Gronwall's inequality

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.

formulation

Given an interval and continuous functions and . The integral inequality also applies ${\displaystyle \ I:=[a,b]}$ ${\displaystyle u,\alpha :I\rightarrow \mathbb {R} }$ ${\displaystyle \beta :I\rightarrow [0,\infty )}$

${\displaystyle u(t)\leq \alpha (t)+\int _{a}^{t}\beta (s)u(s){\rm {d}}s}$

for everyone . Then Gronwall's inequality applies ${\displaystyle t\in I}$

${\displaystyle u(t)\leq \alpha (t)+\int _{a}^{t}\alpha (s)\beta (s)e^{\int _{s}^{t}\beta (\sigma ){\rm {d}}\sigma }{\rm {d}}s}$

for everyone . ${\displaystyle t\in I}$

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 . ${\displaystyle u}$ ${\displaystyle u}$

Special case

If it increases monotonically , the estimate is simplified to ${\displaystyle \alpha }$

${\displaystyle u(t)\leq \alpha (t)e^{\int _{a}^{t}\beta (s){\rm {d}}s}\ .}$

Especially in the case of constant functions and reads the Gronwall inequality ${\displaystyle \alpha \equiv A}$ ${\displaystyle \beta \equiv B\geq 0}$

${\displaystyle u(t)\leq A+\int _{a}^{t}ABe^{B(t-s)}{\rm {d}}s=Ae^{B(t-a)}\ .}$

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 . ${\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}$ ${\displaystyle G\subset \mathbb {R} \times \mathbb {K} ^{n}}$ ${\displaystyle (a,y_{0})\in G}$ ${\displaystyle F=F(x,y):G\rightarrow \mathbb {K} ^{n}}$ ${\displaystyle \ y'=F(x,y),y(a)=y_{0}}$ ${\displaystyle y\in C^{1}([a,b);\mathbb {K} ^{n})}$

Linearly bounded differential equations

Be , , , and steadily. There are also functions such that ${\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}$ ${\displaystyle G\subset [a,b)\times \mathbb {K} ^{n}}$ ${\displaystyle (a,y_{0})\in G}$ ${\displaystyle b<\infty }$ ${\displaystyle F=F(x,y):G\rightarrow \mathbb {K} ^{n}}$ ${\displaystyle \alpha ,\beta \in C([a,b);[0,\infty ))\cap L^{1}([a,b))}$

${\displaystyle \|F(x,y)\|\leq \alpha (x)+\beta (x)\|y\|}$

for everyone . Then every solution is from ${\displaystyle (x,y)\in G}$ ${\displaystyle y}$

${\displaystyle y'=F(x,y)\ ,\ y(a)=y_{0}}$

on limited. ${\displaystyle [a,b)}$

proof

It applies

${\displaystyle \|y(x)\|\leq \|y_{0}\|+\int _{a}^{x}\|F(s,y(s))\|{\rm {d}}s\leq \|y_{0}\|+\int _{a}^{x}\alpha (s){\rm {d}}s+\int _{a}^{x}\beta (s)\|y(s)\|{\rm {d}}s\ .}$

The Gronwall inequality implies

${\displaystyle \|y(x)\|\leq \|y_{0}\|+\int _{a}^{x}\alpha (s){\rm {d}}s+\int _{a}^{x}\left(\|y_{0}\|+\int _{a}^{s}\alpha (\sigma ){\rm {d}}\sigma \right)\beta (s)e^{\int _{s}^{x}\beta (\sigma ){\rm {d}}\sigma }{\rm {d}}s\ ,}$

and this results in the following estimate against a constant:

${\displaystyle \|y(x)\|\leq \|y_{0}\|+\int _{a}^{b}\alpha (s){\rm {d}}s+\int _{a}^{b}\left(\|y_{0}\|+\int _{a}^{b}\alpha (\sigma ){\rm {d}}\sigma \right)\beta (s)e^{\int _{a}^{b}\beta (\sigma ){\rm {d}}\sigma }{\rm {d}}s\ .}$

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 . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( mat.univie.ac.at ).

Web links

Wikibooks: Proof of Gronwall's Inequality  - Learning and Teaching Materials

Wikibooks: Proof of the uniqueness theorem for locally Lipschitz continuous differential equation  - learning and teaching materials

• Gronwall's lemma . In: PlanetMath . (English)