Consider the following ODE: $$ x'(t)=b(x(t)),\quad x(0)=x_0. $$ If $b$ is bounded and Holder continuous, then the Cauchy-Peano theorem ensures that there exists a solution to the above equation (but in general not unique). The question is:
is it possible that there always exists a solution $x_t(x_0)$ which depends Lipschitz continuous in $x_0$? Or $x_0\to x_t(x_0)$ is one-to-one?
Many thanks for the answers!