# 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 vanishes on the segment at most times if the equation possesses this property and when (see ).

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 ).

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

One rich source for obtaining comparison theorems is the Lyapunov comparison principle with a vector function (see ). The idea of the comparison principle is as follows. Let a system of differential equations (1)

and vector functions  be given, where . For any solution of the system (1), the function , , satisfies the equation Therefore, if the inequalities (2) are fulfilled, then on the basis of the properties of the system of differential inequalities (3)

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 ). 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 .