Unique solution to a general second order BVP

Question

Consider the following second order nonlinear differential equation:

$$u''(x)=f(x,u(x))$$

where $f\in C([0,1],\mathbb{R})$ and $u(0)=u(1)=0$

How can we show that this differential equation has a unique solution of form:

$$u(x)=\int_0^1g(x,\zeta)f(\zeta,u(\zeta))d\zeta$$ with $u\in C[0,1]$.

Thanks in advance

Answer 1

Unless some condition is imposed on $f$, we cannot. Consider the equation $$ u''=-\pi^2u,\quad u(0)=u(\pi)=0. $$ $u(t)=C\sin (\pi\,t)$ is a solution for all $C\in\mathbb{R}$.