Gronwall's Inequality - Differential Form

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.