Nonlinear system of integral equations I have encountered a system of nonlinear integral equations in my work. They take the form
$$\int_{0}^{1} \frac{1}{g(y)}e^{f(x)/g(y)}(x+f(x)/g(y)-f(x))dy=0$$
$$\int_{0}^{1}\frac{f(x)}{g(y)^2} e^{f(x)/g(y)}(y+f(x)/g(y)-f(x))dx=0$$
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."
If it's worth anything, I know a few conditions that the functions $f$ and $g$ should satisfy. I know for example that $f$ and $g$ should be non-decreasing; $g(0)=1$ and $f(0)=0$ and of course $g(y)\neq 0$ anywhere. Additionally, it should be the case that $f$ and $g$ both take values in $\mathbb{R_+}$.
I have somewhat limited experience with integral equations. All of my initial attempts at making sense of the system have come up short. At this point, 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? Is there way that you can see to simplify or perhaps decouple the system?

*My only remaining (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 the power series of $f$ and $g$. Does it seem possible that this will give useful information?

*Is there any theorem like the Picard-Linelof theorem (existence and uniqueness of solutions) for integral equations?

Thank you in advance for your help!
 A: Unless you have made a mistake when typing the equations or are willing to abandon the non-decreasing property of $f$ and $g$ (and I'm not sure the latter will help), there is no solution.
Indeed, multiply the first equation by $f(x)$ and integrate over $x$. Then multiply the second one by $g(y)$ and integrate over $y$. You'll get
$$
\int_{[0,1]^2}\Phi(x,y)x\,dxdy=\int_{[0,1]^2}\Phi(x,y)f(x)(1-\tfrac 1{g(y)})\,dxdy=\int_{[0,1]^2}\Phi(x,y)y\,dxdy\,,
$$
where $\Phi(x,y)=\frac{f(x)}{g(y)}e^{f(x)/g(y)}$.
Note now that, since $\Phi$ is non-decreasing in $x$ for fixed $y$, we have
$\int_{[0,1]^2}\Phi(x,y)x\,dxdy\ge \frac 12\int_{[0,1]^2}\Phi(x,y)\,dxdy$ and the inequality is strict unless $f$ is constant. On the other hand, since $\Phi$ is non-increasing in $y$ for fixed $x$, we have
$\int_{[0,1]^2}\Phi(x,y)y\,dxdy\le \frac 12\int_{[0,1]^2}\Phi(x,y)\,dxdy$ and the inequality is strict unless $g$ is constant or $f\equiv 0$. Thus, the only way to have the above equality is to have $f$ and $g$ constant or $f\equiv 0$, which is clearly impossible for the original system.
