A generalized Picard-Lindelöf theorem

Abstract

We generalize the Picard-Lindelöf theorem on the unique solvability of initial value problems $\dot x=f(t,x)$, $x(t_0)=x_0$, by replacing the sufficient classical Lipschitz condition of $f$ with respect to $x$ with a more general Lipschitz condition along hyperspaces of the $(t,x)$-space. A comparison with known results is provided and the generality of the new criterion is shown by an example.

Keywords

• picard-lindelöf theorem
• initial value problem
• generalized lipschitz condition
• unique solvability