Abstract
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 distinct pairs of classical solutions. Finally, an example is presented to illustrate the feasibility of the main results in this paper.
1. Introduction
Consider the multiple solutions of fractional order impulsive systems as follows: where are the left and right Riemann-Liouville fractional integrals of order , are used to denote the left and right Caputo fractional derivatives of order , , , is a given function, is the gradient of at , there are constants , with , for .
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 [4–8].
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 [9–11].
In [12], the authors discussed the following problems:
They proved that there are at least 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 [13–18]).
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 [19–24]).
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), there exist some constants , , such that , , and , for
(H2) and , uniformly on
(H3) uniformly for . There exist four constants such that , ,
(H4) and , for
The key outcomes are as follows.
Theorem 1. Let (H1)–(H3) hold. Then, the problem (1) has at least three classical solutions.
Theorem 2. Let (H1)–(H4) hold. Then, the problem (1) has at least distinct pairs of classical solutions.
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.
2. Preliminaries
Similar to [25], we first convert system (1) into a new format as follows:
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).
Let us recall that for any fixed and , .
Definition 4 (see [22]). Let , we define the fractional derivative space by the closure of with under the norm
Lemma 5 (see [25]). Let , and is a Banach space with reflexive and separable.
Definition 6 (see [25]). We define that the function is a weak solution of the system (6) if the following holds:
for .
We define as
From (H1), (H2), we know the functional is continuously differentiable. So , we have
Remark 7. Obviously, from (10), we know that the critical points of functional 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: (i)(ii) content system (1) on
Lemma 9 (see [25]). The function is a classical solution of system (6) when 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 is the classical solution of the system (1). Therefore, we will directly discuss the critical point of as below.
Lemma 11 (see [8]). Let and , for all , one has Moreover, if , , then In particular, if , then It is easy to prove that the norm is equivalent to , Next, we will use as the norm in .
Lemma 12 (see [8]). Let , , have
Lemma 13 (see [8]). Let . Assume the sequence converges weakly to in . Then, strongly in , , , as .
Definition 14 (see [23]). We say that satisfies the condition in , if any , for which is bounded and as owns a strongly convergent subsequence in .
Lemma 15 (see [26]). Let have a direct sum decomposition , and . Let 0 be a critical point of with , is bounded below and satisfying condition. Suppose that, for some ,
Also, assume that . Then, has at least two nonzero critical points and .
Lemma 16 (see [27]). Let be a real Banach space, ; assume that is even, bounded from below, and satisfying condition. Assume , there exists a set such that is homeomorphic to by an odd map, and . Then, has at least distinct pairs of critical points.
3. Proofs of Main Results
Lemma 17. Suppose (H1), (H2) hold, if is a sequence, then is bounded.
Proof. If is a sequence, that is, From (H2), for some small enough, there is a constant , for any , such that According to (19), for , , one has Because is bounded, by (20), we can get is bounded in and is bounded from below. The proof is completed.
Lemma 18. Assume (H1), (H2) hold, then satisfies the condition.
Proof. If is a sequence, from Lemma 17, we get is a bounded sequence in . By Lemma 5, we get has a weakly convergent subsequence. Without loss of generality, we also assume that converges weakly to in , then from (9) and (18), we know
By Lemma 13, we can obtain that in , as , i.e.,
From (10), we have
By (21), (22), and (23), we can infer that , as , strongly converges to . Therefore, satisfies the condition.
By Lemma 5, we can obtain that there is an orthogonal basis of such that . We define . Then, .
Proof of Theorem 1. From (H1), (H2), one knows and , . We find out has a critical point at 0. Therefore, we can get the linking of at 0.
According to the equivalence of norm of normed space in finite dimension, there exist positive constants , such that
First, let . Because is finite dimensional, there exists small for , such that
For any , we set , , where and are pairwise disjoint.
Set , for , combine (H1), (H3) and Lemma 12, we have
where .
According to (25) and the definition of is empty set, we have , for any . By (H3), one has . On , . From (H3), we can get , as .
Then, , , according to (26), we can get
Hence,
Next, set . Because is continuous compact embedding. Hence, for , there exists small such that , for .
From (H3), , there is , one has
From (H1), , one has
Let , , , by Lemmas 11 and 12 and (29) and (30), we obtain
Hence,
Let , from (28) and (32), we obtain
It follows from Lemmas 17 and 18 that is bounded from below and satisfies the condition. Then, from Lemma 15, we can get has at least two nonzero critical points, and , so is a homological nontrivial point of . Hence, the system (1) has at least three classical solutions.
Proof of Theorem 2. According to (H4), we can deduce that is even. By Lemmas 17 and 18, we know is bounded from below and satisfies the condition. For given , set . By (27), if is small enough, one has . Clearly, . Then, we can conclude that has at least distinct pairs of critical points from Lemma 16. Hence, the system (1) has at least distinct pairs of classical solutions. We complete the proof.
Example 19. According to (34), we can see that , , , .
Let , then the condition (H1) holds with , , , .
Let , . By simple calculations, we can get , ,
then, the condition (H2) holds.
By (H3), we know
Let , , , ; then, for , we have
then, the condition (H3) is satisfied.
It easy to see that the condition (H4) holds.
According to Theorem 1, the system (1) exists at least three classical solutions. According to Theorem 2, the system (1) possesses at least distinct pairs of classical solutions.
4. Conclusions
In this work, we study perturbed fractional differential equation with impulses. We give sufficient conditions of the existence of at least three classical solutions and at least distinct pairs of classical solutions for problems (1), where is the dimension of .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgments
The research is supported by the National Natural Science Foundation of China (No. 61673008) and the Young Backbone Teacher Funding Scheme of Henan (No.2019GGJS079).