Suppose that we have the following
$ u'' = -\lambda \cos u $
with $u(0) = u(1) = 0$. How can it be shown that if $|\lambda|$ is sufficiently small then the problem above has a unique solution?
The hint is to reformulate the problem as a nonlinear integral equation. So here is my attempt:
$ u'(t) = C -\int_{0}^{t} \lambda \cos(u(s)) ds $
If we define the map $T$ as
$ T(u(t)) = C -\int_{0}^{t} \lambda \cos(u(s)) ds $
we can write
$ u' = T(u) $
With the contraction mapping theorem I could show that if $T(x) = x$ and $T$ is a contraction mapping then $T(x) = x$ has a unique solution.
I suppose that if I integrate again:
$ u(x) = C_2 + C_1 x - \int_{0}^{x} \int_{0}^{t} \lambda \cos(u(s)) ds dt $
but that is where I'm running out of ideas.