An extension of the Nagumo uniqueness theorem

Abstract:

The classical Nagumo uniqueness theorem is a best possible result in the sense that if the Nagumo constant is replaced by a number greater than one then the result is false. This classical result uses the continuity of the right-hand side, $f(x,y)$, of the first order ordinary differential equation, and there is no explicit connection shown between the constant and the continuity of $f$; the observation that one makes is that the counterexample originally given by Perron uses a discontinuity at the origin together with a constant greater than one to obtain an initial-value problem with many solutions. This paper shows explicitly that the original theorem remains true with Nagumo constant greater than one provided that $f$ is sufficiently small at the origin, this sufficiency being determined by the actual value of the constant. Moreover, it is shown that the Nagumo inequality need not be imposed on the entire right-hand side of the equation; it suffices that only a certain factor satisfy a weakened form of the inequality.