I have the following question for which I have some intuition but not a complete one.
I have a function $F(x)$ (Lipschitz) and its smooth (asymptotic?) approximation $\tilde{F}_k(x)$ that smoothly depends on a real positive parameter $k$. $\tilde{F}_k(x)$ is an "asymptotic" approximation of $F(x)$ as
$$\lim_{k\to+\infty} \tilde{F}_k(x) = F(x)$$
Is there any relationship between the solutions of $F(x)=0$ and the solutions of $\tilde{F}_k(x)=0$? Or even better, under which conditions the solutions of $\tilde{F}_k(x)=0$ approach the solutions of $F(x)=0$?
My first conjecture
Assume that for each $x^*$ such that $: F(x^*)=0$
- F(x) is locally smooth (probably $\mathcal{C}^1$ would be enough) around $x^*$
- $JF(x^*)$ (Jacobian of $F$) is nonsingular in a neighborhood of $x^*$,
- $J\tilde{F}_k(x^*)$ (Jacobian of $F$) is nonsingular in a neighborhood of $x^*$, $\forall k$
then for all $x^*$ solutions of $F(x^*)=0$, for each neighborhood $B_\delta(x^*)$
$\exists \overline{k}: \forall k> \overline{k}$ there exists a unique $\tilde{x}^* \in B_\delta(x^*)$ such that $\tilde{F}_k (\tilde{x}^*)=0$.
One counterexample where some of the conditions above do not hold
I know that if we don't impose enough conditions, in general there is no 'useful' relationship. For example, consider $F(x)=0$ and $\tilde{F}_k(x)=1/k^2$.