Uniqueness of IVP solution with a condition weaker than Lipschitz?

Question

We know that Lipschitz condition with respect to $x$ in $$x' = f(t,x) , x(t_0)=x_0 $$ implies uniqueness of IVP problem above. Can we have uniqueness with condition less than Lipschitz?

Answer 1

Yes, for example $$|f(t,x_1)-f(t,x_1)|\le C|x_1-x_2|(1+|\log|x_1-x_2||) $$ is weaker than the Lipschitz condition (it allows $f(t,x)=x\log x$, for example), and also implies uniqueness. This is a special case of the Osgood criterion.

Also, there is a book filled with various conditions implying uniqueness: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations.

Answer 2

We have uniqueness in the separable case $$f(x,t) = a(x) \cdot b(t)$$

if $a(x)$, $b(t)$ continuous and $a(x)\ne 0$ since for any solution $x(t)$ we can write

$$\int_{x_0}^{x(t)} \frac{ dx}{a(x)}= \int_{t_0}^t b(t) dt $$