Affine Gronwall estimate ($v(t)\le c(e^{At}-1)/A$)

If a nonnegative continuous function on $[0,h]$ satisfies $v(t)\le ct+A{\int }_0^{t}v(s)ds$ with $c,A>0$, put $W(t)=ct+A{\int }_0^{t}v(s)ds$. Then $v\le W$, $W(0)=0$ and $W^{\prime }\le c+AW$. Multiplication by the integrating factor $e^{-At}$ and integration prove the displayed estimate. For $A=0$ its limiting interpretation is $v(t)\le ct$.