Applications of Morse Theory to Fourth-Order Impulsive Differential Equations

Abstract

This paper is devoted to studying the nontrivial solutions for fourth-order impulsive differential equations. By applying Morse theory and combining it with local linking theory, some new criteria to guarantee that the fourth-order impulsive differential equation has at least one nontrivial solution will be given. Then, we give an example to illustrate the effectiveness and feasibility of the obtained results in our paper. Moreover, some previous corresponding results will be significantly improved in our paper.

1. Introduction

In this paper, we will consider the following fourth-order differential equation with impulsive effects:where and are the real constants, for and , where and , denote the left and the right limits, respectively, of at .

In the last decade, the existence of solutions for boundary value problems with impulses has been dramatically investigated, and the methods to deal with these problems are coincidence degree theory, the method of upper and lower solutions, the monotone iterative technique, and some classical fixed point theorems (see [1–9] and references therein). It is becoming a trend to study boundary value problems with impulses by variational methods and some critical point theory in recent years (see [10–16] and references therein). Especially, the existence of solutions for fourth-order impulsive differential equations has attracted attention of many authors (see [17–22] and references therein).

More precisely, in [17], Afrouzi et al. considered a fourth-order impulsive differential equation with two control parameters as follows:where and are two real constants and and are two control parameters; they obtained that system (2) has at least three solutions under certain conditions by employing a critical point theorem.

In [18], Sun et al. investigated the fourth-order differential equation with impulsive effects as follows:

By applying the variational methods and critical point theory, one nontrivial solution and infinitely many distinct solutions for system (3) have been obtained.

Also, Yue et al. [21] further considered the existence of infinitely many solutions for the above system (3) involving oscillatory behaviors of nonlinearity at infinity.

In [19], Xie and Luo studied the following fourth-order impulsive differential equation:

They obtained at least one solution and infinitely many solutions for system (4) via some existing critical point theorems.

In [22], Kong considered the following nonperiodic fourth-order differential equation:

By using the mountain pass theorem, at least two nontrivial homoclinic solutions for system (5) have been obtained.

Motivated by above facts, it is clear to see the methods to deal with these fourth-order boundary value problems are variational methods and some critical point theory. But as far as we know, Morse theory is an important method in nonlinear analysis which can also be a powerful tool to study the existence of solutions for differential equations (see [23–25] and references therein). However, Morse theory has not received so much attention as variational method and critical point theory, and also it should be pointed out that there is no published paper which has discussed the existence of solutions for fourth-order boundary value problems by using Morse theory. So, the aim in this paper is to establish some criteria of existence of at least one nontrivial solution for the fourth-order differential equations by using Morse theory.

The highlights and major contributions of this paper are reflected in the subsequent key aspects. (1) We firstly consider the existence of solutions for fourth-order impulsive differential equation by employing Morse theory coupled with local linking arguments, which will effectively fill in the blank of Morse theory in fourth-order differential equations. (2) We construct a new suitable inequality in Lemma 1, which will significantly improve some previous results [17–21]. In fact, the restrictive conditions in [17–21] have room for improvement which is why we give a more accurate condition to ensure the feasibility of Lemma 1. (3) Our model is more general than the previous paper [26, 27]. In fact, when we remove the impulsive functions and , system (1) is reduced to the form in [26]; when we remove the impulsive functions and and let , system (1) is reduced to the form in [27].

The rest of the paper is organized as follows. In Section 2, some preliminaries and results which are applied later in the paper are presented. Section 3 is devoted to studying the existence of at least one nontrivial solution to (1). In Section 4, an example is presented to illustrate the effectiveness and feasibility of the obtained results in Section 3.

2. Preliminaries

We will assume for the remainder of the paper that .

In the Sobolev space , we can consider the inner productwhich induces the norm

Then, we can define the norm of as follows:

It is immediate that

The usual norm of is defined as follows:

Lemma 1. Let ; then, , where

Proof. From , we haveTherefore,The above two equalities implywhich shows thatOn the other hand, since and , by the mean value theorem, there exists at least one , such that . So, we haveLet , and we obtain .

Remark 1. It is clear that is more simple in our results than that in [18, 21], which is defined as , where and , and if we assume that , then and , so .

Remark 2. It is also obvious that is better in our results than that in [17, 19], which is defined as . However, in our results if . Thus, our results generalize and improve the results in [17, 19].

Remark 3. It is also clear that is better in our results than that in [20], which is defined as . When , in [20]; however, in our results. Thus, our results generalize and improve the result in [20].

Definition 1. Let be a weak solution of system (1) if the equalityholds for .
Then, we consider the functional defined bywhere .
Owing to the continuity of and , we immediately deduce that is continuous and differential for all andfor any . Then, we can clearly know that the critical points of are weak solutions of system (1).

Lemma 2 (see [18]). If is a weak solution of system (1), then is a classic solution of (1).

Definition 2. The functional satisfies the (C)-condition if for , any sequence such that has a convergent subsequence as .
The following lemmas are crucial for proving the existence results in Section 3.

Lemma 3 (see [28]). If , then for every . It follows that if for some , then must have a nontrivial critical point.

Lemma 4 (see [29]). Let be a critical point of with . Suppose that has a local linking at 0 with respect to , that is, there exists such that

Then, , and hence 0 is a homological nontrivial critical point of .

For more details about the -th critical group of and the singular relative homology group , we refer readers to [25, 30].

3. Main Results

Theorem 1. Suppose that the following conditions hold:(A1) and there exist and such that for any and , we have(A2) and there exist and such that for any and , we have(A3) and , where and , . There exist such that and for any .(H1) and there exist such that where .(H2) for any and uniformly for .(H3) uniformly for .(H4) and there exists a such that for any . Then, system (1) has at least one nontrivial solution.In order to complete the proof, we need to prove the following lemmas and compute the critical groups at zero and at infinity.

Lemma 5. Suppose that satisfies (A1), (A2), (A3), (H1), (H2), and (H4); then, satisfies the (C)-condition.

Proof. Firstly, we will show the boundedness of (C) sequences. Suppose that has an unbounded (C) sequence . Up to a subsequence, we may assume thatas for some .
Let , and thus ; up to a subsequence, we have in and , . .
If , as the same proof of [31], we can extract a sequence such thatLet , where is a positive integer, such that as . Since in and from (H1), we immediately obtain in . That is,By (A1), (A2), and (18), we havewhich implies that .
But from another dimension, we know and ; then,Combining (A3) and (H4), we haveas . This contradicts .
If , we define and . So, and for . as . By (24), we haveThen,where is a positive constant. By (H2), we havefor any . According to Fatou’s lemma, we immediately havewhich is also a contradiction with . So, we deduce that is bounded in .
Since is a reflexive space, we may extract a weakly convergent subsequence, and we denote and in . Then, we have in and , . . By (19), we havesinceas , so we deduce that for , that is, in . Thus, satisfies the (C)-condition.

Lemma 6. Suppose that satisfies (A1), (A2), (A3), (H3), and (H4); then, for all .

Proof. It is similar to the proof of Theorem 1.1 in [31]. Let . By (A1), (A2), (10), and Lemma 1, we haveandCombining (H3) with Fatou’s lemma, we obtainThen,as . Let ; there must exist such that for and . Thus, we immediately obtainBy (A3), (H4), and (40), we haveSo, we clearly know that there exists a unique such that by the implicit function theorem. From [29], we can apply the function to make a strong deformation retract from to . Thus, we obviously obtainThe proof is complete.
Since is a reflexive and separable Banach space, we can extract an orthogonal basis of such that and denote that , , , so .

Lemma 7. Suppose that satisfies (A1), (A2), and (H1); then, there exists such that .

Proof. From (A1), (A2), and (H1), we know that , , and , so we know that the zero function is a critical point of . Then, it is enough to verify that the functional has a local linking at zero with respect to . Since all the norms of finite dimensional normed space are equivalent, by (10), for any , we havewhich means is a norm of and there exists such that

Step 1. Let ; since is finite dimensional, for given , there exists such thatThen, we can divide into three parts for : , , and , that is, . For convenience, we let . By (A1), (A2), (H1), (44), and Lemma 1, we haveAlso, from Lemma 1, we know that . If , then . Since the norms on are equivalent to each other and as , then for all and , we have .

Step 2. Let ; owing to the compactness of the continuous embedding , for given , there exists such thatCombining (A1), (A2), and (H1), we havein which is sufficiently small. So, we complete the proof owing to Lemma 4.
At last, we show the main proofs of Theorem 1.

Proof of Theorem 1. Let in Lemma 4. By Lemma 5, we obtain that satisfies the (C)-condition. Combining Lemma 6 with Lemma 7, we immediately have for some . Thus, all conditions of Lemma 3 hold, that is, must have a nontrivial critical point which is also a classical solution of system (1).

4. An Example

In this part, we give corresponding example to illustrate the main results in our paper.

Example 1. Let ; then, consider the following fourth-order impulsive differential equations:where we denoteBy simple calculations, we haveLet , , and we can clearly see that conditions (A1)–(A3) and (H1)–(H4) all hold. Then, system (49) possesses at least one nontrivial solution by Theorem 1 [32].

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (12071105 and 11571088), the Key Projects of Science Research in University of Anhui Province (KJ2021A1049), the Natural Science Foundation of Anhui Province of China (2108085MA11), and the Talent Foundation of Tongling University (2021tlxyrc24).