Abstract
In this paper we study the existence and the multiplicity of solutions for an impulsive boundary value problem for fourthorder differential equations. The notions of classical and weak solutions are introduced. Then the existence of at least one and infinitely many nonzero solutions is proved, using the minimization, the mountainpass, and Clarke’s theorems.
MSC: 34B15, 34B37, 58E30.
Keywords:
fourthorder differential equations; impulsive conditions; weak solution; classical solution; PalaisSmale condition; mountainpass theorem; Clarke’s theorem1 Introduction
The theory of impulsive boundary value problems (IBVPs) became an important area of studies in recent years. IBVPs appear in mathematical models of processes with sudden changes in their states. Such processes arise in population dynamics, optimal control, pharmacology, industrial robotics, etc. For an introduction to theory of IBVPs one is referred to [1]. Some classical tools used in the study of impulsive differential equations are topological methods as fixed point theorems, monotone iterations, upper and lower solutions (see [24]). Recently, some authors have studied the existence of solutions of IBVPs using variational methods. The pioneering work in this direction is the paper of Nieto and O’Regan [5], where the secondorder impulsive problem
(with
In this paper, we consider the boundary value problem for fourthorder differential equation with impulsive effects
Here,
We look for solutions in the classical sense, as given in the next definition.
Definition 1 A function
Moreover, we introduce, for every
To deduce the existence of solutions, we assume the following conditions:
(H1) The constant a is positive, b and c are continuous functions on
(H2) There exist
A simple example of functions fulfilling the last condition is given by
where
Note that (2) implies that there exist positive constants
In the next section we will prove the following existence result for
Theorem 2Suppose that
Having in mind the case
(H3) There exist positive constants
A simple example of this new situation is given by the functions
The result to be proven is the following.
Theorem 3Suppose that
If we consider the problem
we introduce the following condition.
(H2′) There exist
A simple example now is
The obtained result is the following.
Theorem 4Suppose that
The proofs of the main results are given in Section 3.
2 Preliminaries
Denote by
Denote by
and
Let
and the corresponding norm.
By assumption (H_{1}) an equivalent scalar product and norm in X are given by
and
It is well known (see [[9], Lemma 2.2], [11]) that the following Poincaré and imbedding inequalities hold for all
where M is a positive constant depending on T, a and
We have the following compactness embedding, which can be proved in the standard way.
Proposition 5The inclusion
We define the functional
By assumption (H1), we find that
In the sequel we introduce the concept of a weak solution of our problem.
Definition 6 A function
As a consequence, the critical points of ϕ are the weak solutions of the problem (P). Let us see that they are, actually, strong solutions too.
Lemma 7Ifuis a weak solution of (P) then
Proof Let
This means that for every
and
By a standard regularity argument (see [9,11]) the weak derivative
We have for
Summing the last identities for
Therefore, by (11) and (12), we have
Now, take a test function
Then we obtain
In the proofs of the theorems, we will use three critical point theorems which are the main tools to obtain weak solutions of the considered problems.
To this end, we introduce classical notations and results. Let E be a reflexive real Banach space. Recall that a functional
We have the following wellknown minimization result.
Theorem 8LetIbe a weakly lower semicontinuous operator that has a bounded minimizing sequence on a reflexive real Banach
spaceE. ThenIhas a minimum
Note that a functional
Next, recall the notion of the PalaisSmale (PS) condition, the mountainpass theorem and Clarke’s theorem.
We say that I satisfies condition (PS) if any sequence
Theorem 9 ([[14], p.4])
LetEbe a real Banach space and
(i) there are constants
(ii) there is an
ThenIpossesses a critical value
Theorem 10 ([[14], p.53])
LetEbe a real Banach space and
ThenIpossesses, at least, mdistinct pairs of critical points.
3 Proofs of main results
This section is devoted to the proof of the three theorems enunciated in the introduction of this work.
First consider the case
Lemma 11Suppose that
Proof Let
Then we have
and
for all sufficiently large k,
In particular,
Adding the last inequality with (14), by assumption (H2), we obtain
which implies that
Then, by the compact inclusion
it follows that
and
Then by (16) it follows that
i.e.,
Now, we are in a position to prove the main results of this paper.
Proof of Theorem 2 We find by (H1) and (8) that the following inequalities are valid for every
It is evident that this last expression is strictly positive when
where
Since
Now consider the case
Lemma 12Suppose that
Proof By
Further, if
Now we are in a position to prove the next existence result for the problem (P).
Proof of Theorem 3 By assumption, we know that
Moreover, for
Clearly
and there exist positive constants
where
Arguing as in [[15], pp.1618], one can prove that there exists
Denote
By (H3) we see that for every
and
Denote
where
By the last inequality, it follows that
we obtain
By Clarke’s Theorem 10, there exist at least m pairs of different critical points of the functional ϕ. Since m is arbitrary, there exist infinitely many solutions of the problem (P), which concludes the proof. □
Concerning the problem (P_{1}), one can introduce similarly the notions of classical and weak solutions. In this
case it is not difficult to verify that the weak solutions are critical points of
the functional
Proof of Theorem 4 By the Poincaré inequalities (7) we find that
Since the functional
is sequentially weakly continuous, from the fact that the inclusion
Next, let us see that
where
Then, by Theorem 8, there exists a minimizer of
Let u be a weak solution of (P_{1}), i.e., a critical point of
If
which is a contradiction. Then, for
Suppose now that
Take
where
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors are thankful to the anonymous referees for the careful reading of the manuscript and suggestions.
References

Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations, World Scientific, Singapore (1995)

Agarwal, RP, O’Regan, D: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput.. 114, 51–59 (2000). Publisher Full Text

Georgescu, P, Moroşanu, G: Pest regulation by means of impulsive controls. Appl. Math. Comput.. 190, 790–803 (2007). Publisher Full Text

Franco, D, Nieto, JJ: Maximum principle for periodic impulsive first order problems. J. Comput. Appl. Math.. 88, 149–159 (1998). Publisher Full Text

Nieto, J, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl.. 10, 680–690 (2009). Publisher Full Text

Bonanno, G, Di Bella, B, Henderson, J: Existence of solutions to secondorder boundaryvalue problems with small perturbations of impulses. Electron. J. Differ. Equ.. 2013, (2013) Article ID 126

Xiao, J, Nieto, JJ: Variational approach to some damped Dirichlet nonlinear impulsive differential equations. J. Franklin Inst.. 348, 369–377 (2011). Publisher Full Text

Afrouzi, G, Hadjian, A, Radulescu, V: Variational approach to fourthorder impulsive differential equations with two control parameters. Results Math. (2013). Publisher Full Text

Sun, J, Chen, H, Yang, L: Variational methods to fourthorder impulsive differential equations. J. Appl. Math. Comput.. 35, 323–340 (2011). Publisher Full Text

Xie, J, Luo, Z: Solutions to a boundary value problem of a fourthorder impulsive differential equation. Bound. Value Probl.. 2013, (2013) Article ID 154. http://www.boundaryvalueproblems.com/content/2013/1/154

Tersian, S, Chaparova, J: Periodic and homoclinic solutions of extended FisherKolmogorov equations. J. Math. Anal. Appl.. 260, 490–506 (2001). Publisher Full Text

Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems, Springer, New York (1989)

Berger, M: Nonlinearity and Functional Analysis, Academic Press, New York (1977)

Rabinowitz, P: Minimax Methods in Critical Point Theory with Applications to Differential Equations, Am. Math. Soc., Providence (1986)

Chen, J, Tang, XH: Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory. Abstr. Appl. Anal.. 2012, (2012) Article ID 648635
Article ID 648635
Publisher Full Text