I'm looking for a proof of the following result:
Let I denote an interval of the real line of the form $[a,b]$ with $ a<b $. Let $\beta$ and $u$ be real-valued continuous functions defined on I. If $u$ is differentiable in the interior of I and satisfies, for all $t \in I^o$ $$ u'(t) \leq u(t)\beta(t), $$ then u also satisfies $$ u(t) \leq u(a)exp(\int^t_a\beta(s)ds) $$ for all $t \in I$.
It's basically a summarized version of the one appearing if you look for Gronwall's inequality on wikipedia. I'd like to find it somewhere else so I could understand it better, also I'd need a good reference since I'm in need of this result for an assignement. I've tried and tried googling it yet all I find are proofs of the integral form. Could someone tell me of a good article/book where I could find it?
Thank you in advance.
Perhaps you might try:
E. Coddington & N. Levinson, Theory of ordinary differential equations, McGraw-Hill, 1955.
A copy of a paper on its proof is given here (where the above reference was cited)
https://www.math.wisc.edu/~robbin/angelic/gronwall.pdf
Also, check out
Any other proof for the Gronwall's inequality?
Hope this helps.