I have encountered a system of nonlinear integral equations in my work. They take the form $$\int_{0}^{1} g(y)e^{f(x)g(y)}(x+f(x)g(y)-f(x))dy=0$$ $$\int_{0}^{1}f(x)g(y)^2 e^{f(x)g(y)}(y+f(x)g(y)-f(x))dx=0$$
The goal is to find (or prove the existence of) real functions $f$ and $g$ satisfying the above equations. I have gathered that this would be called a "nonlinear system of homogeneous Fredholm equations of the first kind."
I have somewhat limited experience with integral equations. Some of my questions (I would appreciate insights into any of them!)
- Is it reasonable to expect that there is a method to find an explicit form for $f$ and $g$?
- Can somebody point me to a reference that describes dealing with systems of integral equations? The sources that I have looked at on integral equations so far tend to deal with primarily a single integral equation with a single unknown function.
- Are there techniques that you know of that may be worth applying to such a system?
- My only (naive) idea for approaching it is to Taylor expand everything in sight and see if it is possible to extract conditions on the coefficients for $f$ and $g$. Does it seem possible that this will give useful information?
- Is there any theorem like the Picard-Linelof theorem (existence and uniqueness) for integral equations?
If it's worth anything, I know a few conditions that the functions $f$ and $g$ should satisfy. I know for example that $g$ should be non-increasing and $f$ should be non-decreasing; $g(0)=1$ and $f(0)=0$.
Thank you in advance for your help!