TITLE: Solving or approximating infinitely nested integral
QUESTION [6 upvotes]: Let $f$ given by
$$f(x) = g(x) + \int_0^x\left(g(x_1) + \int_0^{x_1} \left( g(x_2) + \int_0^{x_2} \ldots d_{x_n} \ldots \right) d_{x_2} \right) d_{x_1}$$
where $n \rightarrow \infty$ and $g(x)$ is strictly decreasing in $x$.
How can such an integral be solved or approximated for?

REPLY [3 votes]: Yet another way: using the Cauchy formula for repeated integration, an $n$-fold integral can be replaced by one integral:
$$ (J^k f)(x) = \int_0^{x}\dotsi \int_0^{t_{n-1}} f(t_1) \, dt_n \dotsm dt_1 = \frac{1}{(n-1)!} \int_0^x (x-t)^{n-1} f(t) \, dt, $$
which leads to the right-hand side becoming
$$ g(x) + \sum_{k=1}^{\infty} (J^k g)(x) = g(x) + \int_0^x \sum_{k=1}^{\infty} \frac{(x-t)^{k-1}}{k!} g(t) \, dt = g(x) + \int_0^x e^{x-t} g(t) \, dt; $$
the interchange of the integral and sum using the convergence assumption and that $J$ is a contraction mapping on the space of functions.