Gronwall's inequality

The Gronwall inequality is an inequality that allows, from the implicit information of an integral inequality to derive explicit bounds. Furthermore, it is an important aid to the proof of existence and Einschließungssätzen for solutions of differential and integral equations. It is named after Thomas Hakon Grönwall, who proved in 1919 and described in a scientific publication.

Formulation

Given an interval and continuous functions and. Continues to apply the integral inequality

For everyone. Then the Gronwall inequality holds

For everyone.

Note that the function in the assumed inequality still exists on both sides, in the conclusion but only on the left side, that is, you get a real estimate for.

Special case

Is monotonically increasing as the estimate simplifies to

Especially in the case of constant functions and is the Gronwall inequality

Applications

Uniqueness theorem for initial value problems

It was, and ever, and locally Lipschitz continuous with respect to the second variable. Then the initial value problem has at most one solution.

Linear limited differential equations

For everyone. Then every solution is of

Limited to.

Evidence

It is

The Gronwall inequality implies

And this results in the following estimate for a constant: