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# Infinitely many solutions for a boundary value problem with impulsive effects

Gabriele Bonanno1*, Beatrice Di Bella2 and Johnny Henderson3

Author Affiliations

1 Department of Civil, Information Technology, Construction, Environmental Engineering and Applied Mathematics, University of Messina, Messina, 98166, Italy

2 Department of Mathematics and Computer Science, University of Messina, Messina, 98166, Italy

3 Department of Mathematics, Baylor University, Waco, 76798-7328, USA

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Boundary Value Problems 2013, 2013:278  doi:10.1186/1687-2770-2013-278

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/278

 Received: 24 October 2013 Accepted: 1 December 2013 Published: 20 December 2013

© 2013 Bonanno et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper we are interested in multiplicity results for a nonlinear Dirichlet boundary value problem subject to perturbations of impulsive terms. The study of the problem is based on the variational methods and critical point theory. Infinitely many solutions follow from a recent variational result.

MSC: 34B37, 34B15.

##### Keywords:
impulsive differential equations; critical points; infinitely many solutions

### 1 Introduction

The theory of impulsive differential equations provides a general framework for the mathematical modeling of many real world phenomena; see, for instance, [1-3] and [4]. Indeed, many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. Impulsive differential equations are basic tools for studying these phenomena [5,6].

There are some common techniques to approach these problems: the fixed point theorems [7,8], the method of upper and lower solutions [9], or the topological degree theory [10-12]. On the other hand, in the last few years, some authors have studied the existence of solutions by variational methods; see [13-19].

Here, we use critical point theory to investigate the existence of infinitely many solutions for the following nonlinear impulsive differential problem:

where , , , with and , , , and are continuous for every .

We establish some multiplicity results for problem () under an appropriate oscillation behavior of the primitive of the nonlinearity g and a suitable growth of the primitive of at infinity, for all λ belonging to a precise interval and provided μ is small enough (Theorem 3.3, Theorem 3.4). It is worth noticing that, when the impulsive effects , , are sublinear at infinity, our results hold for all (see Remark 3.1). Here, as an example of our results, we present the following special case of Theorem 3.3.

Theorem 1.1Letbe a continuous function and putfor every. Assume that

Then there is, where, such that for each, the problem

admits infinitely many pairwise distinct classical solutions.

We explicitly observe that in Theorem 1.1 impulsive effects , , (that is, for all ) are linear, contrary to the usual assumption of sublinearity of impulses; see [14,16,20-22] and [23]. The rest of this paper is organized as follows. In Section 2, we introduce some notations and preliminary results. Moreover, the abstract critical point theorem (Theorem 2.1) is recalled. In Section 3, we obtain some existence results. In Section 4, we give some examples to illustrate our results.

### 2 Preliminaries

By a classical solution of () we mean a function

that satisfies the equation in () a.e. on , the limits , , , exist, that satisfies the impulsive conditions and the boundary conditions . Clearly, if a, b and g are continuous, then a classical solution , , satisfies the equation in () for all .

We consider the following slightly different form of problem ():

where , and with .

It is easy to see that, by choosing

the solutions of () are solutions of ().

Let us introduce some notations. In the Sobolev space , consider the inner product

which induces the norm

The following lemmas are useful for proving our main result. Their proofs can be found in [24].

Lemma 1 ([[24], Proposition 2.1])

Let. Then

(1)

where.

Here, and in the sequel, is an L1-Carathéodory function, namely:

(a) is measurable for every ;

(b) is continuous for almost every ;

(c) for every , there exists a function such that

for almost every .

Definition 1 A function is said to be a weak solution of () if u satisfies

(2)

for any .

Lemma 2 ([[24], Lemma 2.1 ])

is a weak solution of () if and only ifuis a classical solution of ().

Now, we define the functionals in the following way:

(3)

for each , where for each . Using the property of f and the continuity of , , we have that and for any , one has

and

So, arguing in a standard way, it is possible to prove that the critical points of the functional are the weak solutions of problem () and so they are classical solutions.

In the next section we shall prove our results applying the following infinitely many critical points theorem obtained in [25]. First, we recall the following definition.

Definition 2 Let X be a real Banach space, two Gâteaux differentiable functionals, . We say that functional satisfies the Palais-Smale condition cut off upper at r (in short -condition) if any sequence , such that

(α) is bounded,

(β) ,

(γ) for all ,

has a convergent subsequence.

When , the previous definition is the same as the classical definition of the Palais-Smale condition, while if , such a condition is more general than the classical one. We refer to [25] for more details.

Theorem 2.1 (see [25], Theorem 7.4)

LetXbe a real Banach space, and letbe two continuously Gâteaux differentiable functionals such that Φ is bounded from below. For every, let us put

and

(a) Ifand for each, the functionalsatisfies the-condition for all, then for each, the following alternative holds: either

(a1) has a global minimum

or

(a2) there exists a sequenceof critical points (local minima) ofsuch that.

(b) Ifand for each, the functionalsatisfies the-condition for all, then for each, the following alternative holds: either

(b1) there exists a global minimum of Φ which is a local minimum of

or

(b2) there exists a sequence of pairwise distinct critical points (local minima) ofsuch that.

We recall that Theorem 2.1 improves [[26], Theorem 2.5] since no assumptions with respect to weak topology of X are made. In particular, the set is not involved in the definition of φ and the sequential weak lower semicontinuity of is not required.

### 3 Main results

In this section, we present our main results. Put

Moreover, let

Our first result is as follows.

Theorem 3.1Assume that

(a1) for all;

(a2) .

Then, for everyand for every continuous function, , whose potential, , satisfies

(i1) ;

(i2) ,

there exists, where

such that for every, problem () has an unbounded sequence of weak solutions.

Proof First, we observe that owing to (a2) the interval Λ is non-empty. Moreover, for each and taking into account that , one has . Now, fix λ and μ as in the conclusion. Our aim is to apply Theorem 2.1. For this end, take and Φ, Ψ as in (3).

We divide our proof into three steps in order to show Theorem 3.1. First, we prove that satisfies the -condition for all . So, fix and let be a sequence such that is bounded, and for all . From , taking into account that Φ is coercive, is bounded in X. Since the embedding of X in is compact (see, for instance, [[27], Theorem 8.8]) and X is reflexive, up to a subsequence, is uniformly convergent to , and is weakly convergent to in X. The uniform convergence of , taking also into account Lebesgue’s theorem, ensures that

that is,

(4)

Now, from , there is a sequence , with , such that

for all with and for all . Setting = , one has

(5)

for all . Moreover, having in mind that , one has

that is,

(6)

From (5) and (6) one has

and owing to (4), one has

Hence, [[27], Proposition III.30] ensures that strongly converges to and our claim is proved.

Second, we wish to prove that

Let be a sequence of positive numbers such that and

Put for all . By Lemma 1, for all , one has

for all such that . Hence, one has

So, from assumptions (a2) and (i2),

Assumption immediately yields

that is, . The previous inequality assures that conclusion (a) of Theorem 2.1 can be used, for which either has a global minimum or there exists a sequence of solutions of problem () such that .

The final step is to verify that the functional has no global minimum. From , and taking into account that , there is such that

(7)

So, there exists a sequence of positive numbers such that and

It follows that there is such that for all , one has

Now, consider a function defined by setting

Clearly, one has

Moreover, bearing in mind (a1) and (i1),

(8)

Putting together (7) and (8), we get that the functional is unbounded from below and so it has no global minimum.

Therefore, Theorem 2.1 assures that there is a sequence of critical points of such that and, taking into account the considerations made in Section 2, the theorem is completely proved. □

Remark 3.1 Assume that . Clearly, condition (a1) holds, and condition (a2) assumes the following simpler form:

()

In particular, if and , then () holds and problem () has an unbounded sequence of weak solutions in X for every pair .

Moreover, under the assumption , Theorem 3.1 guarantees the existence of infinitely many solutions to problem () for every .

As an example, we point out below a special case of Theorem 3.1.

Corollary 3.1Letbe a continuous function, putfor every, and let. Assume that

Then, for each, and for each continuous functionsuch that, , the problem

admits infinitely many pairwise distinct classical solutions.

Replacing the condition at infinity of the potential F by a similar one at zero, and arguing as in the proof of Theorem 3.1 but using conclusion (b) of Theorem 2.1 instead of (a), one establishes the following result. Put

Theorem 3.2Assume that

(a1) for all;

(b2) .

Then, for everyand for every continuous function, , whose potential, , satisfies

(i1) ;

(j2) ,

there exists, where

such that for every, problem () has a sequence of non-zero weak solutions, which strongly converges to 0.

Proof We take X, Φ and Ψ as in the proof of Theorem 3.1. Fix , let be a function that satisfies assumptions (i1) and (j2) and take . Arguing as in the proof of Theorem 3.1, one has . Now, arguing again as in the proof of Theorem 3.1, there is a sequence of positive numbers such that and for all and for some . By choosing as in the proof of Theorem 3.1, the sequence strongly converges to 0 in X and for each . Therefore, taking into account that , 0 is not a local minimum of . The part (b) of Theorem 2.1 ensures that there exists a sequence in X of critical points of such that and the proof is complete. □

Let be a primitive of , an -Carathéodory function and put

Moreover, let

In virtue of Theorems 3.1 and 3.2, we obtain the following results for problem ().

Theorem 3.3Assume that

(c1) for all;

(c2) .

Then, for everyand for every continuous function, , whose potential, , satisfies

(i1) ,

(i2) ,

there exists, where

such that for each, problem () has an unbounded sequence of weak solutions.

Proof As seen in Section 2, we put , and , . Clearly, one has , , , , . Hence, from Theorem 3.1 the conclusion is achieved. □

Remark 3.2 Theorem 1.1 in Introduction is an immediate consequence of Theorem 3.3. In fact, it is enough to observe that (c1) is verified and one has and , for which . Moreover, from , one has and the conclusion is achieved.

Replacing the condition at infinity of the potential G by a similar one at zero, one establishes the following result. Put

Theorem 3.4Assume

(c1) for all;

() .

Then, for everyand for every continuous function, , whose potential, , satisfies

(i1) ,

(j2) ,

there exists, where

such that for each, problem () has a sequence of non-zero weak solutions, which strongly converges to 0.

Proof The conclusion follows from Theorem 3.2 by arguing as in the proof of Theorem 3.3. □

Remark 3.3 We point out that in Theorem 3.3 (as in Theorem 3.4) the assumption can be deleted provided that we assume the constant and the interval .

Finally, we observe that the existence of infinitely many solutions to problem () can be obtained from Theorem 3.3 and Theorem 3.4 even under small perturbations of the nonlinearity. As an example, we point out the following consequence of Theorem 3.3.

Corollary 3.2Letbe an-Carathéodory function satisfying (c1) and (c2) of Theorem 3.3.

Then, for every, for every nonnegative-Carathéodory function, whose potentialsatisfies

and for every continuous function, , whose potential, , satisfies (i1) and (i2) of Theorem 3.3, there existand, where

such that for alland for all, the problem

has an unbounded sequence of weak solutions.

Proof It is enough to apply Theorem 3.3 to the following function:

where is fixed in and is fixed in Λ. In fact, one has

(9)

and

(10)

for which , that is, . Moreover, from (9) one has and from (10) . Hence, and Theorem 3.3 ensures the conclusion. □

### 4 Applications

In many papers [13,20,22,28] and [23], the authors obtain the existence of infinitely many solutions for problem () while the impulsive term is supposed to be odd. The next examples provide problems that admit infinitely many solutions for which those other results cannot be applied.

Example 4.1 Consider the following boundary value problem:

(11)

where is the function defined as follows:

It is easy to see that conditions (a1), (a2), (i1) and (i2) of Theorem 3.1 hold. In particular, and

Then, for each and for every , problem (11) has an unbounded sequence of solutions in X.

Now, we give an application of Theorem 3.4.

Example 4.2 Consider the Dirichlet problem

(12)

where is the function defined as follows:

By a simple calculation, we get and

Then, from Theorem 3.4, for each and for every , problem (12) admits a sequence of pairwise distinct classical solutions strongly converging at 0. We observe that, in this case, as direct computations show, also zero is a solution of the problem.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors have contributed equally to this research work. All authors read and approved the final manuscript.

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