A general Lipschitz uniqueness criterion for scalar ordinary differential equations

Abstract

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 $(1,f(t_{0},x_{0}))$. Considering the vector $v$ depending on $t$, a new general uniqueness result is derived and a short proof based on the implicit function theorem is developed. The advantage of the new criterion is shown by an example. A comparison with known results is given as well.

Keywords

• fundamental theory of ordinary differential equations
• initial value problems
• uniqueness
• lipschitz type conditions