Grönwall's inequality (also called Grönwall's lemma or the Grönwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation. There are two forms of the lemma, a differential form and an integral form.
then u is bounded by the solution of the corresponding differential equation v ′(t) = β(t) v(t):
Note that v satisfies
Thus the derivative of the function u(t)/v(t) is non-positive and the function is bounded above by its value at the initial point a of the interval I
Grönwall's inequality is an important tool to obtain various estimates in the theory of ordinary and stochastic differential equations. In particular, it provides a comparison theorem that can be used to prove uniqueness of a solution to the initial value problem
Let I denote an interval of the real line of the form [a, ∞) or [a, b] or [a, b) with a < b. Let β and u be real-valued continuous functions defined on I. If u is differentiable in the interior Io of I (the interval I without the end points a and possibly b) and satisfies the differential inequality
then u is bounded by the solution of the corresponding differential equation v ′(t) = β(t) v(t):
for all t ∈ I.
Note : There are no assumptions on the signs of the functions β and u
Proof : -
Define the function
Note that v satisfies
➡ with v(a) = 1 and v(t) > 0 for all t ∈ I. By the quotient rule
Thus the derivative of the function u(t)/v(t) is non-positive and the function is bounded above by its value at the initial point a of the interval I
which is Grönwall's inequality.
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