Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $\mathbb{X}$ be a linear space with a complete metric $d:\mathbb{X}\times\mathbb{X}\to [0,+\infty)$. Let's $B[x_o,b]$ is a compact ball of radius $b$ centered at $x_o$.

THEOREM:If $F:[t_o-a,t_o+a]\times B[x_o,b]\subset\mathbb{R}\times\mathbb{X}\to \mathbb{X}$ a limited application, continuous and continuous Lipschiz in $B[x_o,b]$ (note that if $\mathbb {X}$ have finite demension the condition is limited to be redundant).Then there exists a unique solution $$ \varphi : [t_o-\alpha,t_o+\alpha]\to B[x_o,b] $$ to the problem of Cauchy $$ x'(t)=F(x,t)\quad x(t_o)=x_o $$ where $\alpha=\min\{a,b\backslash M\}$ and $M=\sup\{|F(t,x)| : (t,x)\in [t_o-a,t_o+a]\times B[x_o,b] \}$.

DEFINITION:We say that $F$ is $\gamma$-log-Lipschitz in $B[x_o,b]$ if there exist $\gamma \ge 0$, $L>0$ and $C>0$ such that $$ \|F(x,t)- F(y,t) \| \le C{\bigg(\log\frac{L}{\|x-y\|}\bigg)^{\gamma}}\|x-y\|, $$ for all $ x,y \in B[x_o,b]$ and all $t\in [t_o-a,t_o+a]$.

QUESTION 1. There is a version of this theorem for Log-Lipschitz fields?We may waive the conditions of compactness of the ball and the range in this case?

QUESTION 2. There are other more unusual versions of this theorem where the field $ F $ satisfas $$ \|F(x,t)- F(y,t) \| \le |\Psi(x,y)|\cdot\|x-y\|, $$ for some function $ \Psi :\mathbb{X}\times\mathbb{X}\to\mathbb{R}$?We may waive the conditions of compactness of the ball and the range in this case?

Thank you.

share|cite|improve this question
up vote 1 down vote accepted

The necessary and sufficient condition for a modulus of continuity $h(t)$ to give uniqueness in ODEs is that the integral of $dt/h(t)$ is divergent at $0$ (exercice). Hence for your log-lipschitz modulus, you have uniqueness iff $\gamma\leq 1$. The corresponding flow is then Hölder, with exponentially decreasing exponent.

share|cite|improve this answer
A regularity more general than shown in Question 2 above proposed by the @user25917. For every $ t $ the application $ F (\;,t) $ has continidade module $ h $ is $$ \| F(x,t)-F(y,t) \| \leq h (\|x-y\|)\quad \forall x,y\in\mathbb{X} $$ For more see Wikipedia: – MathOverview Feb 28 '12 at 17:53

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.