Integral Form For Continuous Functions
Let I denote an interval of the real line of the form or [a, b) with a < b. Let α, β and u be real-valued functions defined on I. Assume that β and u are continuous and that the negative part of α is integrable on every closed and bounded subinterval of I.
- (a) If β is non-negative and if u satisfies the integral inequality
-
- then
- (b) If, in addition, the function α is non-decreasing, then
Remarks:
- There are no assumptions on the signs of the functions α and u.
- Compared to the differential form, differentiability of u is not needed for the integral form.
- For a version of Grönwall's inequality which doesn't need continuity of β and u, see the version in the next section.
Read more about this topic: Gronwall's Inequality
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