# the solution of Fredholm´s integral equation

Be $\lambda \in \mathbb{R}$ such that $\left | \lambda \right |> \left \| \kappa \right \|_{\infty }(b-a)$.

Prove that the solution $f^*$ of the integral equation of Fredholm $$\lambda f -\int_{a}^{b}\kappa (x,y)f(y)dy=g(x)$$ for all $x\in [a,b]$ satisfies $$\left \| f^*-\sum_{m=1}^{k}\frac{1}{\lambda ^m}\Im ^{m-1}g \right \|_{\infty }\leq \frac{\alpha ^{k}}{(1-\alpha) \left | \lambda \right |} \left \| g \right \|_{\infty }$$ for all $k\in \mathbb{N}$ where $$\alpha :=\frac{\left \| \kappa \right \|(b-a)}{\left |\lambda \right | }$$

i´m really stuck in this problem, I know that I can use that $$\Im :C_{\infty }^{0}[a,b]\rightarrow C_{\infty }^{0}[a,b]$$ is Lipschitz continuous and that linear Fredholm´s operator is linear, can anybody just give a hint please? thanks!