Rather than asking the most general question possible, I will frame it in terms of what I believe is an illustrative example.
Let $\epsilon>0$ be a small parameter, let $a,b>0$ and $x\in [-\epsilon,\epsilon]$. If $y\geq 0$ is sufficiently small in terms of $\epsilon$, then we can solve the equation $a x^2+b x^4=y$ explicitly for $x$. One solution is $x_1=\sqrt{-\frac{a}{2 b} + \frac{\sqrt{a^2 + 4 b y}}{2 b}}.$ Suppose that instead we want to solve the perturbed equation $a x^2+b x^4+\phi(x)=y$ for $x$, where $\phi$ is some smooth function satisfying $\phi(x)=O(x^5)$ as $|x|\rightarrow 0$. In general, an explicit solution like before is not available. How to proceed? In particular, if $\widetilde{x}_1$ denotes the nonnegative solution of the perturbed equation, what can be said about $|x_1-\widetilde{x}_1|$? Is it $O(\epsilon^5)$?
Thank you.