Abstract
This paper deals with the existence of solutions for a new class of nonlinear fractional boundary value systems involving the left and right Riemann-Liouville fractional derivatives. More precisely, we establish the existence of at least three weak solutions for the problem using variational methods combined with the critical point theorem due to Bonano and Marano. In addition, some examples in and are given to illustrate the theoritical results.
1. Introduction
Fractional differential equations (FDEs) are a generalization of ordinary differential equations (ODEs), as they contain fractional derivatives whose degree is not necessarily an integer. This is what makes it receive great attention from researchers due to its ability to model some difficult and complex phenomena in many fields, including engineering, science, biology, economics, and physics (for more information, see [1–22]). One of the most investigated issues is the existence of solutions for the fractional initial and boundary value problems by using some fixed point theorems, coincidence degree theory, and monotone interactive method. Among the most important of these are the works mentioned in Oldham and Spanier and Podlubny’s books (see [13, 23]) and the work of Metzler and Klafter (see [24]). Furthermore, the first to use the critical point theorem was Jiao and Zhou in [6] to study the following problem: where and are the left and right Riemann-Liouville fractional derivatives with , respectively, and is a suitable function satisfying some hypothesis and is the gradient of with respect to
In [22], the authors have used variational methods to investigate the existence of weak solutions for the following system: for and are the left and right Riemann-Liouville fractional derivatives with and denotes the partial derivative of with respect to In [?], Zhao et al. obtained the existence of infinitely many solutions for system (2) with perturbed functions .
Yet, there are a few findings for fractional boundary value problems which were established exploiting this approach due to its difficulty in establishing a suitable space and variational functional for fractional problems.
In this work, we shall study the existence of three weak solutions for the following system: for , where , and are the left and right Riemann–Liouville fractional derivatives of order , respectively, with
, is a measurable function for all and is with respect to for a.e. , denotes the partial derivative of with respect to respectively, and are Lipschitz continuous functions with the Lipschitz constants , for , i.e., for all and , for . In order to state the main results, we introduce the following conditions:
(F0) For all and any
(F1) , for a.e. .
In the present study, motivated by the results introduced in [12, 13, 25], using the three critical point theorems due to Ricceri ([26], see Theorem 2.6 in the next section), we ensure the existence of at least three solutions for system (3). For other applications of Ricceri’s result for perturbed boundary value problems, the interested readers are referred to the papers [11–13, 23–25, 27].
We divided the paper as follows: in the second section, we put some preliminary facts, while in the third section we presented the main result and its proof. Finally, we proposed two practical examples of our theorem.
2. Preliminaries
In this section, introducing some necessary definitions and preliminary facts.
Definition 1 [28]. Let be a function defined on and for The left and right Riemann–Liouville fractional integrals of order for the function are defined by for , provided the RHS are pointwise given on , where is the standard gamma function defined by
Definition 2 [25]. Let for The fractional derivative space is given by the closure , that is with the norm for every and for
We point out that () is a reflexive and separable Banach space (see [22], Proposition 3.1) for details.
For every , set
Definition 3 [27]. We mean by a weak solution of system (3), any such that for all ,
Lemma 4 [27]. Let , for . , we have
Moreover,
From Lemma 4, we easily observe that for , and
By using (15), the norm of (10) is equivalent to
Throughout this paper, let be the Cartesian product of the spaces for , i.e., ; we equip with the norm defined by where is given in (17). We have compactly embedded in
Theorem 5 [25]. Let be a reflexive real Banach space and be a coercive, continuously Gâteaux differentiable sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on , bounded on bounded subsets of a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that
Suppose that and , with , satisfying
(a1) .
(a2) For each , the functional is coercive.
Hence, , the functional has at least three critical points in the space .
3. Main Results
In this section, by applying Theorem 5, we examine the existence of multiple solutions for system (3). For any , let us define
This set will be used in some of our hypotheses with appropriate choices of . For , we define where and , and .
Furthermore, let
Theorem 6. Let , for , and suppose that and the conditions (F0) and (F1) are satisfied. Furthermore, assume that and a function satisfying
Then, setting
system (3) admits at least 3 weak solutions in .
Proof. For each , we introduce the functionals as
It is clear that and are continuously Gâteaux differentiable functionals whose Gâteaux derivatives at the point are defined by
for every
We have , where is the dual space of . And the functional is sequentially weakly lower semicontinuous and its Gâteaux derivative admits a continuous inverse on ; also it is coercive. Now, we show that the functional is sequentially weakly upper semicontinuous and its derivative is a compact operator. Let in , where ; then certainly converges uniformly to on the interval . Then,
which gets that is sequentially weakly upper semicontinuous.
Moreover, we have
Note that . The Lebesgue control convergence theorem implies that strongly, hence yielding that is strongly continuous on . Then, is a compact operator.
We show that required hypothesis follows from (i) and the definition of by taking Indeed, as (5) holds for all and ; one has , for any . It follows from (15) that
From (16), for every , we have
for . Hence,
Assume that and the supposition (i) deduces that and they hold from definitions (25) and (26), which are required assumptions in Theorem 5. Applying relations (16), (17), and (22) gives the following relation: which implies that
Hence, under the condition (ii), we get the following inequality
Thus, the hypothesis (a1) of Theorem 5 holds.
On the other hand, fix . From (iii) into account, there exist constants such that for any and , by using (36) and (15) yields, it follows that, for each ,
And from him,
Moreover, analogous to the case of , we imply that as with . Then, the hypotheses of Theorem 5 hold, which means that system (3) admits at least 3 weak solutions in , which completes the proof.
Now, we present some notations, before the corollary of Theorem 6.
Put
Corollary 7. Let and supposition (iii) in Theorem 6 holds. Suppose that and such that , and also Then, setting
Thus, system (3) admits at least three weak solutions in .
Proof. Choose We derive
Moreover,
Then, , ; hence, , and we have
By (25), for , imply that
Similar to (30) and (46), we have .
Let . From , we have
Thus, the assumption (ii) of Theorem 6 holds.
(i) implies that
Moreover, by condition (ii) we have
Hence, the supposition (ii) of Theorem 6 is verified.
Moreover, the supposition (iii) of Theorem 6 holds under (iii) from . Theorem 6 is successfully employed to ensure the existence of at least 3 weak solutions for system (3). This completes of the proof.
4. Examples
In this section, we propose two practical examples of Theorem 6.
Example 1. Let . Then, system (3) gets the following form:
where , and .
Furthermore, ; put
where
Obviously are three Lipschitz continuous functions with Lipschitz constants and . Clearly, , by the direct calculation, we have , , and Taking
By a simple calculation, we obtain
Select , we find
We deduce that the supposition (i) holds, and
Then, suppositions (ii) and (iii) are verified. Hence, in view of Theorem 6 for every , system (50) has at least 3 weak solutions in the space .
Example 2. Let , ,
Hence, system (3) gives
Taking
Moreover, for all , put
where
Obviously are three Lipschitz continuous functions, and for all with Lipschitz constants and . Clearly, for any , , , , and
The direct calculation, gives
So that
Select ; we find
We deduce that the supposition (i) holds, and
Then, suppositions (ii) and (iii) are verified. Hence, in view of Theorem 6 for every , system (58) has at least 3 weak solutions in the space .
5. Conclusion
In this work, at least 3 weak solutions were obtained for a new class of nonlinear fractional BVPs using a critical three-point theorem due to Bonano and Marano. Some appropriate function spaces and variational frameworks were successfully created for system (3). Finally, we suggested two practical examples of Theorem 6 with a special case discussion . As for case , it was discussed. This makes our results prominent and distinct than previous ones. In the next work, we extend our recent work to the coupled system for this important problem. Also some numerical examples will be given in order to ensure the theory study by using some famous algorithms which are presented in ([28, 29]).
Data Availability
No data were used to support the study.
Conflicts of Interest
This work does not have any conflicts of interest.
Acknowledgments
The sixth author extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Project under Grant No. R.G.P-2/1/42.