A New Proof of the Existence of Nonzero Weak Solutions of Impulsive Fractional Boundary Value Problems

Asma Alharbi 1,†, Rafik Guefaifia 2,† and Salah Boulaaras 1,3,*,† 1 Department of Mathematics, College of Sciences and Arts, ArRass, Qassim University, Buraidah 51452, Saudi Arabia; ao.alharbi@qu.edu.sa 2 Department of Mathematics, Faculty of Exact Sciences, University Tebessa 12002, Tebessa 12002, Algeria; rafik.guefaifia@univ-tebessa.dz 3 Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Oran 31000, Algeria * Correspondence: saleh_boulaares@yahoo.fr or S.Boularas@qu.edu.sa; Tel.: +966-559-618-327 † These authors contributed equally to this work.


Introduction
The paper deals with the existence of weak solutions to the following boundary value problems for impulsive fractional Hamiltonian differential equations: where 0 ≤ α < 1, c 0 D α t and t D α T denote the left Caputo fractional derivative and the right Riemann-Liouville fractional derivative of order α, respectively, T > 0, , A := {1, . . . , N} , B := {1, . . . , L} , u (t) = u 1 (t) , u 2 (t) , . . . , u N (t) , 0 = t 0 < t 1 < t 2 < · · · < t L < t L+1 = T, ∇F (t, x) denotes the gradient of F (t, x) in x, and the operator is defined as and c 0 D α t is the left Caputo fractional derivatives of order α. I ij : R → R, i ∈ A, j ∈ B are continuous with F : [0, T] × R N → R satisfies the following condition: (A) F (t, u) is measurable according to t for any u ∈ R N and continuously differentiable according u for a.e. t ∈ [0, T] . However, there exist a ∈ C (R + , R + ) and b ∈ L 1 ([0, T] , R + ) , such that for any u ∈ R N and a.e. t ∈ [0, T] .
Fractional differential equations have recently proved to be valuable tools for modeling many phenomena in various fields of science and engineering. Indeed, one can find many applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc., for example, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Moreover, recently, the existence of solutions of boundary value problems for Fractional differential equations have widely been studied in many papers and we refer the reader to the papers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein. For instance, in [3], the authors created a variable structure and, using the critical point theory, investigated the existence of multiple solutions for a class of fractional advection-dispersion equations derived from symmetric mass flow transmission. In [20], Torres consider the fractional Hamiltonian system given by where α ∈ 1 2 , 1 , u ∈ R N , and F : [0, T] × R N → R satisfies some conditions. In addition by using a modified version of mountain pass theorem for functional bounded from below due to Bonanno [21], the author studied the existence of at least three different solutions for problem (2) (see Theorems 1 and 2). In addition, under certain conditions and using some critical point theorems In [22] Zhou et al. proved the following fractional Hamiltonian system with impulsive effects has at least one weak solution: The outline of the paper is as follows: In Section 2, we lay down preliminaries and assumptions, some of which will be needed in the body of the paper. Then, in Section 3, the main result is obtained, which gives the result of the existence of at least two non zero weak solutions to problem (1) via Brezis and Nirenberg's Linking theorem (see Lemma 2 in [23]).

Preliminaries and Assumptions
In this section, we present some assumptions, definitions and basic properties of fractional computing, used later in this article. For omitted evidence, we refer the reader to ( [24]) or other texts on the basic fractional calculation.
To state our main results, we set where Γ is the standard gamma function given by Now, we assume that I ij (s) and F (t, u) satisfy the following assumptions: (h1) There exists a constant γ > 0, such that (h2) There exist constants ρ > 0, λ > 0 and k ≥ 1 verify for all |u| ≤ ρ and a.e. t ∈ [0, T] .

Definition 1. ([24]) Let u be a function defined on interval [a, b]
. left and right Riemann-Liouville fractional integrals of order α > 0 for a function u denoted by a D −α t u (t) and t D −α b u (t), respectively, are defined by where n − 1 ≤ α < 1 and n ∈ N. In particular, if 0 ≤ α < 1, then and Definition 3. ( [24]) Let α ≥ 0 and n ∈ N, we have: then the left and right Caputo fractional derivatives of order α for function u denoted by c a D α t u and c t D α b u respectively, are represented by respectively.

Proposition 1. ([24]) We have the following property of fractional integration
The Riemann -Liouville fractional derivative and the Caputo fractional derivative are related to one another by the following relationships.
In our study in this paper, we use Hilbert space H α = H α,2 0 with the inner product and the following corresponding norm In view of (11), we have that, for t ∈ [0, T] and p = q = 2 Definition 5. Let X be a Banach space with J : X → R differentiable. It can be said that J satisfies the Palais-Smale (PS)-condition if for all sequence (u n ) in X which J (u n ) is bounded and J (u n ) → 0 as n → ∞ possesses a convergent subsequence.
Definition 6. Let X be a Banach space and J : X → R differentiable and c ∈ R. It can be said that J satisfies the (PS) c -condition if the existence of a sequence (u n ) in X such that as n → ∞, implies that c is a critical value of J.

Remark 1.
It can be remarked the (PS)-condition implies the (PS) c condition for each c ∈ R.

Lemma 2.
[23] Let X be a Banach space with a direct sum decomposition X = X 1 ⊕ X 2 and k := dim X 2 < ∞.
Let J ∈ C 1 (X, R) with J (0) = 0, satisfying the (PS)-condition, we assume that, for ρ > 0, In addition, we assume also that J is bounded below and inf X J < 0. Thus J has at least two non zero critical points.

Mains Result
We establish the existence of at least two non zero weak solutions to problem (1) via Brezis and Nirenberg's Linking Theorem. where and Proposition 7. The functional J is continuously differentiable on H α and Proof. The proof of this proposition is very simple, we omit it.
It is clear that, the critical points of J are weak solutions of (1). Now, we give the proof of our main results.
Proof of Theorem 1. We apply Lemma 1 to J. Knowing that H α is a Banach space and J ∈ C 1 (H α , R) (see Proposition 5). By (15), it can be easily checked that functional J satisfies J (0) = 0. We decompose the proof of the theorem into the following three steps.
Step 2 We prove that (H1) or (H2) implies that J (u) satisfies the (PS) condition. Suppose that {u n } is a sequence in H α such that J (u n ) is bounded and J (u n ) → 0 as n → ∞. Then {u n } is bounded on H α 0 . In fact, if {u n } is an unbounded sequence, without loss of generality we assume that u n α → ∞ as n → ∞. By Step 1, we know that (H1) or (H2) implies (19). Thus J (u n ) → ∞, which contradicts the boundedness of J (u n ) . Since{u n } ⊂ H α is bounded and H α is a reflexive Banach space and so by passing to a subsequence (for simplicity denoted again by {u n }) if necessary, by Proposition 6, we may assume that By (18), we have I ij u i n t j − I ij u i t j u i n t j − u i t j dt + T 0 (∇F (t, u n (t) − ∇F (t, u (t))) , u n (t) − u (t)) dt,

One has
By (27), we know that and for any i ∈ A, j ∈ B, we have that u i n t j → u i t j , as n → ∞. In fact u i n t j − u i t j ≤ u n t j − u t j for any i ∈ A, j ∈ B, Thus, it follows from the continuity of all I ij that I ij u i n t j − I ij u i t j u i n t j − u i t j → 0, as n → ∞.

Conclusions
Fractional differential equations have recently proved to be valuable tools for modeling many phenomena in various fields of science and engineering. Indeed, one can find many applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc., for example, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Moreover, recently, the existence of solutions of boundary value problems for fractional differential equations have widely been studied in many papers and we refer the reader to the papers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In this work, we can extend the previous mentioned works for proving the existence of at least two non zero weak solutions to a new class of impulsive fractional boundary value problems via Brezis and Nirenberg's Linking Theorem. Finally, an example is presented to illustrate our results. In the next work, we will try to prove the existence of three different weak solutions of the p-Laplacian fractional for an overdetermined nonlinear fractional partial Fredholm-Volterra integro-differential system by using variational methods combined with a critical point theorem due to Bonanno and Marano.