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.
Similar content being viewed by others
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 1.1 Let be a continuous function and put for 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
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
for any .
Lemma 2 ([[24], Lemma 2.1 ])
is a weak solution of () if and only if u is a classical solution of ().
Now, we define the functionals in the following way:
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)
Let X be a real Banach space, and let be two continuously Gâteaux differentiable functionals such that Φ is bounded from below. For every , let us put
and
-
(a)
If and for each , the functional satisfies the -condition for all , then for each , the following alternative holds: either
(a1) has a global minimum
or
(a2) there exists a sequence of critical points (local minima) of such that .
-
(b)
If and for each , the functional satisfies 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) of such 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 3.1 Assume that
(a1) for all ;
(a2) .
Then, for every and 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,
Now, from , there is a sequence , with , such that
for all with and for all . Setting = , one has
for all . Moreover, having in mind that , one has
that is,
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
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),
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 3.1 Let be a continuous function, put for every , and let . Assume that
Then, for each , and for each continuous function such 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 3.2 Assume that
(a1) for all ;
(b2) .
Then, for every and 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 3.3 Assume that
(c1) for all ;
(c2) .
Then, for every and 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 3.4 Assume
(c1) for all ;
() .
Then, for every and 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 3.2 Let be an -Carathéodory function satisfying (c1) and (c2) of Theorem 3.3.
Then, for every , for every nonnegative -Carathéodory function , whose potential satisfies
and for every continuous function , , whose potential , , satisfies (i1) and (i2) of Theorem 3.3, there exist and , where
such that for all and 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
and
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:
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
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.
References
Benchohra M, Henderson J, Ntouyas S Contemporary Mathematics and Its Applications 2. In Theory of Impulsive Differential Equations. Hindawi Publishing Corporation, New York; 2006.
Chen L, Sun J: Nonlinear boundary value problem for first order impulsive functional differential equations. J. Math. Anal. Appl. 2006, 318: 726-741. 10.1016/j.jmaa.2005.08.012
Chu J, Nieto JJ: Impulsive periodic solutions of first-order singular differential equations. Bull. Lond. Math. Soc. 2008, 40(1):143-150. 10.1112/blms/bdm110
Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.
Agarwal RP, Franco D, O’Regan D: Singular boundary value problems for first and second order impulsive differential equations. Aequ. Math. 2005, 69: 83-96. 10.1007/s00010-004-2735-9
Baek H: Extinction and permanence of a three-species Lotka-Volterra system with impulsive control strategies. Discrete Dyn. Nat. Soc. 2008., 2008: Article ID 752403
Lee EL, Lee YH: Multiple positive solutions of two point boundary value problems for second order impulsive differential equations. Appl. Math. Comput. 2004, 158: 745-759. 10.1016/j.amc.2003.10.013
Chen J, Tisdell CC, Yuan R: On the solvability of periodic boundary value problems with impulse. J. Math. Anal. Appl. 2007, 331: 902-912. 10.1016/j.jmaa.2006.09.021
Luo Z, Nieto JJ: New result for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Anal. TMA 2009, 70: 2248-2260. 10.1016/j.na.2008.03.004
Lee YH, Liu X: Study of singular boundary value problems for second order impulsive differential equation. J. Math. Anal. Appl. 2007, 331: 159-176. 10.1016/j.jmaa.2006.07.106
Mawhin J: Topological degree and boundary value problems for nonlinear differential equations. Lecture Notes in Math. 1537. In Topological Methods for Ordinary Differential Equations. Springer, Berlin; 1993:74-142.
Qian D, Li X: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal. Appl. 2005, 303: 288-303. 10.1016/j.jmaa.2004.08.034
Chen H, Li J: Variational approach to impulsive differential equations with Dirichlet boundary conditions. Bound. Value Probl. 2010., 2010: Article ID 3254152
Liu Z, Chen H, Zhou T: Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem. Comput. Math. Appl. 2011, 61: 1687-1699. 10.1016/j.camwa.2011.01.042
Nieto JJ, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 70: 680-690.
Sun J, Chen H, Yang L: The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method. Nonlinear Anal. 2010, 73: 440-449. 10.1016/j.na.2010.03.035
Xiao J, Nieto JJ, Luo Z: Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 426-432. 10.1016/j.cnsns.2011.05.015
Zhang Z, Yuan R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 2010, 11: 155-162. 10.1016/j.nonrwa.2008.10.044
Zhang D, Dai B: Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions. Comput. Math. Appl. 2011, 61: 3153-3160. 10.1016/j.camwa.2011.04.003
Bai L, Dai B: An application of variational methods to a class of Dirichlet boundary value problems with impulsive effects. J. Franklin Inst. 2011, 348: 2607-2624. 10.1016/j.jfranklin.2011.08.003
Chen P, Tang XH: New existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Math. Comput. Model. 2012, 55: 723-739. 10.1016/j.mcm.2011.08.046
Sun J, Chen H: Multiplicity of solutions for class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems. Nonlinear Anal., Real World Appl. 2010, 11: 4062-4071. 10.1016/j.nonrwa.2010.03.012
Wang W, Yang X: Multiple solutions of boundary-value problems for impulsive differential equations. Math. Methods Appl. Sci. 2011, 34: 1649-1657. 10.1002/mma.1472
Bonanno G, Di Bella B, Henderson J: Existence of solutions to second-order boundary-value problems with small perturbations of impulses. Electron. J. Differ. Equ. 2013, 126: 1-14.
Bonanno G: A critical point theorem via the Ekeland variational principle. Nonlinear Anal. TMA 2012, 75(5):2992-3007. 10.1016/j.na.2011.12.003
Ricceri B: A general variational principle and some of its applications. J. Comput. Appl. Math. 2000, 113: 401-410. 10.1016/S0377-0427(99)00269-1
Brezis H: Analyse fonctionnelle; théorie et applications. Masson, Paris; 1983.
Zhou J, Li Y: Existence and multiplicity of solutions for some Dirichlet problems with impulse effects. Nonlinear Anal. TMA 2009, 71: 2856-2865. 10.1016/j.na.2009.01.140
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Bonanno, G., Di Bella, B. & Henderson, J. Infinitely many solutions for a boundary value problem with impulsive effects. Bound Value Probl 2013, 278 (2013). https://doi.org/10.1186/1687-2770-2013-278
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-2770-2013-278