Two initial conditions on the functions Let $g \, : \, [0,1] \, \longrightarrow \, \mathbb{R}$ a function continuous on $[0,1]$. I would like to solve the following problem :
$$
(S) \; \left\{
\begin{array}{l}
f''=g \\
f(0)=a \\
f(1)=b \\
\end{array}
\right.
$$
where $(a,b) \in \mathbb{R}^{2}$ are given. How can I prove that $(S)$ has a solution ? My first idea would be to apply Cauchy-Lipschitz's theorem but I only have initial conditions on $f$. Is there an idea to prove the existence of a solution using Cauchy-Lipschitz ? 

$(S)$ has a unique solution
$$
f(x)=\int_0^x (x-t)\,g(t)\,dt+a+ct,
$$
where
$$
c=b-\int_0^1 (1-t)\,g(t)\,dt-a.
$$
