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A generalization of the Nagumo uniqueness criterion

Author: Thomas C. Gard
Journal: Proc. Amer. Math. Soc. 70 (1978), 167-172
MSC: Primary 34A10
MathSciNet review: 0470288
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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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