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.[edit]

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 [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

$$f_i(t,x_1\dots x_n)\ge g_i(t,x_1\dots x_n),$$

$$x_i^0\ge y_i^0,i=1\dots n,$$

$$\frac{\partial f_i}{\partial x_j}\ge 0,i,j=1\dots n,i\ne j,$$

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

$$\begin{array}{}\text{(1)}& \dot{x}=f(t,x),x=(x_1\dots x_n)\end{array}$$

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

$$\begin{array}{}\text{(2)}& \frac{\partial V_j(t,x)}{\partial t}+\sum _{k=1}^n\frac{\partial V_j(t,x)}{\partial x_k}{f}_k(t,x)\le W_j(t,V(t,x)),\end{array}$$

$$j=1\dots m,$$

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

$$\begin{array}{}\text{(3)}& {\dot{v}}_j\le W_j(t,v_1\dots v_m),j=1\dots m,\end{array}$$

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

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

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 [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 $ \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

$$\frac{\partial V_j(t,x)}{\partial t}+\underset{y\in F(t,x)}{sup}\sum _{k=1}^n\frac{\partial V_j(t,x)}{\partial x_k}{y}_k\le W_j(t,V(t,x)).$$

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

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