Solving a implicit, recursive function

Question

Assuming $F(0,0)=0$, can I solve $F(F(x,y),y)=0$ for $y=f(x)$ near $(0,0).$

Answer 1

Denote $$ G(x,y) = F(F(x,y), y) \tag{1}. $$

Implicit function theorem says that there is a unique solution $y = f(x)$ in some neighborhood of $(0,0)$ if $$ \frac{\partial G}{\partial y}(0,0) \neq 0. $$

Using (1) and chain rule we get $$ \frac{\partial G}{\partial y} = F'_1 \cdot \frac{\partial F}{\partial y} + \frac{\partial F}{\partial y}. $$

Here by $F'_1$ we denoted partial derivative of $F$ with respect to first variable.

Can you proceed now?