I know that given a homogeneous Fredholm equation of the second kind, of the form $u(x)=\int_a^b K(x,y)u(y)\ dy$, under some nice conditions on the kernel $K$ we can solve it iteratively starting with some trial solution $u_0$ and defining $u_n(x)=\int K(x,y)u_{n-1}(y)\ dy$. Then as $n\to\infty$, the sequence $u_n$ converges to the solution $u$. There is also a cute extension of this method to the inhomogeneous case.
However, what I am wondering is does anyone know if a similar iterative approach exists for inhomogeneous Fredholm equations of the first kind, of the form $$f(x)=\int_a^b K(x,y)u(y)\ dy$$ where $f$ is a known function?
So, what I've find from doing some reading is that one can take a small parameter $\epsilon$ to define a new equation $$\epsilon u_{\epsilon}(x)=f(x)-\int_a^b K(x,y)u_{\epsilon}(y)\ dy,$$ solve this equation using iterative techniques for Fredholm integral equations of the second kind and then obtain the answer from the limit $$u(x)=\lim_{\epsilon\to 0}u_{\epsilon}(x).$$