Abstract
In this paper, we consider the existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions. Infinitely many solutions are obtained by using a version of the symmetric mountainpass theorem, and this sequence of solutions converge to zero. Some recent results are extended.
MSC: 34B37, 35B38.
Keywords:
impulsive effects; variational methods; Dirichlet boundary value problem; critical points1 Introduction
In this paper, we study the following nonlinear impulsive differential equations with Dirichlet boundary conditions
where
In recent years, a great deal of works have been done in the study of the existence of solutions for impulsive boundary value problems, by which a number of chemotherapy, population dynamics, optimal control, ecology, industrial robotics and physics phenomena are described. For the general aspects of impulsive differential equations, we refer the reader to the classical monograph [2]. For some general and recent works on the theory of impulsive differential equations, we refer the reader to [313]. Some classical tools or techniques have been used to study such problems in the literature. These classical techniques include the coincidence degree theory [14], the method of upper and lower solutions with a monotone iterative technique [15], and some fixed point theorems in cones [16,17].
On the other hand, in the last few years, many authors have used a variational method to study the existence and multiplicity of solutions for boundary value problems without impulsive effects [1821]. For related basic information, we refer the reader to [22,23].
For a second order differential equation
A new approach via critical point and variational methods is proved to be very effective in studying the boundary problem for differential equations. For some general and recent works on the theory of critical point theory and variational methods, we refer the reader to [2837].
More precisely, in [28] the authors studied the following equations with Dirichlet boundary conditions:
They obtained the existence of solutions for problems by using the variational method. Zhang and Yuan [30] extended the results in [28]. They obtained the existence of solutions for problem (1.2) with a perturbation term. Also, they obtained infinitely many solutions for problem (1.2) under the assumption that the nonlinearity f is a superlinear case. Soon after that, Zhou and Li [29] extended problem (1.2). In all the abovementioned works, the information on the sequence of solutions was not given.
Motivated by the fact above, the aim of this paper is to show the existence of infinitely many solutions for problem (1.1), and that there exists a sequence of infinitely many arbitrarily small solutions, converging to zero, by using a new version of the symmetric mountainpass lemma due to Kajikiya [1]. Our main results extend the existing study.
Throughout this paper, we assume that
(I_{1})
(I_{2}) There exist
(H_{1})
(H_{2})
(H_{3})
The main result of this paper is as follows.
Theorem 1.1Suppose that (I_{1})(I_{2}) and (H_{1})(H_{3}) hold. Then problem (1.1) has a sequence of nontrivial solutions
Remark 1.1 Without the symmetry condition (i.e.,
Remark 1.2 We should point out that Theorem 1.1 is different from the previous results of [2837] in three main directions:
(1) We do not make the nonlinearity f satisfy the wellknown AmbrosettiRabinowitz condition [23];
(2) We try to use LusternikSchnirelman’s theory for
(3) We can obtain a sequence of nontrivial solutions
Remark 1.3 There exist many functions
2 Preliminary lemmas
In this section, we first introduce some notations and some necessary definitions.
Definition 2.1 Let E be a Banach space and
Definition 2.2 Let E be a real Banach space. For any sequence
In the Sobolev space
which induces the norm
It is a consequence of Poincaré’s inequality that
Here,
In this paper, we will assume that
which induces the equivalent norm
Lemma 2.1[29]
If
Lemma 2.2[29]
There exists
where
For
Taking
by v and integrating between 0 and T, we have
Moreover, since
Combining (2.3), we get
Lemma 2.3A weak solution of (1.1) is a function
for any
Consider
where
Thus, the solutions of problem (1.1) are the corresponding critical points of J.
Lemma 2.4If
Proof Obviously, we have
For
By the definition of weak derivative, the equality above implies that
Hence
Combining this fact with (2.8), we get
Hence,
Lemma 2.5If
Proof Let
We have that
This completes the proof. □
Under assumptions (H_{1}) and (H_{2}), we have
which means that for all
Hence, for every positive constant k, we have
where
Lemma 2.6Suppose that (I_{1})(I_{2}) and (H_{1})(H_{3}) hold, then
Proof Let
By condition (I_{1}), we can deduce that
Setting
where
Thus, (2.12) and (2.13) imply that
as
as
Therefore,
3 Existence of a sequence of arbitrarily small solutions
In this section, we prove the existence of infinitely many solutions of (1.1), which tend to zero. Let X be a Banach space and denote
For
If there is no mapping φ as above for any
Proposition 3.1LetAandBbe closed symmetric subsets ofX, which do not contain the origin. Then the following hold.
(1) If there exists an odd continuous mapping fromAtoB, then
(2) If there is an odd homeomorphism fromAtoB, then
(3) If
(4) Thenndimensional sphere
(5) IfAis compact, then
Let
Lemma 3.1LetEbe an infinitedimensional space and
(C_{1})
(C_{2}) For each
Then either (R_{1}) or (R_{2}) below holds.
(R_{1}) There exists a sequence
(R_{2}) There exist two sequences
Remark 3.1 From Lemma 3.1, we have a sequence
In order to get infinitely many solutions, we need some lemmas. Under the assumptions
of Theorem 1.1, let
where
Let
Then it is easy to see
Then
where
From the arguments above, we have the following.
Lemma 3.2Let
(i)
(ii) If
(iii) Suppose that (I_{1})(I_{2}) and (H_{1})(H_{3}) hold, then
Proof It is easy to see (i) and (ii). (iii) are consequences of (ii) and Lemma 2.6. □
Lemma 3.3Assume that (I_{2}) and (H_{3}) hold. Then for any
Proof Firstly, by (H_{3}) of Theorem 1.1, for any fixed
Secondly, from Lemma 5 of [33], we have that for any finite dimensional subspace
Therefore, for any
since
This completes the proof. □
Now, we give the proof of Theorem 1.1 as following.
Proof of Theorem 1.1 Recall that
and define
By Lemma 3.2(i) and Lemma 3.3, we know that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
CZ carried out the theoretical studies, and participated in the sequence alignment and drafted the manuscript. FM participated in the design of the study and performed the statistical analysis. SL conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
Acknowledgements
The authors are supported by the Research Foundation during the 12th FiveYear Plan Period of Department of Education of Jilin Province, China (Grant [2013] No. 252), the China Postdoctoral Science Foundation (Grant No. 2012M520665), the Youth Foundation for Science and Technology Department of Jilin Province (20130522100JH), the open project program of Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University (Grant No. 93K172013K03), the Natural Science Foundation of Changchun Normal University.
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