Theory and Modern Applications

# Multiple solutions to impulsive differential equations

## Abstract

In this paper, we study the existence of a second-order impulsive differential equations depending on a parameter λ. By employing a critical point theorem, the existence of at least three solutions is obtained.

## 1 Introduction

In recent years, the study of the existence of solutions to impulsive differential equation has aroused extensive interest, we refer the reader to  and the references therein.

In , by using some existing critical point theorems, Xie and Luo investigated the existence of multiple solutions of the following Neumann boundary value problem:

\begin{aligned}& -\bigl(p(t)u'(t)\bigr)'+q(t)u(t)=\lambda f\bigl(t, u(t)\bigr), \quad t\neq t_{j}, t\in[0, 1], \\& \Delta p(t_{j})u'(t_{j})=I_{j} \bigl(u(t_{j})\bigr), \quad j=1,2,\ldots, m, \\& u'(0)=u'(1)=0. \end{aligned}
(1.1)

In , Liang and Zhang considered the following boundary value problems:

\begin{aligned}& -\bigl(p(t)u'(t)\bigr)'= f\bigl(t, u(t)\bigr),\quad t \neq t_{j}, t\in[0, T], \\& \Delta p(t_{j})u'(t_{j})=I_{j} \bigl(u(t_{j})\bigr),\quad j=1,2,\ldots, m, \\& u(0)=u(T),\qquad p(0)u'(0)=p(T)u'(T). \end{aligned}
(1.2)

The authors gave some criteria to guarantee that the problem has at least one solution under some different conditions.

In , Li and Shen were concerned with the existence of three solutions for the following boundary value problems:

\begin{aligned}& -u''(t)=\lambda f\bigl( u(t)\bigr), \quad t\neq t_{j}, t\in[0, 1], \\& \Delta u'(t_{j})=I_{j}\bigl(u(t_{j}) \bigr), \quad j=1,2,\ldots,m, \\& u(0)=u(1)=0. \end{aligned}
(1.3)

Motivated by the previous mentioned paper, in this paper, we will study the existence of at least three solutions for the following boundary value problems:

\begin{aligned}& -\bigl(p(t)u'(t)\bigr)'+u(t)=\lambda f\bigl(t, u(t) \bigr), \quad t\neq t_{j}, t\in[0, 1], \\& \Delta p(t_{j})u'(t_{j})=I_{j} \bigl(u(t_{j})\bigr), \quad j=1,2,\ldots, m, \\& u'(0)=u'(1)=0, \end{aligned}
(1.4)

where $$0=t_{0}< t_{1}<\cdots<t_{m}<t_{m+1}=1$$, $$p\in PC^{1}([0,1])$$, $$f\in C([0,1]\times R, R)$$, $$I_{j}\in C(R, R)$$, $$j=1,2,\ldots, m$$, $$\Delta p(t_{j})u'(t_{j})=p(t_{j}^{+})u'(t_{j}^{+})-p(t_{j}^{-})u'(t_{j}^{-})$$, $$p(t_{j}^{+})u'(t_{j}^{+})$$ and $$p(t_{j}^{-})u'(t_{j}^{-})$$ denote the right and the left limits, respectively, $$\lambda\in[0, +\infty)$$ is a real parameter.

## 2 Preliminaries

Let $$p_{0}=\min_{t\in[0, 1]}p(t)>0$$, $$M_{0}=\max\{\frac{1}{p_{0}},1\}$$, $$X=W^{1,2}[0, 1]$$ with the norm

$$\|u\|= \biggl( \int^{1}_{0}\bigl(p(t)\bigl\vert u'(t)\bigr\vert ^{2}+\bigl\vert u(t)\bigr\vert ^{2}\bigr)\,dt \biggr)^{\frac{1}{2}}.$$

Define the norm in $$C([0, 1])$$ by $$\|u\|_{\infty}=\max_{t\in[0, 1]}|u(t)|$$.

### Lemma 2.1

For any $$u\in X$$, we have $$\|u\|_{\infty}\leq \sqrt{2M_{0}}\|u\|$$.

### Proof

For $$u\in X$$ by the mean-value theorem, there exists $$\tau\in(0, 1)$$ such that $$\int^{1}_{0}u(s)\,ds=u(\tau)$$. Hence, for $$t\in[0, 1]$$, we have

\begin{aligned} \bigl\vert u(t)\bigr\vert =&\biggl\vert u(\tau)+ \int^{t}_{\tau}u'(s)\,ds\biggr\vert \leq\bigl\vert u(\tau)\bigr\vert + \int ^{1}_{0}\bigl\vert u'(s)\bigr\vert \,ds \\ \leq& \int^{1}_{0}\bigl\vert u(s)\bigr\vert \,ds+ \int^{1}_{0}\bigl\vert u'(s)\bigr\vert \,ds \\ \leq& \biggl( \int^{1}_{0}\bigl\vert u(s)\bigr\vert ^{2}\,ds\biggr)^{\frac{1}{2}}+\sqrt{1/p_{0}}\biggl( \int ^{1}_{0}p(t)\bigl\vert u'(t) \bigr\vert ^{2}\,dt\biggr)^{\frac{1}{2}} \\ \leq& \sqrt{2M_{0}}\|u\|. \end{aligned}

For every $$u\in X$$, we define the functional $$\varphi(u): X\to R$$ by

$$\varphi(u)=\Phi(u)-\lambda\Psi(u);$$

here

$$\Phi(u)=\frac{1}{2}\|u\|^{2}+\sum^{m}_{j=1} \int^{u(t_{j})}_{0} I_{j}(s)\,ds$$

and

$$\Psi(u)= \int^{1}_{0} F(t, u)\,dt,$$

where $$F(t, u)=\int^{u(t)}_{0}f(t, s)\,ds$$.

We easily show that φ is differentiable at any $$u\in X$$ and

$$\varphi'(u)v= \int^{1}_{0}\bigl(p(t)u'(t)v'(t)+u(t)v(t) \bigr)\,dt +\sum^{m}_{j=1}I_{j} \bigl(u(t_{j})\bigr)v(t_{j})-\lambda \int^{1}_{0}f\bigl(t, u(t)\bigr)v(t)\,dt.$$

Obviously, Φ is a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on $$X^{*}$$, and Ψ is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. □

### Lemma 2.2



If $$u\in X$$ is a critical point of the functional φ, then u is a classical solution of problem (1.4).

Suppose that $$E\subset X$$. We denote $$\overline{E}^{\omega}$$ as the weak closure of E, that is, $$u\in \overline{E}^{\omega}$$ if there exists a sequence $$\{u_{n}\}\subset E$$ such that $$g(u_{n})\to g(u)$$ for every $$g\in X^{*}$$. Our main tool is the following three critical points theorem obtained in .

### Lemma 2.3

, Theorem 2.1

Let X be separable and reflexive real Banach space. $$\Phi: X\to R$$ a nonnegative continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on $$X^{*}$$. $$J: X\to R$$ a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that there exists $$x_{0}\in X$$ such that $$\Phi(x_{0})=J(x_{0})=0$$ and that

1. (i)

$$\lim_{\|x\|\to+\infty}(\Phi(x)-\lambda J(x))=+\infty$$ for all $$\lambda \in[0, +\infty)$$.

Further, assume that there are $$r>0$$, $$x_{1}\in X$$ such that

1. (ii)

$$r<\Phi(x_{1})$$.

2. (iii)

$$\sup_{x\in\overline{\Phi^{-1}((-\infty, r))}^{\omega}} J(x)<\frac{r}{r+\Phi(x_{1})}J(x_{1})$$.

Then, for each

$$\lambda\in\Lambda_{1}= \biggl( \frac{\Phi(x_{1})}{J(x_{1})-\sup_{x\in\overline{\Phi^{-1}((-\infty, r))}^{\omega}} J(x)}, \frac{r}{\sup_{x\in \overline{\Phi^{-1}((-\infty, r))}^{\omega}}J(x)} \biggr),$$

the equation

$$\Phi'(x)-\lambda J'(x)=0$$
(2.1)

has at least three solutions in X and, moreover, for each $$h>1$$, there exist an open interval

$$\Lambda_{2}\subseteq \biggl[0, \frac{hr}{r(J(x_{1})/\Phi(x_{1}))-\sup_{x\in \overline{\Phi^{-1}((-\infty, r))}^{\omega}}J(x)} \biggr)$$

and a positive real number σ such that, for each $$\lambda\in \Lambda_{2}$$, (1.2) has at least three solutions in X whose norms are less than σ.

## 3 Main results

### Theorem 3.1

The following conditions are given.

(H1):

$$\sum^{m}_{j=1}\int^{u(t_{j})}_{0}I_{j}(t)\,dt\geq0$$.

(H2):

Let $$a_{i}>0$$ ($$i=1,2$$), $$M>0$$, and $$0<\mu<2$$ such that

$$F(t, u)\leq a_{1}|u|^{\mu}- a_{2}, \quad \textit{for } |u|\geq M, t\in[0, 1].$$
(H3):

There exist two positive constants c, $$c_{1}$$ with $$c_{1}>\frac{c}{\sqrt{2M_{0}}}$$, such that

$$4M_{0} \int^{1}_{0}\max_{|u|\leq c}F(t, u) \,dt< c^{2} \Biggl(\frac{c^{2}}{4M_{0}}+\frac{c_{1}^{2}}{2}+\sum ^{m}_{j=1} \int ^{c_{1}}_{0}I_{j}(t)\,dt \Biggr)^{-1} \int^{1}_{0}F(t, c_{1})\,dt.$$

Furthermore, put

\begin{aligned} &\lambda_{1}=\frac{4M_{0}\int^{1}_{0}\max_{ |u|\leq c}F(t, u)\,dt}{c^{2}}, \\ &\lambda_{2}=\frac{\int^{1}_{0}F(t, c_{1})\,dt-\int^{1}_{0}\max_{ |u|\leq c} F(t, u)\,dt}{\frac{c_{1}^{2}}{2}+\sum^{m}_{j=1}\int^{c_{1}}_{0}I_{j}(s)\,ds}. \end{aligned}
(3.1)

Then, for each $$\lambda\in(\frac{1}{\lambda_{2}}, \frac{1}{\lambda_{1}})$$, problem (1.4) has at least three solutions in X.

### Proof

Now we show the conditions (i)-(iii) of Lemma 2.3 are satisfied.

For any $$u\in X$$, $$|u|\geq M$$, and $$\lambda\geq0$$, and the assumptions (H1)-(H2) we have

\begin{aligned} \Phi(u)-\lambda \Psi(u) =&\frac{1}{2}\|u\|^{2}+\sum ^{m}_{j=1} \int ^{u(t_{j})}_{0}I_{j}(s)\,ds-\lambda \int^{1}_{0}F\bigl(t, u(t)\bigr)\,dt \\ \geq&\frac{1}{2}\|u\|^{2}-\lambda\bigl[a_{1}|u|^{\mu}-a_{2} \bigr] \\ \geq& \frac{1}{2}\|u\|^{2}-\lambda\bigl[a_{1}(2M_{0})^{\mu/2} \|u\|^{\mu}-a_{2}\bigr], \end{aligned}

$$0<\mu<2$$ implies that

$$\lim_{\|u\|\to\infty}\bigl(\Phi(u)-\lambda J(u)\bigr)=+\infty,$$

which shows the condition (i) of Lemma 2.3 is satisfied.

Let $$u_{1}=c_{1}\in X$$ and $$c_{1}>\frac{c}{\sqrt{2M_{0}}}$$. Then

\begin{aligned} \Phi (u_{1}) =&\frac{1}{2}\|u_{1}\|^{2}+\sum ^{m}_{j=1} \int ^{u_{1}(t_{j})}_{0}I_{j}(s)\,ds \\ =& \frac{1}{2}c_{1}^{2}+\sum^{m}_{j=1} \int ^{c_{1}}_{0}I_{j}(s)\,ds\geq \frac{1}{2}c_{1}^{2}>\frac{c^{2}}{4M_{0}}=r, \end{aligned}

so the condition (ii) of Lemma 2.3 is obtained.

By Lemma 2.1, if $$\Phi(u)\leq r$$, then

$$\bigl\vert u(t)\bigr\vert ^{2}\leq2M_{0}\|u \|^{2}\leq4M_{0}\Phi(u)\leq4M_{0}r=c^{2}, \quad \mbox{for }t\in [0, 1],$$

which implies that

$$\Phi^{-1}(-\infty, r)\subseteq\bigl\{ u\in X, \bigl\vert u(t)\bigr\vert \leq c, t\in[0, 1]\bigr\} .$$

So for any $$u\in X$$, we have

$$\sup_{u\in\overline{\Phi^{-1}(-\infty, r)}^{\omega}} \Psi(u) =\sup_{u\in\Phi^{-1}(-\infty, r)} \Psi(u)\leq \int^{1}_{0}\max_{ |u|\leq c}F(t, u)\,dt.$$

On the other hand, we obtain

$$\frac{r}{r+\Phi(u_{1})}\Psi(u_{1}) =c^{2} \Biggl[4M_{0} \Biggl(\frac{c^{2}}{4M_{0}}+\frac{c_{1}^{2}}{2}+\sum^{m}_{j=1} \int ^{c_{1}}_{0}I_{j}(t)\,dt\Biggr) \Biggr]^{-1} \int^{1}_{0}F(t, c_{1})\,dt.$$

From the assumption (H3) we have

$$\sup_{u\in\overline{\Phi^{-1}(-\infty, r)}^{\omega}} \Psi(u)< \frac{r}{r+\Phi(u_{1})}\Psi(u_{1}),$$

which shows the condition (iii) of Lemma 2.3 is satisfied.

Note that

\begin{aligned}& \frac{\Phi(u_{1})}{\Psi(u_{1})-\sup_{u\in\overline{\Phi^{-1}(-\infty , r)}^{\omega}} \Psi(u)}\leq \frac{\frac{1}{2}c_{1}^{2}+\sum^{m}_{j=1}\int^{c_{1}}_{0}I_{j}(s)\,ds}{\int ^{1}_{0}F(t, c_{1})\,dt-\int^{1}_{0}\max_{ |u|\leq c}F(t, u)\,dt}=\frac{1}{\lambda_{2}}, \\& \frac{r}{\sup_{u\in\overline{\Phi^{-1}(-\infty, r)}^{\omega}} \Psi(u)}\geq\frac{c^{2}}{4M_{0}\int^{1}_{0}\max_{ |u|\leq c}F(t, u)}=\frac{1}{\lambda_{1}}. \end{aligned}

The condition (H3) implies $$\lambda_{2}>\lambda_{1}$$. In the light of Lemma 2.3, the problem (1.4) has at least three solutions in X for each $$\lambda\in (1/\lambda_{2}, 1/\lambda_{1})$$.

The proof is complete. □

## 4 Examples

Consider the following problem:

\begin{aligned}& -\bigl(e^{t}u'(t)\bigr)'+u(t)=\lambda f(t, u), \quad t\in[0, 1], t\neq t_{1}, \\& \Delta\bigl(e^{t_{1}}u'(t_{1}) \bigr)=u(t_{1}),\quad t_{1}=\frac{1}{2}, \\& u'(0)=u'(1)=0, \end{aligned}
(4.1)

where

$$f(t, u)=\left \{ \textstyle\begin{array}{l@{\quad}l} e^{2u},& u\leq4, \\ u^{1/2}+e^{8}-4,& u>4, \end{array}\displaystyle \right .$$

then

$$F(t, u)=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{1}{2}(e^{2u}-1),& u\leq4, \\ \frac{2}{3}u^{3/2}+(e^{8}-4)u+\frac{61}{6}-\frac{7}{2}e^{8}, & u>4. \end{array}\displaystyle \right .$$

Clearly $$M_{0}=1$$. Let $$c=1$$, $$c_{1}=4$$, it follows that

\begin{aligned}& 4M_{0} \int^{1}_{0}\max_{|u|\leq c}F(t, u) \,dt \\& \quad =2\bigl(e^{2}-1\bigr) < c^{2} \Biggl(\frac{c^{2}}{4M_{0}}+\frac{c_{1}^{2}}{2}+\sum ^{m}_{j=1} \int ^{c_{1}}_{0}I_{j}(s)\,ds \Biggr)^{-1} \int^{1}_{0}F(t, c_{1})\,dt= \frac{2(e^{8}-1)}{65}, \\& \frac{1}{\lambda_{1} }=\frac{1}{2(e^{2}-1)}, \qquad \frac{1}{\lambda_{2} }= \frac {32}{e^{8}-e^{2}}, \end{aligned}

which shows that all conditions of Theorem 3.1 are satisfied, so the problem (4.1) admits at least three solutions for $$\lambda\in(\frac{32}{e^{8}-e^{2}}, \frac{1}{2(e^{2}-1)})$$.

## References

1. Xie, J, Luo, Z: Multiple solutions for a second-order impulsive Sturm-Liouville equation. Abstr. Appl. Anal. 2013, Article ID 527082 (2013)

2. Liang, R, Zhang, W: Applications of variational methods to the impulsive equation with non-separated periodic boundary conditions. Adv. Differ. Equ. 2016, 147 (2016)

3. Li, J, Shen, J: Existence of three solutions to impulsive differential equations. J. Integral Equ. Appl. 24, 273-281 (2012)

4. Li, J, Nieto, JJ: Existence of positive solutions for multi-point boundary value problem on the half-line with impulses. Bound. Value Probl. 2009, Article ID 834158 (2009)

5. Li, T, Li, J: Existence results of second-order impulsive neutral functional differential inclusions in Banach spaces. Adv. Differ. Equ. 2015, 309 (2015)

6. Bonanno, G: A critical points theorem and nonlinear differential problems. J. Glob. Optim. 28, 249-258 (2004)

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Correspondence to Hongmei Bao. 