Three Solutions for Fourth-Order Impulsive Differential Inclusions via Nonsmooth Critical Point Theory

Abstract

An existence of at least three solutions for a fourth-order impulsive differential inclusion will be obtained by applying a nonsmooth version of a three-critical-point theorem. Our results generalize and improve some known results.

1. Introduction

In this paper, we will consider the following fourth-order differential inclusion with impulsive effects: where and are real constants, and are real parameters, for , the impulsive functions satisfy that , and the left and right limits of are represented by and , at , respectively. Here, is a locally Lipschitz function defined on satisfying the following: and there exists a Lipschitz positive constant such that  for any There exist constants and , such that for all .

is defined on satisfying is measurable for each ; is locally Lipschitz for each There exist constants and , such that for .

In the last decades, the existence of solutions for boundary value problems has been dramatically investigated, by which a number of physics phenomena, population dynamics, optimal control, ecology, and biotechnology are vividly described and some classical tools have been applied to study this problems, for instance, coincidence degree theory [1], the method of upper and lower solutions [2], and some classical fixed point theorems [3].

After that it is becoming a trend to study boundary value problems with impulses or without impulses by variational methods and some critical point theory in recent years; see [4–8] and references therein. As far as we know, the difficulties dealing with this problems are that their states are discontinuous. In the literature few papers considered the higher-order boundary value problems with impulsive effects. However, the fourth-order boundary value problems are essential in describing a class of elastic deflection; because of this crucial reason, the existence of solutions for fourth-order boundary value problems has attracted much attention of many authors.

More precisely, in [9], the authors have investigated the periodic solutions for semilinear fourth-order differential inclusion without impulsive effects as follows: and they proved that the above system has at least three periodic solutions under certain conditions by using a nonsmooth critical point theorem.

In [10], the authors considered the fourth-order boundary value problem with impulsive effects as follows: By applying the variational methods and critical point theory, one nontrivial solution and infinitely many distinct solutions have been obtained.

In [11], the following boundary value problem for a fourth-order impulsive differential equation has been studied: they obtained that the boundary value problem possesses at least one solution and infinitely many solutions via some existing critical point theorems.

From above several examples, we can clearly see that either the system is a differential equation with impulsive effects, or the system is a differential inclusion without impulsive effects. To the best of our knowledge, there are few papers which have concerned the existence of three solutions for fourth-order differential inclusion with impulsive effects so far. Motivated by above, the main aim in this paper is intended as an attempt to establish some criteria of existence of at least three solutions for the fourth-order differential inclusion by using a nonsmooth version of a three-critical-point theorem.

The arrangement of the rest paper is as follows. In Section 2, some preliminaries and results which are applied in the later paper are presented. In Section 3, we establish a variational approach for problem (1). In Section 4, the main proof of prospective results will be showed. In Section 5, a meaningful example is given to illustrate the obtained results in Section 4.

2. Preliminaries

In this section, we give some definitions and results that we shall use in the rest of the paper. For more details, please refer to [6, 12, 13].

Let be a Banach space, be its topological dual, and be a functional. We denote that is locally Lipschitz if there exist a neighborhood of for each and a real number such that for all . If is locally Lipschitz and , the generalized directional derivative of at along the direction is Meanwhile, the generalized gradient of at is the set for all . So is a multifunction. We also denote that has compact gradient if maps bounded subsets of into relatively compact subsets of .

Lemma 1 (see [13, proposition 1.1]). Let be a functional, then is locally Lipschitz and for all ; for all .

Lemma 2 (see [13, proposition 1.3]). Let be a locally Lipschitz functional, then for , is subadditive and positively homogeneous, and there exists a Lipschitz constant such that for around .

Lemma 3 (see [13, proposition 1.6]). Let be locally Lipschitz functional, then for all .

Lemma 4 (see [12, Lemma 6]). Let be a locally Lipschitz functional with a compact gradient, then is sequentially weakly continuous.

We say that is a critical point of locally Lipschitz functional if

In order to prove our main results, we also need some basic definitions and lemmas.

Definition 5 (see [6, definition 2.1]). An operator is of type , if, for any sequence in , and imply

Theorem 6 (a particular case of [12, Theorem 14]). Let be a reflexive Banach space, be an interval, be a sequentially weakly lower semicontinuous functional and its derivative is of type , be a locally Lipschitz functional with a compact gradient, and . Assume that, for all , we have Then, there exist and with the following characters: for any and any multifunction with compact gradient, then there exists such that the functional possesses at least three critical points with norms in less than for all

The minimax inequality (14) can be derived from the following result.

Lemma 7 (see [5, Proposition 3.1]). Let be a nonempty set, be functions, , and , such that Then, there exists such that (14) holds.

Definition 8. A function is a weak solution of problem (1) if there exists such that for all .

Now, we state our main result.

Theorem 9. Assume that hold. Moreover, the following conditions also hold: There exists satisfying  where  and is defined in Lemma 10. Then, there exist a nondegenerate interval and accompanied by the following property: for any and any multifunction with compact gradient, then there exists such that the functional possesses at least three critical points with norms in less than for all .

3. Variational Approach for the Problem (1)

We will assume for the remainder of the paper that .

Taking , we can define the following norm of : It is immediate that The usual norm of is defined as follows:

Lemma 10. Let , then , where

Proof. From , we have therefore The above two equalities imply which shows On the other hand, since and , by the mean value theorem, there exists at least one , such that . So we have Let ; we obtain .

Remark 11. It is clear that is more simple in our results than that in [10], which is defined as , where and It is easy to see that if we assume , then and , so

Remark 12. It is also obvious that is better in our results than that in [11], which is defined as However, in our results if . Thus our results generalize and improve some known results.

Next we are intended to look for suitable weak solution for system (1). Firstly, we assume that there exists a such that the first inclusion of (1) becomes the following form: Then, let . Take and multiply (30) by and integrate from to ; we have One has and From above, we have and So the weak solutions of (1) is defined as follows: where and .

Lemma 13. If is a weak solution of (1), then is a classic solution of (1).

Proof. Firstly, it is easy to see that for . Since is a weak solution of (1), we have for any Then, choose any and such that if for . Hence . Equation (37) implies that and this also implies that for, any , so is a weak solution of the first equation of (1), and obviously it is a classic solution of (1).
By previous equality (37) and using integration by parts, we obtain since we obtain Next we will verify the second condition of (1); if not, we assume that there exists such that Let and taking a derivation of (44), we have obviously, , , and , which together with (42) yields since , we get which contradicts with (43), so satisfies the second impulsive condition of (1). From (47), we know that (42) can be described as the following form: Let then by the same skills of the above proof, we also obtain So, the third impulsive condition of (1) holds and (42) becomes the following form: since are arbitrary, then we have . Therefore, is a classical solution of (1).

4. Proof of Theorem 9

In order to prove our main results, we will first show some related lemmas. Then, the functional is defined as follows: so where .

Lemma 14. is of type .

Proof. Assume and According to Definition 5, we need to verify that In fact, and Equations (57) and (58) imply Since in , we have in , so and as , and thus we have then from the definition of and the equivalence of with , we have when , which implies .

Lemma 15. The functional is locally Lipschitz; moreover, for each critical point of , is a weak solution of (1).

Proof. Let , where Firstly, we will show is locally Lipschitz on . Since , according to Lemma 1, we know is locally Lipschitz on . On the other hand, from and , we get that is locally Lipschitz on , since is compactly embedded into , so is also locally Lipschitz on . Then, by Lemma 3, we have That is to say, for every , we immediately have for with the character that the functional and for every , so is locally Lipschitz on .
Next we will prove that each critical point of is a weak solution of (1). Let be a critical point of , so According to (62) and Lemma 1, we have where , on . Thus, by (54) and (63), we have for , so is a weak solution of (1).

Lemma 16. is compact.

Proof. Let us define a bounded sequence in and for all , then we choose and such that and , respectively. From , we have so it is clear to see the sequence is bounded, up to a subsequence; if , we can prove .
Suppose the assertion of conclusion is false, assume that there exists such that for all , and then choose with such that Passing if necessary to a subsequence, we suppose that in , then in and . By , we have as , which contradicts with (66).

Proof of Theorem 9. We mainly use Theorem 6 to prove Theorem 9. By the equivalence between usual norm and which is defined in (21), we can easily obtain that the functional defined in (53) is sequentially weak lower semicontinuous; according to above Lemma 14, we have that is of type and is a locally Lipschitz functional with compact gradient by Lemmas 15 and 16.
Firstly, we will verify condition (13) in Theorem 6.
By , for all we have then Lemma 10, , and (68) imply where and
Next, we will verify condition (14) in Theorem 6. According to Lemma 7, choose Obviously, and , so (15) holds. Then from and (68), we have which shows (16) holds.
Finally, we will verify condition (17) in Lemma 7. For any with , from and Lemma 10, we get then and By , (73), and (74), we obtain thus we know that (17) in Lemma 7 holds; that is to say (14) in Theorem 6 holds for some .
With the same methods, the function satisfies , then according to Lemmas 15 and 16, the functional is locally Lipschitz and is compact. From Theorem 6, there exist and accompanied by the following property: for any and any multifunction with compact gradient, then there exists such that the functional possesses at least three critical points with norms in less than for all . By Lemmas 13 and 15, we define the solutions as , which are .

5. An Example

In this part, we will give a meaningful examples to illustrate the main results in our paper, but in [9], the authors do not give the corresponding examples.

We consider the following differential inclusion: let ; let as and as , by Lemma 10; we obtain ; obviously, the assumptions , and hold; then we only need to verify . Let ; by a simple calculation, we have andthus So holds. Let , then ; clearly, the assumptions all hold. By Theorem 9, we can obviously deduce that (76) possesses at least three solutions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by the NSFC (11571088 and 11471109), the Zhejiang Provincial Natural Science Foundation of China (LY14A010024), and Scientific Research Fund of Hunan Provincial Education Department (14A098).