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
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 . 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 , 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:
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.
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 . 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.
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 .
It is easy to see that, by choosing
which induces the norm
The following lemmas are useful for proving our main result. Their proofs can be found in .
Lemma 1 ([, Proposition 2.1])
Lemma 2 ([, Lemma 2.1 ])
In the next section we shall prove our results applying the following infinitely many critical points theorem obtained in . 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
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  for more details.
Theorem 2.1 (see , Theorem 7.4)
We recall that Theorem 2.1 improves [, 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
Our first result is as follows.
Theorem 3.1Assume that
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, [, 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
From (5) and (6) one has
and owing to (4), one has
Hence, [, Proposition III.30] ensures that strongly converges to and our claim is proved.
Second, we wish to prove that
So, from assumptions (a2) and (i2),
Clearly, one has
Moreover, bearing in mind (a1) and (i1),
As an example, we point out below a special case of Theorem 3.1.
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
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. □
Theorem 3.3Assume that
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
Proof The conclusion follows from Theorem 3.2 by arguing as in the proof of Theorem 3.3. □
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.
has an unbounded sequence of weak solutions.
Proof It is enough to apply Theorem 3.3 to the following function:
In many papers [13,20,22,28] and , 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:
Now, we give an application of Theorem 3.4.
Example 4.2 Consider the Dirichlet problem
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.
The authors declare that they have no competing interests.
All authors have contributed equally to this research work. All authors read and approved the final manuscript.
Chen, L, Sun, J: Nonlinear boundary value problem for first order impulsive functional differential equations. J. Math. Anal. Appl.. 318, 726–741 (2006). Publisher Full Text
Chu, J, Nieto, JJ: Impulsive periodic solutions of first-order singular differential equations. Bull. Lond. Math. Soc.. 40(1), 143–150 (2008). Publisher Full Text
Agarwal, RP, Franco, D, O’Regan, D: Singular boundary value problems for first and second order impulsive differential equations. Aequ. Math.. 69, 83–96 (2005). Publisher Full Text
Lee, EL, Lee, YH: Multiple positive solutions of two point boundary value problems for second order impulsive differential equations. Appl. Math. Comput.. 158, 745–759 (2004). Publisher Full Text
Chen, J, Tisdell, CC, Yuan, R: On the solvability of periodic boundary value problems with impulse. J. Math. Anal. Appl.. 331, 902–912 (2007). Publisher Full Text
Luo, Z, Nieto, JJ: New result for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Anal. TMA. 70, 2248–2260 (2009). Publisher Full Text
Lee, YH, Liu, X: Study of singular boundary value problems for second order impulsive differential equation. J. Math. Anal. Appl.. 331, 159–176 (2007). Publisher Full Text
Qian, D, Li, X: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal. Appl.. 303, 288–303 (2005). Publisher Full Text
Liu, Z, Chen, H, Zhou, T: Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem. Comput. Math. Appl.. 61, 1687–1699 (2011). Publisher Full Text
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.. 73, 440–449 (2010). Publisher Full Text
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.. 17, 426–432 (2012). Publisher Full Text
Zhang, Z, Yuan, R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl.. 11, 155–162 (2010). Publisher Full Text
Zhang, D, Dai, B: Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions. Comput. Math. Appl.. 61, 3153–3160 (2011). Publisher Full Text
Bai, L, Dai, B: An application of variational methods to a class of Dirichlet boundary value problems with impulsive effects. J. Franklin Inst.. 348, 2607–2624 (2011). Publisher Full Text
Chen, P, Tang, XH: New existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Math. Comput. Model.. 55, 723–739 (2012). Publisher Full Text
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.. 11, 4062–4071 (2010). Publisher Full Text
Wang, W, Yang, X: Multiple solutions of boundary-value problems for impulsive differential equations. Math. Methods Appl. Sci.. 34, 1649–1657 (2011). Publisher Full Text
Bonanno, G: A critical point theorem via the Ekeland variational principle. Nonlinear Anal. TMA. 75(5), 2992–3007 (2012). Publisher Full Text
Ricceri, B: A general variational principle and some of its applications. J. Comput. Appl. Math.. 113, 401–410 (2000). Publisher Full Text
Zhou, J, Li, Y: Existence and multiplicity of solutions for some Dirichlet problems with impulse effects. Nonlinear Anal. TMA. 71, 2856–2865 (2009). Publisher Full Text