Comparison theorem

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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

vanishes on the segment at most times if the equation

possesses this property and when (see [1]).

2) A differential inequality: The solution of the problem

is component-wise non-negative when if the solution of the problem

possesses this property and if the inequalities

are fulfilled (see [2]).

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

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


and vector functions

be given, where . For any solution of the system (1), the function , , satisfies the equation

Therefore, if the inequalities


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


something can be said about the behaviour of the functions that are solutions of the system (3). Knowing the behaviour of the functions on every solution 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 and satisfy the inequalities (2) and for any , , let a number exist such that

for all , . Furthermore, let every solution of the system of inequalities (3) be defined on . Every solution of the system (1) is then also defined on .

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 [4][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 equations, ordinary, with distributed arguments; Differential inclusion). In particular, for a differential inclusion , , where is a set in dependent on , the role of the inequalities (2) is played by the inequalities

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


[1] C. Sturm, J. Math. Pures Appl. , 1 (1836) pp. 106–186
[2] 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
[3] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)
[4] R.E. Bellman, "Vector Lyapunov functions" J. Soc. Industr. Appl. Math. Ser. A Control. , 1 : 1 (1962) pp. 32–34
[5a] 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
[5b] 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
[5c] 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
[5d] 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
[6] A.A. Martynyuk, "Stability of motion of complex systems" , Kiev (1975) (In Russian)
[7] A.A. Martynyuk, R. Gutovski, "Integral inequalities and stability of motion" , Kiev (1979) (In Russian)
[8] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1–2 , Akad. Verlagsgesell. (1943–1944)



[a1] C.A. Swanson, "Comparison and oscillation theory of linear differential equations" , Acad. Press (1968)
[a2] G.S. Ladde, V. Lakshmikantham, "Random differential inequalities" , Acad. Press (1980)
How to Cite This Entry:
Comparison theorem. E.L. Tonkov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098