Solutions for Impulsive Fractional Differential Equations via Variational Methods

We investigate the boundary value problems of impulsive fractional order differential equations. First, we obtain the existence of at least one solution by the minimization result of Mawhin andWillem. Then by the variational methods and a very recent critical points theorem of Bonanno and Marano, the existence results of at least triple solutions are established. At last, two examples are offered to demonstrate the application of our main results.


Introduction
Fractional differential equations have been an area of great interest recently.This is because of both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various scientific fields such as physics, mechanics, chemistry, and engineering; see [1][2][3] and the references therein.Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attentions.But for almost all the papers above, the main methods are some fixed theorems, the coincidence degree theory, and the monotone iterative methods.
On the other hand, critical point theory and the variational methods have been very useful in dealing with the existence and multiplicity of solutions for integer order differential equations with some boundary conditions.We refer the readers to the books (or surveys) of Mawhin and Willem [4] and Rabinowitz [5] and [6,7] and the references therein.But until now, there are few works that deal with the fractional differential equations via the variational methods; [8,9] are the pioneer in the use of the variational methods to fractional models.By using the variational methods, Jiao and Zhou [8] first considered the following fractional boundary value problems: ( 0     ()) = ∇ (,  ()) , a.e. ∈ [0, ] ,  (0) =  () = 0, where  ∈ (0, 1) and 0    and     are the left and right Riemann-Liouville fractional derivatives, respectively. : [0, ] ×   →  (with  ≥ 1) is a suitable given function and ∇(, ) is the gradient of  with respect to .The problem in [8] and the related problems were further considered by critical point theory and the variational methods in [10][11][12].
By means of critical point theory, Jiao and Zhou [9] considered the following fractional boundary value problems: The above problem arises from the phenomena of advection dispersion and was first investigated by Ervin and Roop in [13].The authors in [14][15][16][17] further studied the existence and multiplicity of solutions for the above problem or related problems by critical point theory.
The impulsive differential equations arise from the real world problems to describe the dynamics of processes in which sudden, discontinuous jumps occur.Such processes are naturally seen in biology, physics, engineering, and so forth.Due to their significance, many authors have established the solvability of impulsive differential equations.There have been many different approaches to the study of the existence of solutions to impulsive fractional differential equations, such as topological degree theory, fixed point theory, upper and lower solutions method, and monotone iterative technique.
To the best of our knowledge, the fractional boundary value problems with impulses using variational methods and critical point theory have received considerably less attention [18][19][20].
In [20], the authors investigated the following fractional differential equations with impulses: By using critical point theory and variational methods, the authors give some criteria of the existence of solutions.Motivated by the work above, we consider the following problem (4) of impulsive fractional differential equations: where  ∈ (0, +∞),  ∈ [0, 1),  = 1 − / for  = 1, . . ., . ( In this paper, the existence results of at least one solution or triple solutions of problem (4) are established.The rest of this paper is organized as follows.In Section 2, some definitions and lemmas which are essential to prove our main results are stated.In Section 3, we give the main results.At last, two examples are offered to demonstrate the application of our main results.

Preliminaries
At first, we present the necessary definitions for the fractional calculus theory and several lemmas which are used further in this paper.
Definition 1 (see [3]).Let  be a function defined by [, ].The left and right Riemann-Liouville fractional integrals of order  for function  denoted by   −  () and   −  () function, respectively, are defined by The left and right Riemann-Liouville fractional derivatives of order  for function  denoted by     () and     () function, respectively, are defined by provided that the right-hand side integral is pointwise defined on [, ].
In view of Definition 1 and Lemma 2, we can easily transfer problem (4) to the following problem: where 4) is equivalent to problem (10).Therefore, a solution of problem (10) corresponds to a solution of BVP (4).
In order to establish a variational structure which enables us to reduce the existence of solution of problem (10) to existence of the critical point of corresponding functional, we construct the following appropriate function spaces.
Let us recall that, for any fixed  ∈ [0, ] and 1 ≤  ≤ ∞, Let 0 <  ≤ 1, and we define the fractional derivative spaces   0 by the closure of  ∞ 0 ([0, ]) with respect to the weighted norm Clearly, the fractional derivative space   0 is the space of functions  ∈  2 ([0, ] \ { 1 ,  2 , . . .,   } having -order Caputo left and right fractional derivatives and Riemann-Liouville left and right fractional derivatives, Lemma 4 (see [9]).Let 0 <  ≤ 1 and 1 <  < ∞.For all  ∈   0 , one has Then we can conclude that In the following, we will consider the fractional derivative spaces   0 with respect to the norm is called a classic solution of problem (10) if for all V() ∈   0 .Similar to the proof of Lemma 2.1 in [18], we have the following Lemma 7.

Lemma 7. The function 𝑢 ∈ 𝐸 𝛼
0 is weak solution of (10), if and only if  is a classical solution of (10).
Lemma 9 (see [9]).Let 1/2 <  ≤ 1.For any  ∈   0 , one has The proofs of the main results in this paper are based on the following critical point theorems.

Theorem 10 (see [4, Theorem 1.1]). If Φ is weakly lower semicontinuous (WLSC) on a reflexive Banach space 𝑋 and has a bounded minimizing sequence, then Φ has a minimum on 𝑋.
Theorem 11 (see [21]).Let  be a reflexive real Banach space, let Φ :  →  be a sequentially weakly lower semicontinuous, coercive, and continuously Gateaux differentiable functional whose Gateaux derivative admits a continuous inverse on  * , and let Ψ :  →  be a sequentially weakly upper semicontinuous and continuously Gateaux differentiable functional whose Gateaux derivative is compact.Assume that there exist  ∈  and  1 ∈  with 0 <  < Φ( 1 ), such that Then, for each  ∈ Λ  the functional Φ − Ψ has at least three distinct critical points in .

Main Results
For convenience, we give the following hypothesis ( 1 ).
Proof.We define Then, for all V() ∈   0 , we know which shows that a critical point of the functional  is a weak solution of problem (10).
Our aim is to apply Theorem 10 to problem (10).We begin by proving that  is weakly lower semicontinuous.Since  is a separable and reflexive real Banach space, we assume that {  } ⊂  converges weakly to  in ⊂ .By Lemma 8, we can obtain that   →  uniformly in ([0, ], ), as  → ∞; that is, Together with ( 1 ), one has Then it implies that  is weakly lower semicontinuous.Now, we are in the position of showing that the functional  is coercive.
From ( 2 ), we know that there exists a positive constant  2 large enough such that On the other hand, from the continuity of (, ), we concluded that (, ) is bounded for || ≤  2 ,  ∈ [0, ].Then there exists a constant  1 > 0 such that for Hence, for all (, ) ∈ [0, ] × , we can get Then it follows from ( 1 ) and Lemmas 4 and 9 that In view of  1 < Γ 2 ( + 1)|cos()|/2 2 , we have Then we know that  is coercive.Thus, by virtue of Theorem 10, the functional  has a minimum, which is a critical point of .It follows that the boundary value problem (10) has one weak solution.By virtue of Lemma 7, we can deduce that the boundary value problem (10) has one solution which implies that the boundary value problem (4) possesses at least one solution.
Remark 13.If the asymptotically quadratic case in ( 2 ) becomes the subquadratic case, that is, then we can get the similar result.Put For  ∈   0 , we define the functional Φ, Ψ :   0 →  as follows: Theorem 14.Let ( 1 ) hold.Suppose that there exist a constant  > 0 and a function  such that ‖‖ 2  > 2/| cos()| and the following assumptions ( 3 )-( 4 ) are satisfied: Then, for each the boundary value problem (4) has at least three distinct solutions in   0 .
On the other hand, if  > 0, from ( 13) and (30), then we have For Hence the functional Φ() − Ψ() is coercive.So condition (ii) of Theorem 11 holds.Then, by virtue of Theorem 11, we can conclude that the equation Φ  () − Ψ  () = 0 has at least three distinct solutions.That is, the boundary value problem (10) has at least three distinct weak solutions.As a consequence of Lemma 7, we deduce that the boundary value problem (10) has at least three distinct solutions which implies that problem (4) possesses at least three distinct solutions.
Finally, we give two examples to illustrate the usefulness of our main result.Consider the following impulsive system of fractional differential equations.( Then it is easy to verify that assumption ( 1 ) holds.