View source for Bellman-Gronwall inequality

An assertion which deduces from a linear integral inequality (cf. also [[Differential inequality|Differential inequality]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b1201101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b1201102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b1201103.png" /> are real continuous functions on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b1201104.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b1201105.png" /> is integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b1201106.png" />, an inequality for the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b1201107.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b1201108.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b1201109.png" /></td> </tr></table>

To prove this assertion, it suffices to establish (see [[#References|[a1]]]) that the integral in (a1) does not exceed the integral in (a2).

A less general form of the Bellman–Gronwall inequality is as follows: If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011010.png" /> is non-decreasing (or constant, in particular), then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011011.png" /></td> </tr></table>

The Bellman–Gronwall inequality can be used to estimate the solution of a linear non-homogeneous (or non-linear) ordinary differential equation in terms of the initial condition and the non-homogeneity (or non-linearity). This allows one to establish the existence and stability of the solution. This inequality can be also used to justify the averaging method, namely to estimate the difference between the solutions of the initial equation and the averaged one.

There is a generalization of the Bellman–Gronwall inequality to the case of a piecewise-continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011012.png" /> satisfying for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011013.png" /> the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011017.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011018.png" /> are the points of discontinuity of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011019.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011020.png" /> satisfies the estimate (see [[#References|[a2]]]):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120110/b12011021.png" /></td> </tr></table>

This form of the inequality plays the same role in the theory of differential equations with impulse effect, as the usual form of the inequality does in the theory of ordinary differential equations.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Hale,   "Theory of functional differential equations" , Springer  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Samoilenko,   N. Perestyuk,   "Impulsive differential equations" , World Sci.  (1995)  (In Russian)</TD></TR></table>