In several papers and other sources, I have seen statements about it being `well-known' that the Fredholm integral equation of the second kind is well-posed, in contrast to a Fredholm integral equation of the first kind. I'm looking for some precise statements that an equation of the form
$$f_i(t)=g_i(t)+\int_a^bK_i(t,s)f_i(s)\ \mathrm ds,\quad i=1,2,$$
has a solution depending on the initial conditions in a continuous manner. In other words, we expect something like: given $\epsilon>0$, there exists $\delta>0$ s.t. $\|g_1-g_2\|,\|K_1-K_2\|<\delta$ $\implies$ $\|f_1-f_2\|<\epsilon$. With respect to which norms do those results hold true? Is this true in $L^1,L^2,L^\infty$? Or even in any $L^p$ space, $p\in[1,\infty]$?
The only result in this spirit that I was able to find is Theorem 2.6.1 from Zemyan, The Classical Theory of Integral Equations, A Concise Treatment, Birkhauser, which deals with $L^\infty$ spaces. However, I'm particularly interested in the $L^1$ case, and where we are able to drop continuity assumptions on the initial conditions. Does such a result hold as well if we use $L^1$ norms?
EDIT It seems that Corollary 3.12 of Gripenberg, Londen, Staffans, Volterra Integral and Functional Equations provides a positive answer to my question.
Any help or reference is much appreciated.