Abstract and Applied Analysis
Volume 2016 (2016), Article ID 5784273, 12 pages
http://dx.doi.org/10.1155/2016/5784273

Research Article

Existence of Infinitely Many Periodic Solutions for Perturbed Semilinear Fourth-Order Impulsive Differential Inclusions

Massimiliano Ferrara,¹ Giuseppe Caristi,² and Amjad Salari³

¹Department of Law and Economics, Mediterranea University of Reggio Calabria, Via dei Bianchi 2, 89131 Reggio Calabria, Italy
²Department of Economics, University of Messina, Via dei Verdi 75, 98122 Messina, Italy
³Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran

Received 28 October 2015; Accepted 11 February 2016

Academic Editor: Agacik Zafer

Copyright © 2016 Massimiliano Ferrara et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper discusses the existence of infinitely many periodic solutions for a semilinear fourth-order impulsive differential inclusion with a perturbed nonlinearity and two parameters. The approach is based on a critical point theorem for nonsmooth functionals.

1. Introduction

The goal of this paper is to establish the existence of infinitely many periodic solutions for the following perturbed semilinear fourth-order impulsive differential inclusion:

where is a positive constant, is continuous positive even -periodic function on , , , , and are generalized gradients of and , respectively, the operator is defined as , with and denoting the right and left limits, respectively, of at , , , , , and is a locally Lipschitz function satisfying the following:

and for all ;there exist two constants and , such that

Also, is a function defined on , satisfying the following:

is measurable for each , is locally Lipschitz for , , and for a.e. and ;there exists a constant , such that

 where is defined in .

We study the existence of solutions, that is, absolutely continuous on every and left continuous at functions which satisfy (1) for a.e. with (possible) jumps (impulses) at .

Fourth-order ordinary differential equations act as models for the bending or deforming of elastic beams, and, therefore, they have important applications in engineering and physical sciences. Boundary value problems for fourth-order ordinary differential equations have been of great concern in recent years (e.g., see [1–4]). On the other hand, impulsive differential equations occur in many applications such as various mathematical models including population dynamics, ecology, biological systems, biotechnology, industrial robotic, pharmacokinetics, and optimal control. For the general aspects of impulsive differential equations, we refer the reader to [5–9]. In association with this development, a theory of impulsive differential equations has been given extensive attention. Very recently, some researchers have studied the existence and multiplicity of solutions for impulsive fourth-order two-point boundary value problems; we refer the reader to [10–12] and references therein. Differential inclusions arise in models for control systems, mechanical systems, economical systems, game theory, and biological systems to name a few.

Recently, multiplicity of solutions for differential inclusions via nonsmooth variational methods and critical point theory has been considered and here we cite the papers [13–19]. For example, in [13] the existence of infinitely many antiperiodic solutions for second-order impulsive differential inclusions has been discussed. In [16], Kristály employing a nonsmooth Ricceri-type variational principle [20], developed by Marano and Motreanu [21], has established the existence of infinitely many radially symmetric solutions for a differential inclusion problem in . Also, in [17], the authors extended a recent result of Ricceri concerning the existence of three critical points of certain nonsmooth functionals. Two applications have been given, both in the theory of differential inclusions; the first one concerns a nonhomogeneous Neumann boundary value problem and the second one treats a quasilinear elliptic inclusion problem in the whole . Tian and Henderson in [18], based on a nonsmooth version of critical point theory of Ricceri due to Iannizzotto [14], have established the existence of at least three solutions for the a second-order impulsive differential inclusion with a perturbed nonlinearity and two parameters. In [19], three periodic solutions with prescribed wavelength for a class of semilinear fourth-order differential inclusions are obtained by using a nonsmooth version critical point theorem.

In the present paper, motivated by [13, 18, 19], employing an abstract critical point result (see Theorem 7 below), we are interested in ensuring the existence of infinitely many periodic solutions for problem (1); see Theorem 12. We refer to [22], in which related variational methods are used for nonhomogeneous problems.

To the best of our knowledge, no investigation has been devoted to establishing the existence of infinitely many solutions to a problem such as (1). As one reference on impulsive differential inclusions, we can refer to [23].

A special case of our main result is the following theorem.

Theorem 1. Assume that and hold, and and , , . Furthermore, suppose that

Then, the problem

admits a sequence of classical solutions.

2. Basic Definitions and Preliminary Results

Let be a real Banach space. We denote by the dual space of , while stands for the duality pairing between and . A function is called locally Lipschitz if, for all , there exist a neighborhood of and a real number such that

If is locally Lipschitz and , the generalized directional derivative of at along the direction is

The generalized gradient of at is the set

So is a multifunction. We say that has compact gradient if maps bounded subsets of into relatively compact subsets of .

Lemma 2 (see [24, Proposition 1.1]). Let be a functional. Then is locally Lipschitz and

Lemma 3 (see [24, Proposition 1.3]). Let be a locally Lipschitz functional. Then is subadditive and positively homogeneous for all , and

with being a Lipschitz constant for around .

Lemma 4 (see [25]). Let be a locally Lipschitz functional. Then is upper semicontinuous and, for all , ,

Moreover, if are locally Lipschitz functionals, then

Lemma 5 (see [24, Proposition 1.6]). Let be locally Lipschitz functionals. Then

Lemma 6 (see [14, Proposition 1.6]). Let be a locally Lipschitz functional with a compact gradient. Then is sequentially weakly continuous.

We say that is a (generalized) critical point of a locally Lipschitz functional if ; that is,

When a nonsmooth functional, , is expressed as a sum of a locally Lipschitz function, , and a convex, proper, and lower semicontinuous function, , that is, , a (generalized) critical point of is every such that

for all (see [24, Chapter 3]).

Henceforth, we assume that is a reflexive real Banach space, is a sequentially weakly lower semicontinuous functional, is a sequentially weakly upper semicontinuous functional, is a positive parameter, is a convex, proper, and lower semicontinuous functional, and is the effective domain of . Write

We also assume that is coercive and

for all . Moreover, owing to (17) and provided that , we can define

If and are locally Lipschitz functionals, in [22, Theorem 2.1] the following result is proved; it is a more precise version of [21, Theorem 1.1] (see also [20]).

Theorem 7. Under the above assumption on , , and , one has the following:

(a)For every and every , the restriction of the functional to admits a global minimum, which is a critical point (local minimum) of in .(b)If , then, for each , the following alternative holds: either

(b1) possesses a global minimum or(b2) there is a sequence of critical points (local minima) of such that .

(c)If , then, for each , the following alternative holds: either

(c1) there is a global minimum of which is a local minimum of or(c2) there is a sequence of pairwise distinct critical points (local minima) of , with , which converges weakly to a global minimum of .

Now we recall some basic definitions and notations. We consider the reflexive Banach space endowed with the norm

Obviously, is a reflexive Banach space and completely embedded in . So there exists a constant , such that . From the positivity of and , it is easy to see that

is also a norm of , which is equivalent to the usual norm. Therefore, there exist two constants and such that

Thus,

Definition 8. A function is said to be a weak solution of problem (1) such that, corresponding to it, there exists a mapping with , for a.e. , and having the property that, for every , and

Definition 9. A solution is called a classical solution of the impulsive differential inclusion (1) if and , for , and

where for a.e. .

Lemma 10. If is a weak solution of (1), then is a classical solution of (1).

Proof. Let be a weak solution of (1). Then there exists with for a.e. , satisfying (23). Using integration by parts (23) becomes

so , and (25) holds for each with , . Through integration by parts, we obtain

Due to , similar to [26, Section 2], we can get and for . Now we show that the boundary conditions are satisfied. Choose any and such that if for . Then, from (25) and by means of integration by parts, we can get

Thus,

which implies that

Since satisfies (26), we have that holds for all , which implies that and in view of we get . Now using a technique similar to the technique of [18, Lemma 3.5] shows that the impulsive conditions are satisfied. From the equality

we have

Substituting (31) into (25), we have

Since satisfies (26), we have

for all . Thus, , . Similarly from the equality

we have . So is a classical solution of (1).

Now we introduce the functionals , , , and by

Thus, and

for all .

Lemma 11. Assume that , , , and hold. Then the functional is locally Lipschitz. Moreover, each critical point of is a weak solution of (1).

Proof. Let , where . Since by Lemma 2, is locally Lipschitz on . From and , we know that is locally Lipschitz on . Moreover, is compactly embedded into . Thus, is locally Lipschitz on [27, Theorem 2.2]. According to Lemma 5, we get

The explanation of (37) is as follows: for every , there is a corresponding mapping for a.e. having the property that, for every , the function and . Therefore, is locally Lipschitz on .
Now we show that each critical point of is a weak solution of (1). Assume that is a critical point of . So

So, by Lemma 2 and (38), we have

and hence a.e. on . It follows from (37) and (39) that for every we have

for all . Thus, by Definition 8, is a weak solution of (1).

3. Main Results

First for every we set

Now we formulate our main result using the following assumptions:

   ;  , , , and .

Theorem 12. Assume that and hold. Let

Then, for every and every nonnegative function satisfying , , and the assumption

  ,

putting

where when , for every problem (1) admits an unbounded sequence of classical solutions in .

Proof. Our goal is to apply Theorem 7(b) to (1). For this purpose, we fix and let be a nonnegative function satisfying ()–(). Since , we have

Now fix , put , and

If , then , , and . If , since , we have

and so

to wit, . Hence, taking into account that , one has . Now, set

for all . Assume that is exactly zero in and for each and put

It is easy to prove that is sequentially weakly lower semicontinuous on . Obviously, . By Lemma 2, is locally Lipschitz on . By Lemma 11, and are locally Lipschitz on . So, is locally Lipschitz on , and since is compactly embedded into , is locally Lipschitz on . In addition, is sequentially weakly upper semicontinuous. For all , by ,

So, we have

for all . Hence, is coercive and Under our hypotheses, we want to show that there exists a sequence of critical points for the functional ; that is, every element satisfies

Now, we prove that . For this, let be a sequence of positive numbers such that and

Put

Then, for all with , taking into account that and , one has for every . Therefore, for all ,

Moreover, from assumptions and , we have

which follows

Therefore,

Since

and taking into account, we get

Moreover, by assumption we have

Therefore, from (60) and (61), we observe that

For the fixed , inequality (58) ensures that condition (b) of Theorem 7 can be applied and either has a global minimum or there exists a sequence of weak solutions of problem (1) such that . Now we prove that for the fixed the functional has no global minimum. Let us verify that the functional is unbounded from below. Since

there exist a sequence of positive numbers and a constant such that and

for each large enough. For all , put

For any fixed , clearly and one has

and so

By (64) and (67) and since is nonnegative we observe that

for every large enough; since and , it follows that

So, the functional is unbounded from below, and it shows that has no global minimum. Therefore, from part (b) of Theorem 7, the functional admits a sequence of critical points such that Since is bounded on bounded sets and taking into account that , then has to be unbounded; that is,

Also, if is a critical point of , clearly, by definition, one has

Finally, by Lemma 11, the critical points of are weak solutions for problem (1), and, by Lemma 10, every weak solution of (1) is a classical solution of (1). Hence, we have the result.

Remark 13. Under the conditions

from Theorem 12, we see that, for every and for each , problem (1) admits a sequence of solutions which is unbounded in . Moreover, if , the result holds for every and .

The following result is a special case of Theorem 12 with .

Theorem 14. Assume that and hold. Then, for each

the problem

has an unbounded sequence of classical solutions in .

Now, we present the following example to illustrate Theorem 14.

Example 15. Let , , and for every . Thus, is a continuous positive even -periodic function on . Define the function

Clearly, is a continuous convex function with . An easy calculation shows that

Hence, assumptions and hold. Moreover, let , , and for every . Thus, is satisfied and

So, we have

Therefore, by applying Theorem 14, the problem

for has an unbounded sequence of classical solutions in .

Now we state the following consequence of Theorem 14, using the following assumptions:

() ;() .

Corollary 16. Assume that , , , , and hold. Then, the problem

has an unbounded sequence of classical solutions in .

Remark 17. Theorem 1 in Introduction is an immediate consequence of Corollary 16.

Here, we give a consequence of the main result.

Corollary 18. Let be a locally Lipschitz function such that , , for all . Furthermore, suppose that

,.

Then, for every function which is locally Lipschitz function such that , , and for all ,  ,     and satisfies the conditions

for each

the problem

has an unbounded sequence of classical solutions in .

Proof. Set for all . Assumption along with the condition

yields

Moreover, assumption together with the condition

ensures

Since

by applying Theorem 14, we have the desired conclusion.

Remark 19. We observe that in Theorem 12 we can replace with , and then by the same argument as in the proof of Theorem 12, but using conclusion (c) of Theorem 7 instead of (b), problem (1) has a sequence of pairwise distinct classical solutions, which strongly converges to in .

We end this paper by presenting the following example.

Example 20. Let , , and for every . Thus, is a continuous positive even -periodic function on . Now let for all and for all . Hence, is locally Lipschitz function and

Therefore, and hold, and also we can simply see that and hold. Moreover, let , , and for every . Thus, is satisfied. An easy computation shows that

for some . So,

Also

Hence, taking Remark 19 into account, the problem

for and has an unbounded sequence of pairwise distinct classical solutions in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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