The Gronwall inequality is an inequality which allows explicit bounds to be derived from the implicit information of an integral inequality. Furthermore, it is an important tool for proving existence and inclusion theorems for solutions of differential and integral equations . It is named after Thomas Hakon Grönwall , who proved it in 1919 and described it in a scientific publication.
Given an interval and continuous functions and . The integral inequality also applies
for everyone . Then Gronwall's inequality applies
for everyone .
Note that the function still occurs on both sides in the assumed inequality, but only on the left side in the conclusion, that is, a real estimate for .
Special case
If it increases monotonically , the estimate is simplified to
Especially in the case of constant functions and reads the Gronwall inequality
Applications
Uniqueness set for initial value problems
It is , , and steadily and locally Lipschitz continuous with respect to the second variable. Then the initial value problem has at most one solution .
Linearly bounded differential equations
Be , , , and steadily. There are also functions such that
for everyone . Then every solution is from
on limited.
proof
It applies
The Gronwall inequality implies
and this results in the following estimate against a constant:
literature
Herbert Amann: Ordinary differential equations . 2nd Edition. de Gruyter textbooks, Berlin / New York 1995, ISBN 3-11-014582-0 .
Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (= Graduate Studies in Mathematics . Volume140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( mat.univie.ac.at ).