Gronwall's inequality

Gronwall's inequality is a useful ODE inequality that controls the growth of a non-negative quantity $Q(t)$ in terms an initial bound, provided that the quantity obeys a linear feedback relation. For instance, given an integral bound of the form

$Q(t)\leq A+\int _{0}^{t}B(t')Q(t')\ dt'$

for all 't, some constant A>0 and some non-negative locally integrable function B, we can conclude that

$Q(t)\leq A\exp(\int _{0}^{t}B(t')\ dt').$

A variant of this inequality is that if Q is continuously differentiable and obeys the differential inequality

$Q'(t)\leq B(t)Q(t)$

for all t, where B is a locally integrable function which may change in sign, then

$Q(t)\leq Q(0)\exp(\int _{0}^{t}B(t')\ dt').$

In nonlinear applications, one often cannot apply Gronwall's inequality directly due to additional nonlinear terms on the right-hand side. However in many cases such terms can be removed by a continuity argument, allowing Gronwall's inequality to be invoked.

Use of Gronwall's inequality typically leads to bounds which grow exponentially in time. Often the true rate of growth is polynomial or smaller, but this tends to require more subtle and specialized tools than Gronwall's inequality to achieve.