Maximum norm of functions; real functional analysis

Question

I am stuck on this exercise. We are studying the boundary value problem \begin{equation} \begin{cases} u(x) - u''(x) = f(x),&x \in (0,1) \\ u(0) = u(1) = 0 \\ \end{cases} \label{prob} \end{equation}

which has the solution (in compact form) \begin{equation} u(x) = \int_{0}^{1}G(x,y)(f(y) - u(y))dy \label{sol} \end{equation} Where $G(x,y)$ is the Greens functions for this specific boundary value problem. Note, the maximum norm of this Greens functions is 1/8. I want to show that \begin{equation} ||u||_{\infty} \leq \frac{1}{7}||f||_{\infty} \end{equation} However, when arriving at $$||u||_{\infty} \leq \frac{1}{8}||f-u||_{\infty}$$ I'm not sure how to treat the maximum norm of the difference between the two functions. Any advice and/or supplementary literature would be very helpful.

Answer 1

It is like this:

$||u||_{\infty} \leq ||Gf ||_{L^1} + ||Gu ||_{L^1} \leq ||Gf||_{L^1} + \frac{||u||_{\infty}}{8}$. I'll let you finish the rest.

($||Gf||_{L^1} \leq \frac{1}{8}||f||_{\infty}$)