A generalization of the Nagumo uniqueness criterion

Gard, Thomas C.

doi:10.1090/S0002-9939-1978-0470288-8

A generalization of the Nagumo uniqueness criterion
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by Thomas C. Gard PDF

Proc. Amer. Math. Soc. 70 (1978), 167-172 Request permission

Abstract:

The classical Nagumo criterion for uniqueness of solutions of the initial value problem $x’ = F(t,x),x({t_0}) = {x_0}$, assumes continuity of F as well as a modulus of continuity condition involving $1/t$. Recently, it has been shown that $1/t$ can be replaced by $\alpha /t$, for $\alpha > 1$, or by $1/{t^2}$ provided that the continuity condition on F is suitably modified; in each case it is required that $F(t,x)$ at least approach a limit at a certain minimal rate as $(t,x)$ tends to the initial value. In this note we extend both of these results, and in so doing, clarify the relationship between the two types of conditions needed.

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