A Fredholm integral equation of the first kind has the following shape: $$ \int a(x,y)f(y)\mathrm dy = b(x)\tag{1} $$ where $f$ is an unknown function. I wonder whether contraction principle can be somehow applied to find the solution of this equation. My issue is that it is of course a linear equation so we can write it as $$ f(x) = -b(x) + \int (\delta(x-y) + a(x,y))f(y)\mathrm dy $$ but to study the contractivity of the operator $I+A$ where $$ A:f \mapsto \int af $$ I need to deal with the Dirac $\delta$ which does not seem to lead to any interesting result so far. I guess for the contractivity of $I+A$ the spectrum of $A$ must be a subset of $(-2,0)$, however I am not sure how to translate this on the conditions on the kernel function $a$. Any thoughts?

Regarding metric for the contraction, ideally it shall be sup-norm, but $L^p$ is interesting as well if there are no results for the former. Functions $a$ and $b$ can be assumed to be $C^\infty$.

Aside from the main question, I am interested in the literature on the existence/uniqueness and numerical schemes for $(1)$.


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