I ran across the following problem with a friend while we were studying for quals. Neither of us are really quite sure where to start. It feels like a differential equation. This is probably easy, but we were not able to get a handle on how to proceed. I wish I could tell you what I tried, but after thinking on this problem for some time, I simply do not have ideas of any real substance (other than what I mention after the problem statement).
Here is the question as it appears on the old qual:
"Let $K$ be a continuous function on the unit square $0\leq x,y\leq1$ satisfying $|K(x,y)|<1$ for all $x$ and $y$. Show that there is a continuous function $f(x)$ on $[0,1]$ such that we have
$$f(x) + \int_0^1 f(y)K(x,y)dy=\sin(x^2)$$
where $0\leq x \leq 1$. Can there be more than one such function $f$?"
I will say that I was able to show that given $K$ as it is, $\exists\,C\in(0,1)$ such that $|K|\leq C$ on the square, and that a function defined as
$$G(x)=\int_0^1 g(y)K(x,y) dy$$
will be continuous, assuming that $g$ is continuous on $[0,1]$.
Any suggestions or possible solutions would be greatly appreciated.