View source for Differential inequality

An inequality which interconnects the argument, the unknown function and its derivatives, e.g.

<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/d/d032/d032210/d0322101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d0322102.png" /> is an unknown function of the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d0322103.png" />. The principal problem in the theory of differential inequalities is to describe, starting from a known differential inequality and additional (initial or boundary) conditions, all its solutions.

Differential inequalities obtained from differential equations by replacing the equality sign by the inequality sign — which is equivalent to adding some non-specified function of definite sign to one of the sides of the equation — form a large class. A comparison of the solutions of such inequalities with the solutions of the corresponding differential equations is of interest. Thus, the following estimates [[#References|[1]]] are valid for any solution of (1):

<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/d/d032/d032210/d0322104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</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/d/d032/d032210/d0322105.png" /></td> </tr></table>

where

<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/d/d032/d032210/d0322106.png" /></td> </tr></table>

on any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d0322107.png" /> of existence of both solutions. This simple statement is extensively employed in estimating the solutions of differential equations (by passing to the respective differential inequality with a particular solution which is readily found), the domain of extendability of solutions, the difference between two solutions, in deriving conditions for the uniqueness of a solution, etc. A similar theorem [[#References|[2]]] is also valid for a differential inequality (Chaplygin's inequality) of the type

<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/d/d032/d032210/d0322108.png" /></td> </tr></table>

Here, estimates of the type (2) for solutions satisfying identical initial conditions at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d0322109.png" /> are only certainly true on some interval determined by the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221010.png" />. E.g., this is the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221012.png" />.

For a system of differential inequalities

<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/d/d032/d032210/d03221013.png" /></td> </tr></table>

it has been shown [[#References|[3]]] that if each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221014.png" /> is non-decreasing with respect to the arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221015.png" /> (for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221016.png" />), the estimate

<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/d/d032/d032210/d03221017.png" /></td> </tr></table>

resembling (2), is valid. The development of these considerations leads to the theory of differential inequalities in spaces with a cone.

A variant of differential inequalities is the requirement that the total derivative of a given function is of constant sign:

<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/d/d032/d032210/d03221018.png" /></td> </tr></table>

This requirement is used in stability theory.

A representative of another class is the 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/d/d032/d032210/d03221019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221020.png" /> is given), which was first studied in the context of the general idea of an approximate description of a real problem by differential equations [[#References|[4]]]. Here the description of the integral funnel, i.e. the set of all points of all solutions which satisfy the given initial conditions, in particular, the behaviour of the funnel as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221021.png" />, is of interest. A natural generalization of the differential inequality (3) is a differential equation in contingencies, specified by a field of cones, which generalizes the concept of a field of directions.

The theory of boundary value problems was also studied for differential inequalities. The inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221023.png" /> is the [[Laplace operator|Laplace operator]], defines subharmonic functions; the differential inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221024.png" /> defines subparabolic functions. Studies were also made of differential inequalities of a more general type (in both the above classes) with partial derivatives for differential operators of various types.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Petrovitsch,   "Sur une manière d'étendre le théorème de la moyence aux équations différentielles du premier ordre"  ''Math. Ann.'' , '''54''' :  3  (1901)  pp. 417–436</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.A. Chaplygin,   "Fundamentals of a new method of approximate integration of differential equations" , Moscow  (1919)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T. Wazewski,   "Systèmes des équations et des inégualités différentielles ordinaires aux deuxièmes membres monotones et leurs applications"  ''Ann. Soc. Polon. Math.'' , '''23'''  (1950)  pp. 112–166</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Bohl,   "Ueber Differentialungleichungen"  ''J. Reine Angew. Math.'' , '''144'''  (1914)  pp. 284–313</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Haar,   "Ueber Eindeutigkeit und analyzität der Lösungen partieller Differentialgleichungen" , ''Atti congress. internaz. mathematici (Bologna, 1928)'' , '''3''' , Zanichelli  (1930)  pp. 5–10</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  W. Walter,   "Differential- und Intergralgleichungen und ihre Anwendung bei Abschätzungs- und Eindeutigkeitsproblemen" , Springer  (1964)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J. Szarski,   "Differential inequalities" , PWN  (1965)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V. Lakshmikantham,   S. Leela,   "Differential and integral inequalities" , '''1–2''' , Acad. Press  (1969)</TD></TR></table>



====Comments====
More generally one considers functional inequalities and integral inequalities of the form

<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/d/d032/d032210/d03221025.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221026.png" /> is a mapping of some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221027.png" /> of functions on an interval into itself. Two useful uniqueness theorems in this setting are as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221028.png" /> be the space of non-negative continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221029.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221030.png" /> be continuous and non-negative. Now, if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221031.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/d/d032/d032210/d03221032.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221033.png" /> is identically zero. Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221034.png" /> be such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221036.png" />. Then if

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also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221038.png" /> (Nagumo's lemma). let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221039.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221040.png" /> and suppose

<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/d/d032/d032210/d03221041.png" /></td> </tr></table>

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/d/d032/d032210/d03221042.png" /></td> </tr></table>

The last result is known as [[Gronwall lemma|Gronwall's lemma]] (Gronwall's inequality). The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221043.png" /> is important. Another variant of Gronwall's lemma is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221044.png" /> and for some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221045.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/d/d032/d032210/d03221046.png" /></td> </tr></table>

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/d/d032/d032210/d03221047.png" /></td> </tr></table>

This last result is useful in discussing, e.g., the stability of (constantly acting) perturbations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221048.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221049.png" /> constant) in terms of the stability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032210/d03221050.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.W. Jordan,   P. Smith,   "Nonlinear ordinary differential equations" , Clarendon Press  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Hille,   "Ordinary differential equations in the complex plane" , Wiley (Interscience)  (1976)</TD></TR></table>