## Taiwanese Journal of Mathematics

### SIGN-CHANGING SOLUTIONS FOR A CLASS OF DAMPED VIBRATION PROBLEMS WITH IMPULSIVE EFFECTS

#### Abstract

In this paper, some sufficient conditions are obtained for theexistence and multiplicity of sign-changing solutions for the dampedvibration problem with impulsive effects \begin{eqnarray*} \left\{ \begin{array}{ll} -u''(t)+g(t)u'(t)=f(t,u(t)), & \hbox{a.e. $t\in [0,T]$;} \\ u(0)=u(T)=0, & \hbox{} \\ \Delta u'(t_{j})=u'(t_{j}^{+})-u'(t_{j}^{-})=I_{j}(u(t_{j})), & \hbox{$j=1,2,\ldots,p,$} \end{array} \right. \end{eqnarray*} where $t_{0}=0\lt t_{1}\lt t_{2}\lt \ldots\lt t_{p}\lt t_{p+1}=T,g\in L^{1}(0,T;\mathbb{R}),I_{j}:\mathbb{R}\rightarrow\mathbb{R},j=1,2,\ldots,p$ are continuous, $f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function with subcritical growth condition:

$(A) |f(t, u)| ≤ C(1 + |u|^{s-1}), \forall t \in[0,T], u\in \mathbb{R}, s\in [2,+\infty)$.

The sign-changing solutions are sought by means of some sign-changing critical point theorems and two examples are presented to illustrate the feasibility and effectiveness of our results.

#### Article information

Source
Taiwanese J. Math., Volume 18, Number 6 (2014), 1863-1877.

Dates
First available in Project Euclid: 21 July 2017

https://projecteuclid.org/euclid.twjm/1500667500

Digital Object Identifier
doi:10.11650/tjm.18.2014.4033

Mathematical Reviews number (MathSciNet)
MR3284035

Zentralblatt MATH identifier
1357.34058

Subjects

#### Citation

Zhou, Jianwen; Li, Yongkun; Wang, Yanning. SIGN-CHANGING SOLUTIONS FOR A CLASS OF DAMPED VIBRATION PROBLEMS WITH IMPULSIVE EFFECTS. Taiwanese J. Math. 18 (2014), no. 6, 1863--1877. doi:10.11650/tjm.18.2014.4033. https://projecteuclid.org/euclid.twjm/1500667500

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