Existence and multiplicity of solutions for a class of damped vibration problems with impulsive effects

Jianwen Zhou¹ ; Yongkun Li¹

¹Department of Mathematics Yunnan University Kunming, Yunnan 650091, People's Republic of China

Annales Polonici Mathematici, Tome 100 (2011) no. 1, pp. 87-98

Résumé

Some sufficient conditions on the existence and multiplicity of solutions for the damped vibration problems with impulsive effects $$ \left\{ \eqalign{ u''(t)+g(t)u'(t)+f(t,u(t))=0,\quad \hbox{a.e.\ $t\in [0,T]$,}\cr (0)=u(T)=0,\cr \Delta u'(t_{j})=u'(t_{j}^{+})-u'(t_{j}^{-})=I_{j}(u(t_{j})),\quad \hbox{$j=1,\ldots,p,$} \cr} \right. $$ are established, where $t_{0}=0\!\!t_{1}\!\!\cdots\!\!t_{p}\!\!t_{p+1}=T$, $g\in L^{1}(0,T;\mathbb{R})$, $f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ is continuous, and $I_{j}:\mathbb{R}\rightarrow\mathbb{R}$, $j=1,\ldots,p$, are continuous. The solutions are sought by means of the Lax–Milgram theorem and some critical point theorems. Finally, two examples are presented to illustrate the effectiveness of our results.

Détail

DOI : 10.4064/ap100-1-8

Keywords: sufficient conditions existence multiplicity solutions damped vibration problems impulsive effects eqalign t t quad hbox delta u quad hbox ldots right established where cdots mathbb times mathbb rightarrow mathbb continuous mathbb rightarrow mathbb ldots continuous solutions sought means lax milgram theorem critical point theorems finally examples presented illustrate effectiveness results