1. Introduction
Fractional differential equations have proved to be valuable tools in modeling many phenomena in various fields of physics, chemistry, biology, engineering and economics. There was a significant development in fractional differential equations. We can see the studies of Miller and Ross [
1], Samko et al. [
2], Podlupny [
3], Hilfer [
4], Kelpas et al. [
5] and papers [
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16] and references therein.
The critical point theory was very useful in determining the existence of solutions to complete differential equations with certain boundary conditions; see for example, in the extensive literature on the subject, classical books [
17,
18,
19] and references appearing there. However, so far, some problems have been created for fractional marginal value problems (briefly BVP) by exploiting this approach, where it is often very difficult to create a suitable space and a suitable function for fractional problems.
In [
20], the authors investigated the following nonlinear fractional differential equation depending on two parameters:
for
where
,
and
are the left and right Riemann-Liouville fractional derivatives of order
respectively,
with
are positive parameters,
are measurable functions with respect to
for every
and are
with respect to
for a.e.
and
denotes the partial derivative of
F and
G with respect to
respectively, and
:
are Lipschitz continuous functions with the Lipschitz constants
for
i.e.,
for every
and
for
Motivated by [
21,
22], using a three critical points theorem obtained in [
23] which we recall in the next section (Theorem 2.6), we ensure the existence of at least three solutions for this system.
For example, according to some assumptions, in [
24], by using variational methods the authors obtained the existence of at least one weak solution for the following
Laplacian fractional differential equation [
24]
where
and
are the left and right Riemann-Liouville fractional derivatives with
, respectively, the function
. Taking a class of fractional differential equation with
Laplacian operator as a model, Li et al. investigated the following equation recently [
25]
with
is a non-negative real parameter.
The functions
is continuous
is a Lipschitz continuous function.
By using the mountain pass theorem combined with iterative technique, the authors obtained the existence of at least one solution for problem (3).
In this paper, we are interested in ensuring the existence of three weak solutions for the following system
where
with
and
are the left and right Riemann-Liouville fractional derivatives of order
respectively, for
is positive parameter, and
is measurable function with respect to
for every
and are
with respect to
for a.e.
,
denote the partial derivative of
F with respect to
respectively,
for
F : be a function such that is continuous in for every , is a function in
In the present paper, motivated by [
26,
27], using a three critical points theorem obtained in [
23], which we recall in the next section (Theorem
), we ensure the existence of at least three solutions for system (4). This theorem has been successfully employed to establish the existence of at least three solutions for perturbed boundary value problems in the papers ([
26,
27,
28,
29,
30]).
This paper is organized as follows. In
Section 2, we present some necessary preliminary facts that will be needed in the paper. In
Section 3, we prove our main result. In
Section 4, we give two numerical examples in order to support the theory of our contribution.
2. Preliminaries
In this section, we first introduce some necessary definitions and preliminary facts are introduced for fractional calculus which are used in this paper.
For , let be the real space of all continuous functions with norm and be the space of functions for which the th power of the absolute value is Lebesgue integrable with norm
Definition 1 (Kilbas et al. [
5]).
Let u be a function defined on The left and right Riemann-Liouville fractional derivatives of order for a function u are defined by and
for every
provided the right-hand sides are pointwise defined on
where
and
Here,
is the standard gamma function given by
Setting the space of functions such that and . Here, as usual, denotes the set of mappings being times continuously differentiable on In particular, we denote
Definition 2 ([
25]).
Let , for , The fractional derivative spaceThen, for any , we can define the weighted norm for as Definition 3 ([
31]).
We mean by a weak solution of system (4), any such that for all Lemma 1 ([
31]).
Let for , For any we haveMoreover, if and then From Lemma 1, we easily observe thatfor , andfor and By using (8), the norm of (5) is equivalent to
Throughout this paper, we let
X be the Cartesian product of the
n spaces
for
i.e.,
equipped with the norm
where
is defined in (10) Obviously,
X is compactly embedded in
Lemma 2 ([
32]).
For and , the fractional derivative space X is a reflexive separable Banach space. Lemma 3 ([
33]).
Let A be a monotone, coercive and hemicontinuous operator on the real, separable, reflexive Banach space Assume is a basis in X. Then the following assertion holds:If A is strictly monotone, then the inverse operator exists. This operator is strictly monotone, demicontinuous and bounded. If A is uniformly monotone, then is continuous. If A is strongly monotone, then it is Lipschitz continuous.
Theorem 1 ([
34]).
Let X be a reflexive real Banach space; be a coercive ,continuously Gateaux differentiable sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on bounded on bounded subsets of a continuously Gateaux differentiable functional whose Gateaux derivative is compact such thatAssume that there exists and with such that
For each the functional is coercive.
Then, for any , the functional has at least three critical point in X.
3. The Main Results
In the present section ,the existence of multiple solutions for system (4) is examined by using Theorem 1.
First and foremost, we define the functionals
as
and
Lemma 4. Let Functionals Φ and are defined in (11) and (12). Then, is a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functionl whose Gâteaux derivative admits a continuous inverse on , and is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.
Proof. For each
, define
as
and
Clearly,
and
are continuously Gateaux differentiable functionals whose Gateaux derivatives at the point
are given by
for every
In addition, according to (11), one has which means that is a coercive functional.
Next, we claim that
admits a continuous inverse on
Let
Recalling (13),we get
According to the well-known inequality
Hence, when
one has
which means that
Then, combining with (18), yields
For every
Then, reapplying inequality (15), we always have
and
That is,
for every
. Therefore, by using (14) and (20), the following inequality holds
which means that
is strictly monotone. Furthermore, in view of
X being reflexive, for
in
X strongly, as
one has
in
as
Thus, we say that is demicontinuous. Then, according to lemma 2 and 3, we obtain that the inverse operator of exists and is continuous.
Moreover , let
owing to the sequentially weakly lower semicontinuity of
we observe that
is sequentially weakly lower semicontinuous in
X.
Considering the functional we will point out that is a Gâteaux differentiable, sequentially weakly upper semicontinuous functional on X.
Indeed, for
, assume that
in
i.e.,
uniformly converges to
u on
as
. By using reverse Fatou’s lemma, one has
whereas
which implies that
is sequentially weakly upper semicontinuous. Furthermore, since
F is continuously differentiable with respect to
u and
v for almost every
then based on the Lebesgue control convergence theorem, we obtain that
strongly , that is
is strongly continuous on
Hence, we confirm that
is a compact operator.
Moreover, it is easy to prove that the functional with the Gâteaux derivative
at the point
for any
The proof is completed. □
In order to facilitate the proof of our main result, some notations are given.
Theorem 2. Let for Assume that there exists a positive constant r and a function such that
Then, setting
for each
system (4) admits at least three weak solutions in
X.
Proof. Considering Theorem 1 and Lemma 5, in order to obtain that system (4) possesses at least three weak solutions in
X, we only need to guarantee the assumptions
and
of Theorem 1 are satisfied. Choose
and
with
Due to (12) and
, we get
and
which satisfy the requirement of Theorem 1. Then, combining (11) and (9) , yields
which implies that
Then, the following inequality is obtained under condition
Thus the hypothesis of Theorem 1 holds.
On the other hand, taking
into account , there exist constants
with
such that
for any
and
, when
by using (11), (22) and (8) yields
Furthermore, analogous to the case of , we can deduce that as with . Hence, all the hypotheses of Theorem 1 hold, then , system (4) admits at least three weak solutions in The proof is completed.
For simplicity, before giving a corollary of Theorem 2, some notations are presented.
Corollary 1. Let Assume that there exist and with and such that
Thus, system (4) admits at least three weak solutions in X.
Proof. Obviously
. Owing to Definition 1 , we derive,
where
and
That is
where (23) is used. Hence
.
Take
, then
for every
which means that
Thus, the assumption of Theorem 2 holds.
On the other hand, based on (7) and (23), yields
Then, from (25) and
we can obtain the following inequality
which means that the hypothesis
of Theorem 2 is satisfied.
Furthermore, the condition of Theorem 2 holds under since Theorem 2 is successfully employed to ensure the existence of at least three weak solutions for system (4), the proof is completed. □
4. Numerical Examples
Now, we give the following two examples to illustrate the applications of our result:
Example 1. Let . Then, system (4) gets the following form Clearly, and for any .
By the direct calculation, we have
and
Take
. We easily obtain that
which implies that the condition
holds, and
and
Thus, conditions and are satisfied. Then, in view of Theorem 2 for each , the system (4) has at least three weak solutions in X.
Example 2. Let and . Then , system (4) gets the following form Clearly, and for any
By the direct calculation, we have
and
Take
We easily obtain that
which implies that the condition
holds, and
and
Thus, conditions and are satisfied. Then, in view of Theorem 2 for each , the system (4) admits at least three weak solutions in X.
5. Conclusions
Fractional differential equations have been carefully investigated. Such problems were studied in many scientific and engineering applications such as models for various processes in plasma physics, biology, medical science, chemistry, chemical engineering, as well as population dynamics, and control theory. In the current contribution, motivated by work in ([
21,
22]) and using a three critical points theorem obtained in [
23], we could ensure the existence of at least three solutions for system (4). Note that some appropriate function spaces and variational frameworks were successfully created for the system (4). Furthermore, we have given two examples to illustrate the application of Theorem 2 we have discussed for the special case of
, and the discussions presented with respect to the case of
, which highlighted the superiority of our results. In next work we will use two control parameters to study a class of perturbed nonlinear fractional
p-Laplacian differential systems where we will try to prove the existence of three weak solutions by using the variational method and Ricceri’s critical points theorems respecting some necessary conditions.