# Existence and uniqueness theorems for ODE. Log-Lipschitz regularity.

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.

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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.
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: en.wikipedia.org/wiki/Modulus_of_continuity – MathOverview Feb 28 '12 at 17:53