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Multiple solutions to impulsive differential equations
Advances in Difference Equations volume 2017, Article number: 20 (2017)
Abstract
In this paper, we study the existence of a secondorder 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 [1–5] and the references therein.
In [1], by using some existing critical point theorems, Xie and Luo investigated the existence of multiple solutions of the following Neumann boundary value problem:
In [2], Liang and Zhang considered the following boundary value problems:
The authors gave some criteria to guarantee that the problem has at least one solution under some different conditions.
In [3], Li and Shen were concerned with the existence of three solutions for the following boundary value problems:
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:
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
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 meanvalue 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
For every \(u\in X\), we define the functional \(\varphi(u): X\to R \) by
here
and
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
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
[1]
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 [6].
Lemma 2.3
[6], 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

(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

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

(iii)
\(\sup_{x\in\overline{\Phi^{1}((\infty, r))}^{\omega}} J(x)<\frac{r}{r+\Phi(x_{1})}J(x_{1})\).
Then, for each
the equation
has at least three solutions in X and, moreover, for each \(h>1\), there exist an open interval
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.
 (H_{1}):

\(\sum^{m}_{j=1}\int^{u(t_{j})}_{0}I_{j}(t)\,dt\geq0\).
 (H_{2}):

Let \(a_{i}>0\) (\(i=1,2\)), \(M>0\), and \(0<\mu<2\) such that
$$F(t, u)\leq a_{1}u^{\mu} a_{2}, \quad \textit{for } u\geq M, t\in[0, 1]. $$  (H_{3}):

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
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 (H_{1})(H_{2}) we have
\(0<\mu<2\) implies that
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
so the condition (ii) of Lemma 2.3 is obtained.
By Lemma 2.1, if \(\Phi(u)\leq r\), then
which implies that
So for any \(u\in X\), we have
On the other hand, we obtain
From the assumption (H_{3}) we have
which shows the condition (iii) of Lemma 2.3 is satisfied.
Note that
The condition (H_{3}) 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:
where
then
Clearly \(M_{0}=1\). Let \(c=1\), \(c_{1}=4\), it follows that
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
Xie, J, Luo, Z: Multiple solutions for a secondorder impulsive SturmLiouville equation. Abstr. Appl. Anal. 2013, Article ID 527082 (2013)
Liang, R, Zhang, W: Applications of variational methods to the impulsive equation with nonseparated periodic boundary conditions. Adv. Differ. Equ. 2016, 147 (2016)
Li, J, Shen, J: Existence of three solutions to impulsive differential equations. J. Integral Equ. Appl. 24, 273281 (2012)
Li, J, Nieto, JJ: Existence of positive solutions for multipoint boundary value problem on the halfline with impulses. Bound. Value Probl. 2009, Article ID 834158 (2009)
Li, T, Li, J: Existence results of secondorder impulsive neutral functional differential inclusions in Banach spaces. Adv. Differ. Equ. 2015, 309 (2015)
Bonanno, G: A critical points theorem and nonlinear differential problems. J. Glob. Optim. 28, 249258 (2004)
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Bao, H. Multiple solutions to impulsive differential equations. Adv Differ Equ 2017, 20 (2017). https://doi.org/10.1186/s1366201710794
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DOI: https://doi.org/10.1186/s1366201710794