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

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

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

$$ \frac{\partial f _ {i} }{\partial x _ {j} } \geq 0,\ \ i, j = 1 \dots n,\ i \neq j, $$

are fulfilled (see [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

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

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 [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

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

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

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