# Proof that boundary value problem has solution

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?