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# Multiplicity of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions

- Chenxing Zhou
^{1}, - Fenghua Miao
^{1}and - Sihua Liang
^{1, 2}Email author

**2013**:200

https://doi.org/10.1186/1687-2770-2013-200

© Zhou et al.; licensee Springer 2013

**Received:**17 June 2013**Accepted:**20 August 2013**Published:**5 September 2013

## 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 mountain-pass 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 points

## 1 Introduction

where $p>2$, $T>0$, $\mu >0$, $f:[0,T]\times \mathbb{R}\to \mathbb{R}$ is continuous, $a\in {L}^{\mathrm{\infty}}[0,T]$, *N* is a positive integer, $0={t}_{0}<{t}_{1}<{t}_{2}<\cdots <{t}_{N}<{t}_{N+1}=T$, $\mathrm{\Delta}{u}^{\prime}({t}_{j})={u}^{\prime}({t}_{j}^{+})-{u}^{\prime}({t}_{j}^{-})={lim}_{t\to {t}_{j}^{+}}{u}^{\prime}(t)-{lim}_{t\to {t}_{j}^{-}}{u}^{\prime}(t)$, ${I}_{j}:\mathbb{R}\to \mathbb{R}$ are continuous. With the help of the symmetric mountain-pass lemma due to Kajikiya [1], we prove that there are infinitely many small weak solutions for equations (1.1) with the general nonlinearities $f(t,u)$.

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 [3–13]. 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 [18–21]. For related basic information, we refer the reader to [22, 23].

For a second order differential equation ${u}^{\u2033}=f(t,u,{u}^{\prime})$, one usually considers impulses in the position *u* and the velocity ${u}^{\prime}$. However, in the motion of spacecraft, one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in the position [24–27].

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 [28–37].

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 above-mentioned 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 mountain-pass lemma due to Kajikiya [1]. Our main results extend the existing study.

Throughout this paper, we assume that ${I}_{j}:\mathbb{R}\to \mathbb{R}$ is continuous, and $f(t,u)$ satisfies the following conditions:

_{1}) ${I}_{j}$ ($j=1,2,\dots ,N$) are odd and satisfy

_{2}) There exist ${\delta}_{j}>0$, $j=1,2,\dots ,N$ such that

(H_{1}) $f(t,u)\in C([0,T]\times \mathbb{R},\mathbb{R})$, $f(t,-u)=-f(t,u)$ for all $u\in \mathbb{R}$;

(H_{2}) ${lim}_{|u|\to \mathrm{\infty}}\frac{f(t,u)}{{|u|}^{p-1}}=0$ uniformly for $t\in [0,T]$;

(H_{3}) ${lim}_{|u|\to {0}^{+}}\frac{f(t,u)}{u}=\mathrm{\infty}$ uniformly for $t\in [0,T]$.

The main result of this paper is as follows.

**Theorem 1.1** *Suppose that* (I_{1})-(I_{2}) *and* (H_{1})-(H_{3}) *hold*. *Then problem* (1.1) *has a sequence of nontrivial solutions* $\{{u}_{n}\}$ *and* ${u}_{n}\to 0$ *as* $n\to \mathrm{\infty}$.

**Remark 1.1** Without the symmetry condition (*i.e.*, $f(x,-u)=-f(x,u)$ and $I(-s)=-I(s)$), we can obtain at least one nontrivial solution by the same method in this paper.

**Remark 1.2**We should point out that Theorem 1.1 is different from the previous results of [28–37] in three main directions:

- (1)
We do not make the nonlinearity

*f*satisfy the well-known Ambrosetti-Rabinowitz condition [23]; - (2)
We try to use Lusternik-Schnirelman’s theory for ${Z}_{2}$-invariant functional. But since the functional is not bounded from below, we could not use the theory directly. So, we follow [38] to consider a truncated functional.

- (3)
We can obtain a sequence of nontrivial solutions $\{{u}_{n}\}$ and ${u}_{n}\to 0$ as $n\to \mathrm{\infty}$.

**Remark 1.3** There exist many functions ${I}_{j}$ and $f(t,u)$ satisfying conditions (I_{1})-(I_{2}) and (H_{1})-(H_{3}), respectively. For example, when $p=4$, ${I}_{j}(s)=s$ and $f(t,u)={e}^{t}{u}^{1/3}$.

## 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 $J:E\to \mathbb{R}$. *J* is said to be sequentially weakly lower semi-continuous if ${lim}_{n\to \mathrm{\infty}}infJ({u}_{n})\ge J(u)$ as ${u}_{n}\rightharpoonup u$ in *E*.

**Definition 2.2** Let *E* be a real Banach space. For any sequence $\{{u}_{n}\}\subset E$, if $\{J({u}_{n})\}$ is bounded and ${J}^{\prime}({u}_{n})\to 0$ as $n\to \mathrm{\infty}$ possesses a convergent subsequence, then we say *J* satisfies the Palais-Smale condition (denoted by (*PS*) condition for short).

**Lemma 2.1** [29]

*If* $ess{inf}_{t\in [0,T]}a(t)=m>-{\lambda}_{1}$, *then the norm* $\parallel \cdot \parallel $ *and the norm* ${\parallel \cdot \parallel}_{{H}_{0}^{1}(0,T)}$ *are equivalent*.

**Lemma 2.2** [29]

*There exists*${c}_{\ast}$

*such that if*$u\in {H}_{0}^{1}(0,T)$,

*then*

*where* ${\parallel u\parallel}_{\mathrm{\infty}}={max}_{t\in [0,T]}|u(t)|$.

For $u\in {H}^{2}(0,T)$, we have that *u* and ${u}^{\prime}$ are both absolutely continuous, and ${u}^{\u2033}\in {L}^{2}(0,T)$, hence, $\mathrm{\Delta}{u}^{\prime}({t}_{j})={u}^{\prime}({t}_{j}^{+})-{u}^{\prime}({t}_{j}^{-})$ for any $t\in [0,T]$. If $u\in {H}_{0}^{1}(0,T)$, then *u* is absolutely continuous and ${u}^{\prime}\in {L}^{2}(0,T)$. In this case, the one-side derivatives ${u}^{\prime}({t}_{j}^{+})$ and ${u}^{\prime}({t}_{j}^{-})$ may not exist. As a consequence, we need to introduce a different concept of solution. Suppose that $u\in C[0,T]$ satisfies the Dirichlet condition $u(0)=u(T)=0$. Assume that, for every $j=1,2,\dots ,N$, ${u}_{j}=u{|}_{({t}_{j},{t}_{j+1})}$ and ${u}_{j}\in {H}^{2}({t}_{j},{t}_{j+1})$. Let $0={t}_{0}<{t}_{1}<{t}_{2}<\cdots <{t}_{N}<{t}_{N+1}=T$.

*v*and integrating between 0 and

*T*, we have

**Lemma 2.3**

*A weak solution of*(1.1)

*is a function*$u\in {H}_{0}^{1}(0,T)$

*such that*

*for any* $v\in {H}_{0}^{1}(0,T)$.

*f*and ${I}_{j}$, $j=1,2,\dots ,N$, we obtain the continuity and differentiability of

*J*and $J\in {C}^{1}({H}_{0}^{1}(0,T),\mathbb{R})$. For any $v\in {H}_{0}^{1}(0,T)$, one has

Thus, the solutions of problem (1.1) are the corresponding critical points of *J*.

**Lemma 2.4** *If* $u\in {H}_{0}^{1}(0,T)$ *is a weak solution of problem* (1.1), *then* *u* *is a classical solution of problem* (1.1).

*Proof*Obviously, we have $u(0)=u(T)=0$ since $u\in {H}_{0}^{1}(0,T)$. By the definition of weak solution, for any $v\in {H}_{0}^{1}(0,T)$, one has

*u*satisfies the equation in (1.1) a.e. on $[0,T]$. By integrating (2.7), we have

Hence, $\mathrm{\Delta}{u}^{\prime}({t}_{j})={I}_{j}(u({t}_{j}))$ for every $j=1,2,\dots ,N$, and the impulsive condition in (1.1) is satisfied. This completes the proof. □

**Lemma 2.5** *If* $ess{inf}_{t\in [0,T]}a(t)=m>-{\lambda}_{1}$, *then the functional* *J* *is sequentially weakly lower semi*-*continuous*.

*Proof*Let $\{{u}_{n}\}$ be a weakly convergent sequence to

*u*in ${H}_{0}^{1}(0,T)$, then

*u*on $C[0,T]$. Then

This completes the proof. □

_{1}) and (H

_{2}), we have

*k*, we have

where $c(\epsilon )>0$.

**Lemma 2.6** *Suppose that* (I_{1})-(I_{2}) *and* (H_{1})-(H_{3}) *hold*, *then* $J(u)$ *satisfies the* (*PS*) *condition*.

*Proof*Let $\{{u}_{n}\}$ be a sequence in ${H}_{0}^{1}(0,T)$ such that $\{J({u}_{n})\}$ is bounded and ${J}^{\prime}({u}_{n})\to 0$ as $n\to \mathrm{\infty}$. First, we prove that $\{{u}_{n}\}$ is bounded. By (2.5), (2.6) and (2.11), one has

_{1}), we can deduce that

*M*is a positive constant. On the other hand, by (I

_{1}), (2.5) and (2.10), we have

Therefore, $\parallel {u}_{n}-u\parallel \to 0$ as $n\to +\mathrm{\infty}$. That is $\{{u}_{n}\}$ converges strongly to *u* in ${H}_{0}^{1}(0,T)$. That is *J* satisfies the (*PS*) condition. □

## 3 Existence of a sequence of arbitrarily small solutions

*X*be a Banach space and denote

If there is no mapping *φ* as above for any $m\in N$, then $\gamma (A)=+\mathrm{\infty}$. We list some properties of the genus (see [1]).

**Proposition 3.1**

*Let*

*A*

*and*

*B*

*be closed symmetric subsets of*

*X*,

*which do not contain the origin*.

*Then the following hold*.

- (1)
*If there exists an odd continuous mapping from**A**to**B*,*then*$\gamma (A)\le \gamma (B)$; - (2)
*If there is an odd homeomorphism from**A**to**B*,*then*$\gamma (A)=\gamma (B)$; - (3)
*If*$\gamma (B)<\mathrm{\infty}$,*then*$\gamma \overline{(A\setminus B)}\ge \gamma (A)-\gamma (B)$; - (4)
*Then**n*-*dimensional sphere*${S}^{n}$*has a genus of*$n+1$*by the Borsuk*-*Ulam theorem*; - (5)
*If**A**is compact*,*then*$\gamma (A)<+\mathrm{\infty}$*and there exists*$\delta >0$*such that*${U}_{\delta}(A)\in \mathrm{\Sigma}$*and*$\gamma ({U}_{\delta}(A))=\gamma (A)$,*where*${U}_{\delta}(A)=\{x\in X:\parallel x-A\parallel \le \delta \}$.

Let ${\mathrm{\Sigma}}_{k}$ denote the family of closed symmetric subsets *A* of *X* such that $0\notin A$ and $\gamma (A)\ge k$. The following version of the symmetric mountain-pass lemma is due to Kajikiya [1].

**Lemma 3.1** *Let* *E* *be an infinite*-*dimensional space and* $I\in {C}^{1}(E,R)$, *and suppose the following conditions hold*.

(C_{1}) $I(u)$ *is even*, *bounded from below*, $I(0)=0$ *and* $I(u)$ *satisfies the Palais*-*Smale condition*;

(C_{2}) *For each* $k\in N$, *there exists an* ${A}_{k}\in {\mathrm{\Sigma}}_{k}$ *such that* ${sup}_{u\in {A}_{k}}I(u)<0$.

*Then either* (R_{1}) *or* (R_{2}) *below holds*.

(R_{1}) *There exists a sequence* $\{{u}_{k}\}$ *such that* ${I}^{\prime}({u}_{k})=0$, $I({u}_{k})<0$ *and* $\{{u}_{k}\}$ *converges to zero*;

(R_{2}) *There exist two sequences* $\{{u}_{k}\}$ *and* $\{{v}_{k}\}$ *such that* ${I}^{\prime}({u}_{k})=0$, $I({u}_{k})<0$, ${u}_{k}\ne 0$, ${lim}_{k\to \mathrm{\infty}}{u}_{k}=0$, ${I}^{\prime}({v}_{k})=0$, $I({v}_{k})<0$, ${lim}_{k\to \mathrm{\infty}}{v}_{k}=0$, *and* $\{{v}_{k}\}$ *converges to a nonzero limit*.

**Remark 3.1** From Lemma 3.1, we have a sequence $\{{u}_{k}\}$ of critical points such that $I({u}_{k})\le 0$, ${u}_{k}\ne 0$ and ${lim}_{k\to \mathrm{\infty}}{u}_{k}=0$.

*P*is not bounded below), we have to perform some sort of truncation. To this end, let ${R}_{0}$, ${R}_{1}$ be such that $m<{R}_{0}<M<{R}_{1}$, where

*m*is the local minimum of $P(s)$, and

*M*is the local maximum and $P({R}_{1})>P(m)$. For these values ${R}_{1}$ and ${R}_{0}$, we can choose a smooth function $\chi (t)$ defined as follows

From the arguments above, we have the following.

**Lemma 3.2**

*Let*$G(u)$

*is defined as in*(3.1).

*Then*

- (i)
$G\in {C}^{1}({H}_{0}^{1}(0,T),\mathbb{R})$

*and**G**is even and bounded from below*; - (ii)
*If*$G(u)<m$,*then*$\overline{P}(\parallel u\parallel )<m$,*consequently*, $\parallel u\parallel <{\rho}_{0}$*and*$J(u)=G(u)$; - (iii)
*Suppose that*(I_{1})-(I_{2})*and*(H_{1})-(H_{3})*hold*,*then*$G(u)$*satisfies the*(*PS*)*condition*.

*Proof* It is easy to see (i) and (ii). (iii) are consequences of (ii) and Lemma 2.6. □

**Lemma 3.3** *Assume that* (I_{2}) *and* (H_{3}) *hold*. *Then for any* $k\in N$, *there exists* $\delta =\delta (k)>0$ *such that* $\gamma (\{u\in {H}_{0}^{1}(0,T):G(u)\le -\delta (k)\}\setminus \{0\})\ge k$.

*Proof*Firstly, by (H

_{3}) of Theorem 1.1, for any fixed $u\in {H}_{0}^{1}(0,T)$, $u\ne 0$, we have

*ρ*small enough, we have

This completes the proof. □

Now, we give the proof of Theorem 1.1 as following.

*Proof of Theorem 1.1*Recall that

By Lemma 3.2(i) and Lemma 3.3, we know that $-\mathrm{\infty}<{c}_{k}<0$. Therefore, assumptions (C_{1}) and (C_{2}) of Lemma 3.1 are satisfied. This means that *G* has a sequence of solutions $\{{u}_{n}\}$ converging to zero. Hence, Theorem 1.1 follows by Lemma 3.2(ii). □

## Declarations

### Acknowledgements

The authors are supported by the Research Foundation during the 12th Five-Year 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.

## Authors’ Affiliations

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