I have a system of two equations in two unknown functions, and I'm trying to figure out if a solution exists, and under which conditions it is unique. Let $p:[0,1]^2\to[0,1]$ be a probability density. Let $F:[0,1]\to\mathbb{R}_+$ and $G:[0,1]\to\mathbb{R}_+$ be any two bounded, integrable functions. I want to find functions $u:[0,1]\to\mathbb{R}_+$ and $w:[0,1]\to\mathbb{R}_+$ such that the following system holds for any $x,y\in[0,1]$:
$$u(x)=\frac{F(x)}{\int_0^1p(x,y)w(y)dy}$$ $$w(y)=\frac{G(y)}{\int_0^1p(x,y)u(x)dx}$$
My intuition tells me that probably I can think of the solution of this system as a fixed point problem. Likewise, from the expression for $w$, I can write
$$\int_0^1p(x,y)w(y)dy=\int_0^1\bigg(\frac{p(x,y)G(y)}{\int_0^1p(x,y)u(x)dx}\bigg)dy$$
And then have one equation with one unknown function $u$ to be
$$u(x)=\frac{F(x)}{\int_0^1\bigg(\frac{p(x,y)G(y)}{\int_0^1p(x,y)u(x)dx}\bigg)dy}.$$
However, I don't know how to obtain the solution to this last equation. I feel I should be able to link it to a differential equation or something. Any help solving this problem is greatly appreciated: especially necessary conditions and fixed point theorems or other results/approaches that help me guarantee a unique solution exists.