View source for Comparison theorem

''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

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vanishes on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c0236402.png" /> at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c0236403.png" /> times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c0236404.png" /> if the equation

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possesses this property and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c0236406.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c0236407.png" /> (see [[#References|[1]]]).

2) A differential inequality: The solution of the problem

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is component-wise non-negative when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c0236409.png" /> if the solution of the problem

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possesses this property and if the inequalities

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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

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and vector functions

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be given, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364017.png" />. For any solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364018.png" /> of the system (1), the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364020.png" />, satisfies the equation

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Therefore, if the inequalities

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

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something can be said about the behaviour of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364025.png" /> that are solutions of the system (3). Knowing the behaviour of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364026.png" /> on every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364027.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364029.png" /> satisfy the inequalities (2) and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364031.png" />, let a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364032.png" /> exist such that

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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364035.png" />. Furthermore, let every solution of the system of inequalities (3) be defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364036.png" />. Every solution of the system (1) is then also defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364037.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364040.png" /> is a set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364041.png" /> dependent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023640/c02364042.png" />, the role of the inequalities (2) is played by the inequalities

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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>