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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.

3 Answers 3