Is there a fixed point theorem I could use to solve this problem? let $E = C([0,1]),\,\,$ $K : E \to E, \,\,
(Kf)(x) = \int_0^1K(x,y)f(y)dy$
also $\|K\| \leq a < 1$
I want to prove that there for $g \in E$ there exists a unique $f_g \in E$ that satisfies the following equation : 
$f_g + Kf_g = g$
which is equivalent to showing that $T : E \to E,\,\,T(f) = g-Kf$ has a fixed point.
with what I have in hands I feel like there must be some theorem I'm missing.
any help will be greatly appreciated !  

You can apply the Contraction mapping, a.k.a. Banach's Fixed Point Theorem. Given $f,h\in C([0,1])$,
$$
\|Tf-Th\|\le\int_0^1|K(x,y)|\,|f(y)-h(y)|\,dy\le\|K\|\,\|f-h\|<a\,\|f-h\|,
$$
with $0<a<1$.
