I'm kinda stuck with a problem right now. I have the boundary problem
$$\left\{
\begin{array}{l}
-u''(x)+\mu u(x)= f(x), \quad x\in (0,T) \\
u'(0)=u'(T)=0
\end{array}
\right.$$
and I have to obtain the values of $\mu\in\mathbb{R}$ such that the equation has solution for every $f\in L^2(0,T)$, and write that solution as $u=Kf$, where $K$ is an operator of the form
$$[Ku](x)=\int\_0^T k(x,y)f(y)dy.$$

The problem is that my formation on differential equations is quite limited, and the only example the proffesor gave us is for the case $\mu=0$, which is quite trivial. Can I get help from you guys? 

Thanks a lot!