Let's consider such a problem: $$\left\{\begin{matrix} &u''-u=f(u)+g(x) & \\ &u'(0)=u'(1)=0 & \end{matrix}\right.$$ for $x\in(0,1)$, where $u=u(x)$ isn't known and $f \in C(\mathbb{R}),g\in C^{1}([0,1])$ are given.
I need to show:
If $|f(u)|\leq L|u|^p$ and $|f(u)-f(v)| \leq L(|u|^{p-1}+|v|^{p-1})|u-v|$ for $L>0$ and $p>1$ for all $u,v \in \mathbb{R}$ and $f \in C^{1}(\mathbb{R})$ then we have a solution $u=u(x)$ when $||g||_{\infty}$ is small.
I tried to rewrite the equation in the integral form (the hint is: $u(x)=\int_{0}^{1}G(x,y)f(u(y))dy+\int_{0}^{1}G(x,y)g(y)dy$) and use the Banach fixed point theorem (for a certain ball). This would correspond to what we've done during the course. As the hint said I tried to find such a functon $G$ but I couldn't solve the second order equation to get the function and I also couldn't do anything with the two integrals from the hint...
Hint: $G(x,y)$, the Green's kernel, is composed of $\sinh(x)$ and $\sinh(1-x)$ and the same in $y$. With a kink along the diagonal. Try $$\sinh(\min(x,y))\sinh(1-\max(x,y)).$$
In general, Green's function for Sturm-Liouville problems has the form $$ G(x,y)=u(\min(x,y))·v(\max(x,y))= \begin{cases} u(x)·v(y)&x\le y\\ u(y)·v(x)&x\ge y \end{cases} $$ where $u,v$ are solutions of the homogeneous S-L BVP with $u$ satisfying the left and $v$ satisfying the right boundary condition. The general solution here for $u''-u=0$ is $$ u=A·e^x+B·e^{-x}=C·\cosh(x)+D·\sinh(x)=E·\cosh(1-x)+F·\sinh(1-x) $$