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

$$\dot{y} dot + p ( t) y = 0,\ \ p ( \cdot ) \in C [ t _ {0} , t _ {1} ] ,$$

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].

References

 [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)