Since you are requiring, among other things, some stability for the existence of solution of $F(x)=0$ under perturbation, it seems quite natural to think in terms of topological degree. 

- Assume that there is an open bounded nbd of $x,$ $\Omega\subset\mathbb{R}^d,$ such that $x$ is the only solution of $F=0$ in $\bar\Omega.$ In particular this implies that $\deg(F,\Omega,0)$ is well defined. 
- Assume further  $\deg(F,\Omega,0)\ne 0.$ (several different facts may ensure this: $F$ is an odd map, or it is injective; or you can compute its degree etc). 
- Assume that $F_n$ converge to $F$ uniformly on $\bar\Omega$ (several diffferent facts may ensure this too: for instance, $F_n$ converges pointwise to $F, $ and it is uniformly equicontinuous on  $\bar\Omega$, etc). 

Then, $\deg(F_n,\Omega,0)$ is eventually defined and different to $0$. So there exists $x_n\in\Omega$ solving $F_n(x_n)=0$ (not necessarily unique). By the compactness of $\bar\Omega$ together with the uniform convergence of $F_n$ to $F$ on $\bar\Omega$, we can conclude $x_n\to x$ as we wish.


*PS*: all maps are assumed to be continuous, of course.