Proof that boundary value problem has solution for every $\lambda$ and every $f \in C[0, 1]$:
$y'' + \lambda \sin(y) = f(x)$
$y(0) = y(1) = 0$
I think we should use one of the fixed-point theorems, such as Brouwer's. I tried to solve it using Green's function, but it's not easy to find a solution of the ordinary equation. Any ideas?