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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?

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