Y is a projection such that ImL
=
KerQ
.
Then the equation Lx = Nx has at least one solution in domL ∩
.
In this paper, we denote by
X
and
Y
the Banach spaces:
X
=
C
^{2}
([0, 1], IR) equipped with the norm ║
u
║
_{X}
=
max
{║
u
║
_{∞}
; 
u^{′}
║
_{∞}
, ║
u^{′′}
║∞} and
Y
=
C
([0, 1], IR) equipped with the norm ║
y
║
_{Y}
= ║
y
║
_{∞}
, where ║
u
║
_{∞}
=
max_{t}
∈[0,1] 
u
(
t
).
Define the operator
L
:
domL
⊂
X
→
Y
by
where
Let
N
:
X
→
Y
be the operator
Then BVP (1.1) is equivalent to the operator equation
It is known
[12]
that the coincidence equation
Lu
=
Nu
is equivalent to
3. Main results
We can now state our result on the existence of a solution for the BVP (1.1).
Theorem 3.1.
Let f
: [0, 1] × IR
^{3}
→ IR
be continuous. Assume that
(
H
_{1}
)
there exists nonnegative functions p
,
q
,
r
,
z
∈
C
[0, 1]
with
Γ(
α
−1)−2
_{q1}
−2
_{r1}
− 2
_{z1}
> 0
such that

f
(
t
,
u
,
v
,
w
) ≤
p
(
t
) +
q
(
t
)
u
 +
r
(
t
)
v
 +
z
(
t
)
w
, ∀
t
∈ [0, 1], (
u
,
v
,
w
) ∈ IR
^{3}
where
(
H
_{2}
)
there exists a constant B
> 0
such that for all c
∈ IR
with
2
c
 >
B either
or
Then BVP(1.1) has at least one solution in X
.
In order to prove Theorem 3.1, we need to prove some lemmas below.
Lemma 3.2.
Let L be defined by (2.1), then
Proof
. By Lemma 2.3,
has solution
Combining with boundary condition of BVP (1.1), (3.1) holds. For
y
∈
ImL
, there exists
u
∈
domL
such that
y
=
Lu
∈
Y
. We have
Differentiating two times and using the boundary conditions for BVP (1.1), we get
On the other hand, let
y
∈
Y
satisfying
, so
y
∈
ImL
. □
Lemma 3.3.
If L be defined by (2.1), then L is a Fredholm operator of index zero and the linear continuous projector operators P
:
X
→
X and Q
:
Y
→
Y can be defined as
Furthermore, the operator K_{p}
:
ImL
→
domL
∩
KerP can be written as
Proof
. Obviously,
ImP
=
KerL
and
P
^{2}
u
=
Pu
. It follows from
u
= (
u
−
Pu
) +
Pu
that
X
=
KerP
+
KerL
. By simple calculation, we get that
KerL
∩
KerP
= 0. Hence
For
y
∈
Y
, it yields
Let
y
= (
y
−
Qy
)+
Qy
, where
y
−
Qy
∈
KerQ
=
ImL
,
Qy
∈
ImQ
. It follows from
KerQ
=
ImL
and
Q
^{2}
y
=
Qy
that
ImQ
∩
ImL
= 0. Then, we have
Thus
This means that
L
is a Fredholm operator of index zero. From the definitions of
P
and
K_{P}
, it is easy to see that the generalized inverse of
L
is
K_{P}
. In fact, for
y
∈
ImL
, we have
Moreover, for
u
∈
domL
∩
KerP
, we get
u
(0) =
u^{′}
(0) =
u^{′′}
(0) = 0. By Lemma 2.3, we obtain that
This together with
u
(0) =
u^{′}
(0) =
u^{′′}
(0) = 0 yields
Combining (3.3) with (3.4), we get
K_{P}
= (
L

_{domL∩KerP}
)
^{−1}
. □
Lemma 3.4.
If
Ω ⊂
X is an open bounded subset such that
,
then N is L − compact on
.
Proof
. By the continuity of
f
, we conclude that
QN
(
) and
K_{P}
(
I
−
Q
)
N
(
) are bounded. In view of ArzelàAscoli theorem, we need only to prove that
K_{P}
(
I
−
Q
)
N
(
) ⊂
X
is equicontinuous. The continuity of
f
implies that exists a constant
A
> 0 such that (
I
−
Q
)
Nu
 ≤
A
, ∀
u
∈
, ∀
t
∈ [0, 1]. Denote
K_{P,Q}
=
K_{P}
(
I
−
Q
)
N
and let 0 ≤
t
_{1}
≤
t
_{2}
≤ 1,
u
∈ (
), then
and
Since
t^{α}
,
t
^{α−1}
and
t
^{α−2}
are uniformly continuous on [0, 1], we get that
K_{P,Q}
(
), (
K_{P,Q}
)
^{′}
(
)(
K_{P,Q}
)
^{′′}
(
) ∈
C
[0, 1] are equicontinuous. Hence
K_{P,Q}
:
→
X
is compact. □
Lemma 3.5.
Suppose
(
H
_{1}
)
holds, then the set
is bounded
.
Proof
. Take
u
∈ Ω
_{1}
, then
Nu
∈
ImL
. By (3.2), it yields
By the integral mean value theorem, we conclude that there exists a constant
h
∈ (0, 1) such that
f
(
h
,
u
(
h
),
u^{′}
(
h
),
u^{′′}
(
h
)) = 0. Then from (
H
_{2}
), we have 
u^{′′}
(
h
) ≤
B
.
Since
u
∈
domL
, then
u
(0) =
u^{′}
(0) = 0. Therefore
and
That is
By
Lu
=
λNu
and
u
∈
domL
, we have
and
Take
t
=
h
, we get
Together with 
u^{′′}
(
h
) ≤
B
, (
H
_{1}
) and (3.5), we have
Then we have
Since Γ(α − 1) − 2
_{q1}
− 2
_{r1}
− 2
_{z1}
> 0, we obtain
and
Therefore, ║
u
║
_{X}
≤
M
, consequently Ω
_{1}
is bounded. □
Lemma 3.6.
Suppose
(
H
_{2}
)
holds, then the set
is bounded
.
Proof
. For
u
∈ Ω
_{2}
, we have
u
(
t
) =
ct
^{2}
,
c
∈ IR, and
Nu
∈
ImL
. Then we get
this together with (
H
_{2}
) implies 2
c
 ≤
B
. Thus, we have ║
u
║
X
≤
B
. Hence, Ω
_{2}
is bounded. □
Lemma 3.7.
Suppose the first part of
(
H
_{2}
)
holds, then the set
is bounded. Where J
:
KerL
→
ImQis an isomorphism defined by J
(
ct
^{2}
) =
ct
^{2}
; ∀
c
∈ IR;
t
∈ [0, 1]:
Proof
. For
u
∈ Ω
_{3}
, we have
u
(
t
) =
ct
^{2}
,
c
∈ IR, and
If
λ
= 0, then
f
(
s
,
cs
^{2}
, 2
cs
, 2
c
)
ds
= 0, thus 2
c
 ≤
B
in view of the first part of (
H
_{2}
). If
λ
∈ (0, 1], we can also obtain 2
c
 ≤
B
. Otherwise, if 2
c
 >
B
, in view of the first part of (
H
_{2}
), one has
which contradicts (3.6). Therefore, Ω
_{3}
is bounded. □
Lemma 3.8.
Suppose the second part of
(
H
_{2}
)
hold, the set
is bounded
.
Proof
. Using similar argument as in the proof of Lemma 3.7, we prove that Ω
^{′}
_{3}
is bounded. □
Now we are able to give the proof of Theorem 3.1, which is an immediate consequence of Lemmas 3.23.8 and Lemma 2.5.
Proof
. Set Ω = {
u
∈
X
 ║
u
║
_{X}
<
max
{
M
,
B
} + 1}. It follows from Lemma 3.2 and Lemma 3.3 that
L
is a Fredholm operator of index zero and
N
is
L−compact
on
. By Lemma 3.4 and Lemma 3.5, we get that the following two conditions are satisfied
(1)
Lu
≠
Nu
for every (
u
,
λ
) ∈ [(
domL
KerL
)] ∩ ∂Ω × (0, 1);
(2)
Nu
∉
ImL
for every
u
∈
KerL
∩ ∂Ω;
Define
H
(
u
,
λ
) =
λIdu
+ (1 −
λ
)
JQNu
: According to Lemma 3.6 (or Lemma 3.7), we see that
H
(
u
,
λ
) ≠ 0 for
u
∈
KerL
∩ ∂Ω. By the degree property of invariance under a homotopy, it yields
So that, the condition (3) of Lemma 2.1 is satisfied. By Lemma 2.1, we can get that
Lu
=
Nu
has at least one solution in
domL
∩
. Therefore by Lemma 2.5, the BVP (1.1) has at least one solution. The proof is complete. □
Example 3.9.
Let us consider the following fractional boundary value problem
We have
f
(
t
,
u
,
v
,
w
) = (
, then
where
p
(
t
) = 3
t
,
q
(
t
) = 0,
. We get that
q
_{1}
= 0,
and
If we choose
B
= 15 then for 2
c
 >
B
it yields
Then, the conditions of Theorem 3.1 are satisfied, so BVP(3.7) has at least one solution.
BIO
A. GuezaneLakoud Her research interests are on differential equations and their applications.
Department of Mathematics, Faculty of Sciences, Badji MokhtarAnnaba University, P.O. Box 12, 23000 Annaba, Algeria.
email: a_guezane@yahoo.fr
S. Kouachi Her research interests are on differential equations and their applications.
Department of Mathematics, Faculty of Sciences, Badji MokhtarAnnaba University, P.O. Box 12, 23000 Annaba, Algeria.
email: sa.kouachi@gmail.com
F. Ellaggoune His research interests in difference equations and its applications.
Department of Mathematics, University 8 mai 1945  Guelma P.O. Box 401, Guelma 24000, Algeria.
email: fellaggoune@gmail.com
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