Since my original question is difficult. I would like to ask following simplier question. I want to give the bonus to a good answer.
Let $f,g, \bar f,\bar g$ be convex and increasing functions from $\mathbb{R}$ to $\mathbb{R}_+$. Assume that $\bar f, \bar g$ are an approximated functions of $f$ and $g$, respectively. Let $\alpha,\bar \alpha$ be the largest solution of $x+g\circ(f(c-x)) = d$ and $x+\bar g\circ(\bar f(c-x)) = d$, where $c, d$ are positive constant. What kind of conditions on $f, g,\bar f, \bar g$ if I want $|\alpha-\bar \alpha| \leq \epsilon$ for arbitrary $\epsilon > 0$.
I think we need some condition like Lipchitz but I do not think it is enough. Does anyone have any idea? Thank you in advance!