While reading the fairly famous paper by Brezzi et al from 1980 (link), I got slightly puzzled by one of the conditions, in particular in Theorem 1 (Implicit Function Theorem basically). They consider a function $F \in C^1(X\times Y,Z)$ and in many places require conditions such as: there exists a monotonically increasing function $L_1\,:\,\mathbb{R}_+ \to \mathbb{R}_+$ such that for all $(x,y) \in S(x_0,y_0,\xi)$, where $S(x_0,y_0,\xi) := \{(x,y) \in X \times Y\,|\, \|x-x_0\|_X + \|y-y_0\|_Y \leq \xi\}$, it holds that $$ \|Df(x,y) - Df(x_0,y_0)\| \leq L_1(\xi)(\|x-x_0\|_X + \|y-y_0\|_Y). $$ It is fairly obvious from the proofs that when they mean 'monotonically increasing', they do not require 'strictly increasing', hence $L_1(\xi) = C$ (constant) would do (unless I'm mistaken?). Now, since $f$ is assumed to be $C^1$, I guess a constant would not work because $DF$ is simply continuous, but not uniformly continuous. Would it be always be true if, say, $f \in C^2$, as then we can argue that $$ \|Df(x,y) - Df(x_0,y_0)\| \leq \|D^2f(x_*,y_*)\|(\|x-x_0\|_X + \|y-y_0\|_Y), $$ where $(x_*,y_*) \in S(x_0,y_0,\xi)$? Are there any subtleties here that I fail to grasp? Is it a strong assumption? I will greatly appreciate any comments, thanks!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.