Affine Gronwall estimate
= Affine Gronwall estimate
{title2=$v(t)\leq c(e^{At}-1)/A$}
If a nonnegative <continuous function> on $[0,h]$ satisfies $v(t)\leq ct+A\int_0^t v(s)\,ds$ with $c,A>0$, put $W(t)=ct+A\int_0^t v(s)\,ds$. Then $v\leq W$, $W(0)=0$ and $W'\leq c+AW$. Multiplication by the <integrating factor> $e^{-At}$ and <integration> prove the displayed estimate. For $A=0$ its limiting interpretation is $v(t)\leq ct$.