There is an equation $$ w(x) = g(x)+\int\limits_0^M w(y)f(x-y)\,dy $$ where $f\geq 0$, $f\in C^\infty(\mathbb R\setminus\{c\})$ for some point $c$ and $\int\limits_{-\infty}^\infty f(t)\,dt\leq 1$. With regards to $g$ we know that $0\leq g(t)\leq 1$. This equation should be solved for $w(x)$ on $[0,M]$. Functions $g,f$ are given. I also know a priori that $w\in C([0,M])$ and bounded by $0$ and $1$.
I guess that it is impossible to solve it analytically. For the numerical methods I know just one method - Neumann series, moreover $$\sup\limits_{x\in[0,M]}\,\,\,\,\,\,\int\limits_0^M f(x-y)dy = \alpha<1$$ but $1-\alpha\approx 0.001$ so the convergence of these series is very slow. Could you advise me any other method for the solution of this problem - or maybe you can refer me to the appropriate literature?
Edited: Thanks to Paul's comment I made my question more explicit. Unknown is function $w$ on $[0,M]$ rather than the point $x$. I would like to stress that the solution I need must be $\varepsilon$-precise, i.e. $\|w-w^*\|\leq \varepsilon$ for the numerical solution $w^*$ . Here $\|\cdot\|$ is a sup-norm.
I also posted this question on mathoverflow.