Differential and Integral Equations: An International Journal for Theory and Applications
For the third order differential equation (see PDF), subintervals of (a,b) of maximal length are characterized, in terms of the Lipschitz coefficients (see PDF) on which certain boundary value problems possess unique solutions. The techniques for determining best interval length involve applications of the Pontryagin Maximum Principle along with uniqueness implies existence arguments.
Eloe, Paul W. and Henderson, Johnny, "Optimal intervals for third order Lipschitz equations" (1989). Mathematics Faculty Publications. 96.