Existence of solutions to a series of integral equations

Question

I am trying to solve the following integral equation analytically:

$$ \sum_{n \geq 1} \left( \int_0^te^{-n^2(t-s)} f_n(s) \, ds \right) = g(t), \quad t \in [0, T], $$ where $(f_n(t))_n$ is the unknown functions to be determined and $g(t)$ is a given function in $L^2(0,T)$. I am interested in the existence of the solution for $g(t)$ in $L^2(0,T)$.

Could someone provide guidance on references or techniques that could be useful for handling this problem? I have tried with Fourier series and Laplace transform but they didn't lead me anywhere.

Thank you in advance!

Answer 1

Absent details about the function space, I'll assume that everything is sufficiently smooth such that we can infer from the integral equation that $g(0)=0$. Then, there is a whole set of solutions of the form $f_n =0$ for all $n\neq m$, with $m$ chosen at will; and $f_m (t) = m^2 g(t) + g'(t)$, as can be readily verified by integration by parts. Of course, one can then also form linear combinations of these.