A uniqueness theorem for 𝑦’=𝑓(𝑥,𝑦),𝑦(𝑥₀)=𝑦₀

Majorana, Armando

doi:10.1090/S0002-9939-1991-1028290-X

A uniqueness theorem for $y’=f(x,y),\;y(x_ 0)=y_ 0$
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by Armando Majorana

Proc. Amer. Math. Soc. 111 (1991), 215-220

DOI: https://doi.org/10.1090/S0002-9939-1991-1028290-X

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Abstract:

Consider the initial value problem for a first-order differential equation \[ y’ = f(x,y),\quad y({x_0}) = {y_0}.\] In this paper a new uniqueness criterion is proved. This criterion is related to the numeric equation \[ u = {y_0} + (t - {x_0})f(t,u).\] It is also shown that some well-known uniqueness theorems are consequences of our result.

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Differential and integral inequalities : theory and application