There are a number of different statements of Gronwall’s inequality. In this post, we will consider only one of them, perhaps the weakest of all.
Proposition 1 Let be a non-negative continuous function on such that there are positive constants and satisfying
for all . Then,
for all .
Proof: Define . Note that, by definition, and is a strictly positive differentiable function. Also, we have . In other words, which means the relative rate of change of is less than . Hence, the growth of is slower than an exponential function with relative rate of change . That is (if you did not like this reasoning, you may integrate both sides of the previous inequality from to ). So, we have the desired result .