Optimal version of the Picard-Lindel\"of theorem

TL;DR

This paper identifies the minimal conditions on the Lipschitz constant of the function F(x,y) that ensure the existence and uniqueness of global solutions to the differential equation y'=F(x,y) for all initial values.

Contribution

It establishes the weakest possible upper bound on the Lipschitz constant that guarantees global uniqueness and existence of solutions for the differential equation.

Findings

Derived the minimal Lipschitz bound for global solution guarantees

Extended classical Picard-Lindel"of theorem to optimal conditions

Provided theoretical proof of the bounds' sufficiency and necessity

Abstract

Consider the differential equation $y'=F(x,y)$. We determine the weakest possible upper bound on $|F(x,y)-F(x,z)|$ which guarantees that this equation has for all initial values a unique solution, which exists globally.