Differential Form
Let I denote an interval of the real line of the form or [a, b) with a < b. Let β and u be real-valued continuous functions defined on I. If u is differentiable in the interior Io of I (the interval I without the end points a and possibly b) and satisfies the differential inequality
then u is bounded by the solution of the corresponding differential equation y ′(t) = β(t) y(t):
for all t ∈ I.
Remark: There are no assumptions on the signs of the functions β and u.
Read more about this topic: Gronwall's Inequality
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