On Existence of Multiplicity of Weak Solutions for a New Class of Nonlinear Fractional Boundary Value Systems via Variational Approach

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 ℝ 3 and ℝ 4 are given to illustrate the theoritical results.


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 ). 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 0 D α T and t D α T are the left and right Riemann-Liouville fractional derivatives with 0 < α ≤ 1, respectively, and F : ½0, T × ℝ ⟶ ℝ n is a suitable function satisfying some hypothesis and Fðt, xÞ is the gradient of F with respect to x: In [22], the authors have used variational methods to investigate the existence of weak solutions for the following system: Þ , a:e t ∈ 0, T ½ , for 0 D α T and t D α T are the left and right Riemann-Liouville fractional derivatives with 0 < α ≤ 1 and F s denotes the par-tial derivative of F with respect to s: In [?], Zhao et al. obtained the existence of infinitely many solutions for system (2) with perturbed functions h i , i = 1, 2.
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: T are the left and right Riemann-Liouville fractional derivatives of order α i , respectively, a i ∈ L ∞ ð½0, TÞ with λ > 0, F : ½0, T × ℝ n ⟶ ℝ is a measurable function for all ðx 1 , ⋯, x n Þ ∈ ℝ n and is C 1 with respect to ðx 1 , ⋯, x n Þ ∈ ℝ n for a.e. t ∈ ½0, T, F u i denotes the partial derivative of F with respect to u i , respectively, and h i : ℝ ⟶ ℝ are Lipschitz continuous functions with the Lipschitz constants L i > 0, for 1 ≤ i ≤ n, i.e., for all x 1 , x 2 ∈ ℝ and h i ð0Þ = 0, for 1 ≤ i ≤ n. In order to state the main results, we introduce the following conditions: (F0) For all C > 0 and any 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.

Preliminaries
In this section, introducing some necessary definitions and preliminary facts.
Definition 1 [28]. Let u be a function defined on ½0, T and α i > 0 for 1 ≤ i ≤ n: The left and right Riemann-Liouville fractional integrals of order α i for the function u are defined by for 1 ≤ i ≤ n, provided the RHS are pointwise given on ½0, T, where Γðα i Þ is the standard gamma function defined by Definition 2 [25]. Let 0 < α i ≤ 1 for 1 ≤ i ≤ n: The fractional derivative space H α i 0 is given by the closure C ∞ 0 ð½0, T, ℝÞ, that is with the norm for every u i ∈ H α i 0 and for 1 ≤ i ≤ n: is a reflexive and separable Banach space (see [22], Proposition 3.1) for details.
For every u i ∈ H Definition 3 [27]. We mean by a weak solution of system (3), any u = ðu 1 , u 2 , ⋯, u n Þ ∈ X such that for Advances in Mathematical Physics Lemma 4 [27]. Moreover, From Lemma 4, we easily observe that for 0 < α i ≤ 1, and By using (15), the norm of (10) is equivalent to Throughout this paper, let X be the Cartesian product of the n spaces H where ku i k H α i 0 is given in (17). We have X compactly embedded in Cð½0, T, ℝÞ n : Theorem 5 [25]. Let X be a reflexive real Banach space and Φ : X ⟶ ℝ be a coercive, continuously Gâteaux differentiable sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X * , bounded on bounded subsets of X,Ψ : X ⟶ ℝ a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Suppose that ∃r > 0 and x ∈ X, with r < Φð xÞ, satisfying (a 1 ) sup ΦðuÞ≤r ðΨðuÞ/rÞ < ðΦð xÞ/Ψð xÞÞ.

Main Results
In this section, by applying Theorem 5, we examine the existence of multiple solutions for system (3). For any σ > 0, let us define This set will be used in some of our hypotheses with appropriate choices of σ. For u = ðu 1 , u 2 , ⋯, u n Þ ∈ X, we define where Theorem 6. Let 1/2 < α i ≤ 1, for 1 ≤ i ≤ n, and suppose that M > 0 and the conditions (F0) and (F1) are satisfied. Furthermore, assume that ∃r > 0 and a function ω = ðω 1 , Then, setting ∀λ ∈ Λ system (3) admits at least 3 weak solutions in X.
It is clear that Φ and Ψ are continuously Gâteaux differentiable functionals whose Gâteaux derivatives at the point u ∈ X are defined by for every v = ðv 1 , v 2 , ⋯, v n Þ ∈ X: We have Φ ′ ðuÞ, Ψ ′ ðuÞ ∈ X * , where X * is the dual space of X. And the functional Φ is sequentially weakly lower semicontinuous and its Gâteaux derivative admits a continuous inverse on X * ; also lim kuk X ⟶+∞ ΦðuÞ = +∞ it is coercive. Now, we show that the functional Ψ is sequentially weakly upper semicontinuous and its derivative Ψ′ : X ⟶ X * is a compact operator. Let u m ⇀ u in X, where u m ðtÞ = ðu m,1 ðtÞ, u m,2 ðtÞ, ⋯, u m,n ðtÞÞ; then certainly u m converges uniformly to u on the interval ½0, T. Then, which gets that Ψ is sequentially weakly upper semicontinuous.
On the other hand, fix 0 < ε < ð1/2TkλÞ. From (iii) into account, there exist constants τ ε ∈ ℝ such that for any t ∈ ½0, T and ðx 1 , ⋯, x n Þ ∈ ℝ n , by using (36) and (15) yields, it follows that, for each u ∈ X, And from him, Moreover, analogous to the case of τ ε > 0, we imply that ΦðuÞ − λΨðuÞ ⟶ +∞ as kuk X ⟶+∞ with τ ε ≤ 0. Then, the hypotheses of Theorem 5 hold, which means that system (3) admits at least 3 weak solutions in X, which completes the proof. Now, we present some notations, before the corollary of Theorem 6. Put iii ′ lim Then, setting Thus, system (3) admits at least three weak solutions in X .
6 Advances in Mathematical Physics 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 Λ ′ ⊆ Λ. Theorem 6 is successfully employed to ensure the existence of at least 3 weak solutions for system (3). This completes of the proof.

Examples
In this section, we propose two practical examples of Theorem 6.