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An inequality which interconnects the argument, the unknown function and its derivatives, e.g.
 
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>
+
$$ \tag{1 }
 +
y  ^  \prime  ( x)  > f ( x , y ( x) ) ,
 +
$$
  
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.
+
where $  y $
 +
is an unknown function of the argument $  x $.  
 +
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):
 
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>
+
$$ \tag{2 }
 +
y ( x)  < z ( x) \  \textrm{ if }  x _ {1} \leq  x < x _ {0} ,
 +
$$
  
<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>
+
$$
 +
y ( x)  > z ( x) \  \textrm{ if }  x _ {0} < x \leq  x _ {2} ,
 +
$$
  
 
where
 
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>
+
$$
 +
z  ^  \prime  = f ( x , z) ,\  z ( x _ {0} )  = y ( x _ {0} ) ,
 +
$$
  
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
+
on any interval $  [ x _ {1} , x _ {2} ] $
 +
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>
+
$$
 +
y  ^ {(} m) + a _ {1} ( x) y  ^ {(} m- 1) + \dots + a _ {m} ( x) y  > f ( x) .
 +
$$
  
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" />.
+
Here, estimates of the type (2) for solutions satisfying identical initial conditions at $  x = x _ {0} $
 +
are only certainly true on some interval determined by the coefficients $  a _ {1} \dots a _ {m} $.  
 +
E.g., this is the interval $  [ x _ {0} - \pi , x _ {0} + \pi ] $
 +
for $  y  ^ {\prime\prime} + y > f $.
  
 
For a system of differential inequalities
 
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>
+
$$
 +
y _ {i}  ^  \prime  ( x)  > f _ {i} ( x , y _ {1} \dots y _ {n} ),\ \
 +
i = 1 \dots n ,
 +
$$
  
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
+
it has been shown [[#References|[3]]] that if each function $  f _ {i} $
 +
is non-decreasing with respect to the arguments $  y _ {j} $(
 +
for all $  j \neq i $),  
 +
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>
+
$$
 +
y _ {i} ( x)  > z _ {i} ( x) \  \textrm{ if }  x _ {0} < x \leq  x _ {2} ; \ \
 +
i = 1 \dots n ,
 +
$$
  
 
resembling (2), is valid. The development of these considerations leads to the theory of differential inequalities in spaces with a cone.
 
resembling (2), is valid. The development of these considerations leads to the theory of differential inequalities in spaces with a cone.
Line 33: Line 70:
 
A variant of differential inequalities is the requirement that the total derivative of a given function is of constant sign:
 
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>
+
$$
 +
 
 +
\frac{d}{dx}
 +
F ( x , y _ {1} \dots y _ {n} )  \equiv 
 +
\frac{\partial  F }{\partial  x }
 +
+
 +
\frac{\partial  F }{\partial  y _ {1} }
 +
y _ {1}  ^  \prime  + \dots +
 +
\frac{\partial  F }{\partial  y _ {n} }
 +
y _ {n}  ^  \prime  \leq  0 .
 +
$$
  
 
This requirement is used in stability theory.
 
This requirement is used in stability theory.
Line 39: Line 86:
 
A representative of another class is the differential inequality
 
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>
+
$$ \tag{3 }
 +
\max _ {i = 1 \dots n }  | y _ {i}  ^  \prime  - f _ {i} ( x , y _ {1} \dots y _ {n} ) |  \leq  \epsilon
 +
$$
  
(<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.
+
( $  \epsilon > 0 $
 +
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 $  x \rightarrow \infty $,  
 +
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.
+
The theory of boundary value problems was also studied for differential inequalities. The inequality $  \Delta u \geq  0 $,  
 +
where $  \Delta $
 +
is the [[Laplace operator|Laplace operator]], defines subharmonic functions; the differential inequality $  \partial  u / \partial  t - \Delta u \leq  0 $
 +
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====
 
====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>
 
<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====
 
====Comments====
 
More generally one considers functional inequalities and integral inequalities of the form
 
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>
+
$$
 +
f ( t)  \leq  T ( f  ) ( t) ,
 +
$$
  
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" />,
+
where $  T $
 +
is a mapping of some space $  X $
 +
of functions on an interval into itself. Two useful uniqueness theorems in this setting are as follows. Let $  C  ^ {+} [ 0 , a ] $
 +
be the space of non-negative continuous functions on $  [ 0 , a ] $.  
 +
Let $  K ( t) \in L ( 0 , a ) $
 +
be continuous and non-negative. Now, if for 0 \leq  t \leq  a $,
  
<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>
+
$$
 +
f ( t)  \leq  \int\limits _ { 0 } ^ { t }
 +
K ( s) f ( s)  d s ,
 +
$$
  
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
+
then $  f $
 +
is identically zero. Now let $  f \in C  ^ {+} [ 0 , a ] $
 +
be such that $  f ( 0) = 0 $
 +
and $  \lim\limits _ {h \downarrow 0 }  h  ^ {-} 1 f ( h) = 0 $.  
 +
Then if
  
<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/d03221037.png" /></td> </tr></table>
+
$$
 +
f ( t)  \leq  \int\limits _ { 0 } ^ { t }
 +
f ( s) 
 +
\frac{ds}{s}
 +
,
 +
$$
  
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
+
also $  f ( t) \equiv 0 $(
 +
Nagumo's lemma). let $  K \in C  ^ {+} [ a , b ] \cap L ( a , b ) $,  
 +
let $  f , g \in C  ^ {+} [ a , b ] $
 +
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>
+
$$
 +
f ( t)  \leq  g ( t) +
 +
\int\limits _ { a } ^ { t }  K ( s)
 +
f ( s)  d s .
 +
$$
  
 
Then
 
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>
+
$$
 +
f ( t)  \leq  g ( t) +
 +
\int\limits _ { a } ^ { t }
 +
K ( s)  \mathop{\rm exp} \left [ \int\limits
 +
_ { s } ^ { t }  K ( u)  d u \right ]
 +
g ( s)  d s .
 +
$$
  
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" />,
+
The last result is known as [[Gronwall lemma|Gronwall's lemma]] (Gronwall's inequality). The case $  K = \textrm{ constant } $
 +
is important. Another variant of Gronwall's lemma is as follows. Let $  f , K \in C  ^ {+} [ a , b ] $
 +
and for some constant $  c $,
  
<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>
+
$$
 +
f ( t)  \leq  c + \int\limits _ { a } ^ { t }
 +
K ( s) f ( s)  d s,
 +
$$
  
 
then
 
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>
+
$$
 +
f ( t)  \leq  c  \mathop{\rm exp} \left ( \int\limits _ { a } ^ { t }
 +
K ( s)  d s \right ) .
 +
$$
  
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" />.
+
This last result is useful in discussing, e.g., the stability of (constantly acting) perturbations $  \dot{x} = A x + B ( t) x $(
 +
with $  A $
 +
constant) in terms of the stability of $  \dot{x} = A x $.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:35, 5 June 2020


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

$$ \tag{1 } y ^ \prime ( x) > f ( x , y ( x) ) , $$

where $ y $ is an unknown function of the argument $ x $. 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 [1] are valid for any solution of (1):

$$ \tag{2 } y ( x) < z ( x) \ \textrm{ if } x _ {1} \leq x < x _ {0} , $$

$$ y ( x) > z ( x) \ \textrm{ if } x _ {0} < x \leq x _ {2} , $$

where

$$ z ^ \prime = f ( x , z) ,\ z ( x _ {0} ) = y ( x _ {0} ) , $$

on any interval $ [ x _ {1} , x _ {2} ] $ 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 [2] is also valid for a differential inequality (Chaplygin's inequality) of the type

$$ y ^ {(} m) + a _ {1} ( x) y ^ {(} m- 1) + \dots + a _ {m} ( x) y > f ( x) . $$

Here, estimates of the type (2) for solutions satisfying identical initial conditions at $ x = x _ {0} $ are only certainly true on some interval determined by the coefficients $ a _ {1} \dots a _ {m} $. E.g., this is the interval $ [ x _ {0} - \pi , x _ {0} + \pi ] $ for $ y ^ {\prime\prime} + y > f $.

For a system of differential inequalities

$$ y _ {i} ^ \prime ( x) > f _ {i} ( x , y _ {1} \dots y _ {n} ),\ \ i = 1 \dots n , $$

it has been shown [3] that if each function $ f _ {i} $ is non-decreasing with respect to the arguments $ y _ {j} $( for all $ j \neq i $), the estimate

$$ y _ {i} ( x) > z _ {i} ( x) \ \textrm{ if } x _ {0} < x \leq x _ {2} ; \ \ i = 1 \dots n , $$

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:

$$ \frac{d}{dx} F ( x , y _ {1} \dots y _ {n} ) \equiv \frac{\partial F }{\partial x } + \frac{\partial F }{\partial y _ {1} } y _ {1} ^ \prime + \dots + \frac{\partial F }{\partial y _ {n} } y _ {n} ^ \prime \leq 0 . $$

This requirement is used in stability theory.

A representative of another class is the differential inequality

$$ \tag{3 } \max _ {i = 1 \dots n } | y _ {i} ^ \prime - f _ {i} ( x , y _ {1} \dots y _ {n} ) | \leq \epsilon $$

( $ \epsilon > 0 $ is given), which was first studied in the context of the general idea of an approximate description of a real problem by differential equations [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 $ x \rightarrow \infty $, 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 $ \Delta u \geq 0 $, where $ \Delta $ is the Laplace operator, defines subharmonic functions; the differential inequality $ \partial u / \partial t - \Delta u \leq 0 $ 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

[1] 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
[2] S.A. Chaplygin, "Fundamentals of a new method of approximate integration of differential equations" , Moscow (1919) (In Russian)
[3] 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
[4] P. Bohl, "Ueber Differentialungleichungen" J. Reine Angew. Math. , 144 (1914) pp. 284–313
[5] A. Haar, "Ueber Eindeutigkeit und analyzität der Lösungen partieller Differentialgleichungen" , Atti congress. internaz. mathematici (Bologna, 1928) , 3 , Zanichelli (1930) pp. 5–10
[6] W. Walter, "Differential- und Intergralgleichungen und ihre Anwendung bei Abschätzungs- und Eindeutigkeitsproblemen" , Springer (1964)
[7] J. Szarski, "Differential inequalities" , PWN (1965)
[8] V. Lakshmikantham, S. Leela, "Differential and integral inequalities" , 1–2 , Acad. Press (1969)

Comments

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

$$ f ( t) \leq T ( f ) ( t) , $$

where $ T $ is a mapping of some space $ X $ of functions on an interval into itself. Two useful uniqueness theorems in this setting are as follows. Let $ C ^ {+} [ 0 , a ] $ be the space of non-negative continuous functions on $ [ 0 , a ] $. Let $ K ( t) \in L ( 0 , a ) $ be continuous and non-negative. Now, if for $ 0 \leq t \leq a $,

$$ f ( t) \leq \int\limits _ { 0 } ^ { t } K ( s) f ( s) d s , $$

then $ f $ is identically zero. Now let $ f \in C ^ {+} [ 0 , a ] $ be such that $ f ( 0) = 0 $ and $ \lim\limits _ {h \downarrow 0 } h ^ {-} 1 f ( h) = 0 $. Then if

$$ f ( t) \leq \int\limits _ { 0 } ^ { t } f ( s) \frac{ds}{s} , $$

also $ f ( t) \equiv 0 $( Nagumo's lemma). let $ K \in C ^ {+} [ a , b ] \cap L ( a , b ) $, let $ f , g \in C ^ {+} [ a , b ] $ and suppose

$$ f ( t) \leq g ( t) + \int\limits _ { a } ^ { t } K ( s) f ( s) d s . $$

Then

$$ f ( t) \leq g ( t) + \int\limits _ { a } ^ { t } K ( s) \mathop{\rm exp} \left [ \int\limits _ { s } ^ { t } K ( u) d u \right ] g ( s) d s . $$

The last result is known as Gronwall's lemma (Gronwall's inequality). The case $ K = \textrm{ constant } $ is important. Another variant of Gronwall's lemma is as follows. Let $ f , K \in C ^ {+} [ a , b ] $ and for some constant $ c $,

$$ f ( t) \leq c + \int\limits _ { a } ^ { t } K ( s) f ( s) d s, $$

then

$$ f ( t) \leq c \mathop{\rm exp} \left ( \int\limits _ { a } ^ { t } K ( s) d s \right ) . $$

This last result is useful in discussing, e.g., the stability of (constantly acting) perturbations $ \dot{x} = A x + B ( t) x $( with $ A $ constant) in terms of the stability of $ \dot{x} = A x $.

References

[a1] D.W. Jordan, P. Smith, "Nonlinear ordinary differential equations" , Clarendon Press (1977)
[a2] E. Hille, "Ordinary differential equations in the complex plane" , Wiley (Interscience) (1976)
How to Cite This Entry:
Differential inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_inequality&oldid=46691
This article was adapted from an original article by A.D. Myshkis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article