Weak Solutions for a Second Order Impulsive Boundary Value Problem

Abstract

In this work we use the topological degree theory and critical point theory to investigate the existence of weak solutions for the second order impulsive boundary value problem
\[
\begin{cases}
-x''(t)-\lambda x(t)=f(t), t\not =t_j, t\in (0,\pi),\\
\Delta x'(t_j)=x'(t_j^+)-x'(t_j^-)=I_j(x(t_j)),j=1,2,\ldots,p,\\
x(0)=x(\pi)=0,
\end{cases}
\]
where $\lambda$ is a positive parameter,
$0=t_0<t_1<t_2<\cdots<t_p<t_{p+1}=\pi$, $f\in L^2(0,\pi)$ is a given function and $I_j\in C(\mathbb R,\mathbb R)$ for $j=1,2,\ldots,p$.