Two initial conditions on the functions

Question

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 ?

Answer 1

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