# Multidimensional (Fredholm) integral equation (of first kind)

Suppose, \begin{align*} g(t_1,t_2)=\int f(s) \left[K_1(t_1,t_2,s)h(t_1) + K_2(t_1,t_2,s)\right]ds %\\ %g(t_1,t_2)=\int f(s) \left[K_1(t_1,t_2,s) + K_2(t_1,s)h(t_2)\right]ds \end{align*} The problem is to solve for $f(s)$ and $h(t_2)$, given the continuous kernel functions $K_1(t_1,t_2,s)$ and $K_2(t_1,t_2,s)$ and the function $g(t_1,t_2)$.

I am interested (1) in solutions to $f(s)$ in this multidimensional (Fredholm) equation (of the first kind), (2) is the solution to $f(s)$ unique, and (3) more generally, are there good references to integral equations when there is more than one unknown function and $g(\cdot)$ is multimdimensional.

Note: I am only interested in solutions to $f(s)$ i.e. $h(t_1)$ is an unknown nuisance function.

Here is my attempt at a solution that I'm not sure how to make rigorous (and if it's correct). Assume there exists $L_2(t_1,t_2,w)$ such that for all $f(\cdot)$, we have $$\int \int f(s) L_2(t_1,t_2,w) K_2(t_1,t_2,s)dt_2ds=f(w)$$ Then, \begin{align*} \int L_2(t_1,t_2,w) g(t_1,t_2) dt_2&= \int L_2(t_1,t_2,w) \int f(s) K_1(t_1,t_2,s)h(t_1) dt_2d_s \\ & \qquad \qquad + \int \int f(s) L_2(t_1,t_2,w) K_2(t_1,t_2,s)dt_2ds \\ &=\int L_2(t_1,t_2,w) f(s) K_1(t_1,t_2,u)h(t_1)dsdt_2 + f(w) \qquad (*) \end{align*} Normalize $h(0)=1$ and let $K_3(w,u)=\int L_2(0,t_2,w) K_1(0,t_2,u)h(0)dt_2$. Then, \begin{align*} \int L_2(0,t_2,w) g(0,t_2) dt_2 &=\int L_2(0,t_2,w) f(u) K_1(0,t_2,u)h(0)dudt_2 + f(w) \\ &=\int f(u) K_3(w,u)du + f(w) \end{align*} which is a Fredholm equation of the second kind and can be solved using standard techniques.

I don't know how to make (*) rigorous.