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''in the theory of differential equations''

A theorem that asserts the presence of a specific property of solutions of a differential equation (or system of differential equations) under the assumption that an auxiliary equation or inequality (system of differential equations or inequalities) possesses a certain property.

==Examples of comparison theorems.==

1) Sturm's theorem: Any non-trivial solution of the equation

$$ 
\dot{y} dot + p ( t) y  =  0,\ \ 
p ( \cdot )  \in  C [ t _ {0} , t _ {1} ] ,
$$

vanishes on the segment  $  [ t _ {0} , t _ {1} ] $
at most  $  m $
times  $  ( m \geq  1) $
if the equation

$$ 
\dot{z} dot + q ( t) z  =  0,\ \ 
q ( \cdot )  \in  C [ t _ {0} , t _ {1} ] ,
$$

possesses this property and  $  q ( t) \geq  p ( t) $
when  $  t _ {0} \leq  t \leq  t _ {1} $(
see [[#References|[1]]]).

2) A differential inequality: The solution of the problem

$$ 
\dot{x} _ {i}  = \ 
f _ {i} ( t, x _ {1} \dots x _ {n} ),\ \ 
x _ {i} ( t _ {0} )  = \ 
x _ {i}  ^ {0} ,\ \ 
i = 1 \dots n ,
$$

is component-wise non-negative when  $  t \geq  t _ {0} $
if the solution of the problem

$$ 
\dot{y} _ {i}  = \ 
g _ {i} ( t, y _ {1} \dots y _ {n} ),\ \ 
y _ {i} ( t _ {0} )  =  y _ {i}  ^ {0} ,\ \ 
i = 1 \dots n
$$

possesses this property and if the inequalities

$$ 
f _ {i} ( t, x _ {1} \dots x _ {n} )  \geq  \ 
g _ {i} ( t, x _ {1} \dots x _ {n} ),\ \ 
$$

$$ 
x _ {i}  ^ {0}  \geq   y _ {i}  ^ {0} ,\  i = 1 \dots n,
$$

$$ 

\frac{\partial  f _ {i} }{\partial  x _ {j} }

 \geq   0,\ \ 
i, j = 1 \dots n,\  i \neq j,
$$

are fulfilled (see [[#References|[2]]]).

For other examples of comparison theorems, including the Chaplygin theorem, see [[Differential inequality|Differential inequality]]. For comparison theorems for partial differential equations see, for example, [[#References|[3]]].

One rich source for obtaining comparison theorems is the Lyapunov comparison principle with a vector function (see [[#References|[4]]]–[[#References|[7]]]). The idea of the comparison principle is as follows. Let a system of differential equations

$$ \tag{1 }
\dot{x}  =  f ( t, x),\ \ 
x = ( x _ {1} \dots x _ {n} )
$$

and vector functions

$$ 
V ( t, x)  =  ( V _ {1} ( t, x) \dots V _ {m} ( t, x)),
$$

$$ 
W ( t, v)  =  ( W _ {1} ( t, v) \dots W _ {m} ( t, v))
$$

be given, where  $  v = ( v _ {1} \dots v _ {m} ) $. 
For any solution  $  x ( t) $
of the system (1), the function  $  v _ {j} ( t) = V _ {j} ( t, x ( t)) $, 
$  j = 1 \dots m $, 
satisfies the equation

$$ 
\dot{v} _ {j} ( t)  = \ 

\frac{\partial  V _ {j} ( t, x ( t)) }{\partial  t }
 +
\sum _ {k = 1 } ^ { n } 

\frac{\partial  V _ {j} ( t, x ( t)) }{\partial  x _ {k} }

f _ {k} ( t, x ( t)).
$$

Therefore, if the inequalities

$$ \tag{2 }

\frac{\partial  V _ {j} ( t, x) }{\partial  t }
 +
\sum _ {k = 1 } ^ { n } 

\frac{\partial  V _ {j} ( t, x) }{\partial  x _ {k} }

f _ {k} ( t, x)  \leq  \ 
W _ {j} ( t, V ( t, x)),
$$

$$ 
j  =  1 \dots m,
$$

are fulfilled, then on the basis of the properties of the system of differential inequalities

$$ \tag{3 }
\dot{v} _ {j}  \leq  \ 
W _ {j} ( t, v _ {1} \dots v _ {m} ),\ \ 
j = 1 \dots m,
$$

something can be said about the behaviour of the functions  $  V _ {j} ( t, x ( t)) $
that are solutions of the system (3). Knowing the behaviour of the functions  $  V _ {j} ( t, x) $
on every solution  $  x ( t) $
of the system (1), in turn, enables one to state assertions on the properties of the solutions of the system (1).

For example, let the vector functions  $  V ( t, x) $
and  $  W ( t, v) $
satisfy the inequalities (2) and for any  $  t _ {1} \geq  t _ {0} $, 
$  \gamma > 0 $, 
let a number  $  M > 0 $
exist such that

$$ 
\sum _ {j = 1 } ^ { m } 
| V _ {j} ( t, x) |
 \geq   M
$$

for all  $  t \in [ t _ {0} , t _ {1} ] $, 
$  \| x \| \geq  \gamma $. 
Furthermore, let every solution of the system of inequalities (3) be defined on  $  [ t, \infty ) $. 
Every solution of the system (1) is then also defined on  $  [ t, \infty ) $.

A large number of interesting statements have been obtained on the basis of the comparison principle in the theory of the stability of motion (see [[#References|[4]]]–[[#References|[6]]]). The Lyapunov comparison principle with a vector function is successfully used for abstract differential equations, differential equations with distributed argument and differential inclusions (cf. [[Differential equation, abstract|Differential equation, abstract]]; [[Differential equations, ordinary, with distributed arguments|Differential equations, ordinary, with distributed arguments]]; [[Differential inclusion|Differential inclusion]]). In particular, for a differential inclusion  $  \dot{x} \in F ( t, x) $, 
$  x \in \mathbf R  ^ {n} $, 
where  $  F ( t, x) $
is a set in  $  \mathbf R  ^ {n} $
dependent on  $  ( t, x) \in \mathbf R  ^ {1} \times \mathbf R  ^ {n} $, 
the role of the inequalities (2) is played by the inequalities

$$ 

\frac{\partial  V _ {j} ( t, x) }{\partial  t }
 +
\sup _ {y \in F ( t, x) } \ 
\sum _ {k = 1 } ^ { n } 

\frac{\partial  V _ {j} ( t, x) }{\partial  x _ {k} }

y _ {k}  \leq  \ 
W _ {j} ( t, V ( t, x)).
$$

A large number of comparison theorems are given in [[#References|[8]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Sturm,   ''J. Math. Pures Appl.'' , '''1'''  (1836)  pp. 106–186</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T. Waźewski,   "Systèmes des équations et des inégalités différentielles ordinaires aux deuxième members monotones et leurs applications"  ''Ann. Soc. Polon. Math.'' , '''23'''  (1950)  pp. 112–166</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.E. Bellman,   "Vector Lyapunov functions"  ''J. Soc. Industr. Appl. Math. Ser. A Control.'' , '''1''' :  1  (1962)  pp. 32–34</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top">  V.M. Matrosov,   "The comparison principle with a Lyapunov vector-function I"  ''Differential Equations'' , '''4''' :  8  (1968)  pp. 710–717  ''Differentsial'nye Uravneniya'' , '''4''' :  8  (1968)  pp. 1374–1386</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top">  V.M. Matrosov,   "Principle of comparison with the Lyapunov vector-functions II"  ''Differential Equations'' , '''4''' :  10  (1968)  pp. 893–900  ''Differentsial'nye Uravneniya'' , '''4''' :  10  (1968)  pp. 1739–1752</TD></TR><TR><TD valign="top">[5c]</TD> <TD valign="top">  V.M. Matrosov,   "Comparison principle with vector-valued Lyapunov functions III"  ''Differential Equations'' , '''5''' :  7  (1969)  pp. 853–864  ''Differentsial'nye Uravneniya'' , '''5''' :  7  (1969)  pp. 1171–1185</TD></TR><TR><TD valign="top">[5d]</TD> <TD valign="top">  V.M. Matrosov,   "The principle of comparison with a Lyapunov vector-function IV"  ''Differential Equations'' , '''5''' :  12  (1969)  pp. 1596–1607  ''Differentsial'nye Uravneniya'' , '''5''' :  12  (1969)  pp. 2129–2143</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.A. Martynyuk,   "Stability of motion of complex systems" , Kiev  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.A. Martynyuk,   R. Gutovski,   "Integral inequalities and stability of motion" , Kiev  (1979)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1–2''' , Akad. Verlagsgesell.  (1943–1944)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.A. Swanson,   "Comparison and oscillation theory of linear differential equations" , Acad. Press  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.S. Ladde,   V. Lakshmikantham,   "Random differential inequalities" , Acad. Press  (1980)</TD></TR></table>