Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Abstract

Some sufficient conditions on the existence and multiplicity of solutions for the damped vibration problems with impulsive effects $$\left\{ \eqalign{& u''(t)+g(t)u'(t)+f(t,u(t))=0,\quad \hbox{a.e.\ $t\in [0,T]$,}\cr &u(0)=u(T)=0,\cr & \Delta u'(t_{j})=u'(t_{j}^{+})-u'(t_{j}^{-})=I_{j}(u(t_{j})),\quad \hbox{$j=1,\ldots,p,$} \cr} \right.$$ 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.