Multiple Solution Results for Perturbed Fractional Differential Equations with Impulses

The multiplicity of classical solutions for impulsive fractional differential equations has been studied by many scholars. Using Morse theory, Brezis and Nirenberg’s Linking Theorem, and Clark theorem, we aim to solve this kind of problems. By this way, we obtain the existence of at least three classical solutions and k distinct pairs of classical solutions. Finally, an example is presented to illustrate the feasibility of the main results in this paper.


Introduction
Consider the multiple solutions of fractional order impulsive systems as follows: Þ, t ≠ t j , a:e:t ∈ 0, T ½ , where β ∈ ½0, 1Þ, α = 1 − β/2 ∈ ð1/2, 1 ; 0 D −β t , t D −β T are the left and right Riemann-Liouville fractional integrals of order β, c 0 D α t , c t D α T are used to denote the left and right Caputo fractional derivatives of order α, 0 = t 0 < t 1 < ⋯<t l < t l+1 = T, a ∈ C½0, T, F : ½0, T × R N ⟶ R is a given function, ∇Fðt, xÞ is the gradient of F at x, there are constants a 0 , a 1 with 0 < a 0 ≤ aðtÞ ≤ a 1 , for j = 1, 2, ⋯, l. The problem (1) arises from the phenomena of advection dispersion and was first scrutinized by Erwin and Roop in [1]. From then on, more and more scholars began to pay attention to the problem in [1] and the related problems.
Fractional calculus is different from integral calculus in nature. It has nonlocal characteristics and is very suitable for describing materials and processes with memory effect and genetic properties. Therefore, fractional differential equations are widely used in many domains, for instance, biomedicine, economic mathematics, and technology science [2,3]. In recent years, the variational methods and critical point theory have been widely used to study fractional differential equations [4][5][6][7][8].
In [8], the authors discussed the following fractional order differential systems: They used the critical point theory and other tools to verify the existence of solutions. From then on, a number of scholars began to use such methods for research, as shown in [9][10][11].
In [12], the authors discussed the following problems: They proved that there are at least k pairs of weak solutions and two weak solutions by using the Clark Theorem and other methods.
An impulsive phenomenon is a common phenomenon in nature and engineering applications. The models reflected in mathematics are impulsive differential equations. The most prominent feature of impulsive differential equation is that it can fully consider the impact of instantaneous mutation on the state. Therefore, in recent decades, impulsive differential equation theory has been widely used in biological mathematics, theoretical mechanics, biomedicine, and economic mathematics (see [13][14][15][16][17][18]).
For the past few years, very few scholars used the variational method and critical point theory to discuss impulsive fractional differential equations and their boundary value problems. Moreover, few papers discuss the fractional order system by using Morse theory (see [19][20][21][22][23][24]).
In [23], the authors discussed the following problems: The multiple solutions of this problem are verified with Morse theory and the Clark theorem by the authors.
In [25], the sufficient conditions for the existence of infinite solutions to the system (1) are obtained by using the variational method.
Based on the above literatures, in the present paper, we will discuss the existence of multiple classical solutions for (1) by using Morse theory, Clark theorem, and Brezis and Nirenberg's Linking Theorem.
First of all, we give some assumptions. (H1)I j ∈ Cð½0, T, RÞ, I j ð0Þ = 0, there exist some constants The key outcomes are as follows. Note that the methods in this article are distinct from [25] and our results are richer. The problems in this paper we studied are different from the problems in [23]. Compared with [23], classical solutions are investigated in this paper.
The structure of this article is as below. In Section 2, we provide some preliminary knowledge, which are helpful to the proof the key outcomes. We prove the key outcomes in Section 3. Finally, an example is given to illustrate the main results.

Preliminaries
Similar to [25], we first convert system (1) into a new format as follows: d dt Journal of Function Spaces Remark 3. Because of the equivalence of system (1) and system (6), we know that the solutions of system (6) are the solutions of system (1).
We first build the function spaces as below, the goal of which is to establish the variational framework of system (6).
Definition 6 (see [25]). We define that the function u ∈ E α 0 is a weak solution of the system (6) if the following holds: From (H1), (H2), we know the functional Φ is continu- Remark 7. Obviously, from (10), we know that the critical points of functional Φ are the weak solutions of system (6).
Definition 8 (see [25]). We define as a classic solution of system (6) if it satisfies the following conditions: (ii) u content system (1) a:e: on t ∈ ½0, T \ ft 1 , t 2 , ⋯, t l g Lemma 9 (see [25]). The function u ∈ E α 0 is a classical solution of system (6) when u is a weak solution of system (6).
Remark 10. Combine Remarks 3 and 7 and Lemma 9, we know that the critical point of functional Φ is the classical solution of the system (1). Therefore, we will directly discuss the critical point of Φ as below.
Definition 14 (see [23]). We say that Φ satisfies the ðPSÞ condition in E α 0 , if any fu n g n∈N ⊂ E α 0 , for which fΦðu n Þg n∈N is bounded and Φ ′ ðu n Þ ⟶ 0 as n ⟶ ∞ owns a strongly convergent subsequence in E α 0 .
Lemma 15 (see [26]). Let E have a direct sum decomposition E = V ⊕ W, and k = dim V < ∞. Let 0 be a critical point of Φ with Φð0Þ = 0, Φ is bounded below and satisfying ðPSÞ condition. Suppose that, for some ρ > 0, Also, assume that inf E Φ < 0. Then, Φ has at least two nonzero critical points and C k ðΦ, 0Þ≅0.
Lemma 16 (see [27]). Let E be a real Banach space, Φ ∈ C 1 ðE, RÞ; assume that Φ is even, bounded from below, and satisfying ðPSÞ condition. Assume Φð0Þ = 0, there exists a set E ′ ⊂ E such that E ′ is homeomorphic to S k−1 by an odd map, and sup E ′ Φ < 0. Then, Φ has at least k distinct pairs of critical points.

Proofs of Main Results
Lemma 17. Suppose (H1), (H2) hold, if fu n g is a ðPSÞ sequence, then fu n g is bounded.
Proof. If fu n g is a ðPSÞ sequence, that is, From (H2), for some ξ > 0 small enough, there is a constant C ξ > 0, for any u ∈ R, t ∈ ½0, T such that F t, u ð Þ j j≤ 1 2 According to (19) Because ΦðuÞ is bounded, by (20), we can get fu n g is bounded in E α 0 and Φ is bounded from below. The proof is completed. Proof. If fu n g is a ðPSÞ sequence, from Lemma 17, we get fu n g is a bounded sequence in E α 0 . By Lemma 5, we get fu n g has a weakly convergent subsequence. Without loss of generality, we also assume that u n converges weakly to u 0 in E α 0 , then from (9) and (18), we know By Lemma 13, we can obtain that u n ⟶ u 0 in Cð½0, T, RÞ, as n ⟶ ∞, i.e., From (10), we have Journal of Function Spaces By (21), (22), and (23), we can infer that ku n − u 0 k 2 α ⟶ 0, as n ⟶ ∞, i:e:, u n strongly converges to u 0 . Therefore, Φ satisfies the ðPSÞ condition.