A general Lipschitz uniqueness criterion for scalar ordinary differential equations - Josef Diblik (Electronic Journal of Qualitative Theory of Differential Equations, 2014)
The classical Lipschitz-type criteria guarantee unique solvability of the scalar initial value problem $\dot x=f(t,x)$, $x(t_0)=x_0,$ by putting restrictions on $|f(t,x)-f(t,y)|$ in dependence of $|x-y|$. Geometrically it means that the field differences are estimated in the direction of the $x$-axis. In 1989, Stettner and the second author could establish a generalized Lipschitz condition in both arguments by showing that the field differences can be measured in a suitably chosen direction $v=(d_{t},d_{x})$, provided that it does not coincide with the directional vector $(1,f(t_{0},x_{0}))$.