# Chaplygin theorem

on differential inequalities

If in the differential inequality

$$\tag{* } L [ y] \equiv y ^ {(} m) + a _ {1} ( x) y ^ {( m - 1 ) } + \dots + a _ {m} ( x) y > f ( x)$$

all the $a _ {i}$ and $f$ are summable on $[ x _ {0} , x _ {1} ]$, then there exists an $x ^ {*} \in ( x _ {0} , x _ {1} ]$, independent of $f$, such that $y ( x) > z ( x)$, $x _ {0} < x \leq x ^ {*}$, where

$$L [ z] = f ( x),$$

$$z ( x _ {0} ) = y ( x _ {0} ) \dots z ^ {( n - 1 ) } ( x _ {0} ) = y ^ {( n - 1 ) } ( x _ {0} ) .$$

Here

$$x ^ {*} = \max \{ {x \in [ x _ {0} , x _ {1} ] } : {\forall \xi \in [ x _ {0} , x ] ,\ \forall s \in [ \xi , x ] \Rightarrow G ( s ; \xi ) \geq 0 } \} ,$$

where $G ( x ; \xi )$ is the corresponding Cauchy function, i.e. the solution of the equation $L [ G] = 0$, $\xi \leq x \leq x _ {1}$, that satisfies the initial conditions

$$\left . G \right | _ {x = \xi } = \dots = \ \left . G _ {x} ^ {( m - 2 ) } \right | _ {x = \xi } = 0 ,\ \ \left . G _ {x} ^ {( m - 1 ) } \right | _ {x = \xi } = 1 .$$

Thus, for $m = 1$, and also for the inequality $y ^ {\prime\prime} - y > f ( x)$, one obtains $x ^ {*} = x _ {1}$, while for the inequality $y ^ {\prime\prime} + y > f ( x)$ one obtains

$$x ^ {*} = \min \{ x _ {1} , x _ {0} + \pi \} .$$

Analogous statements hold: for weak inequalities; for the comparison of $y ^ {(} i) ( x)$ with $z ^ {(} i) ( x)$, $i = 1 \dots m - 1$; for initial conditions of the form

$$y ( x _ {0} ) \geq z ( x _ {0} ) \dots y ^ {( n - 1 ) } ( x _ {0} ) \geq z ^ {( n - 1 ) } ( x _ {0} ) ;$$

and for solutions of the inequality (*) with $x < x _ {0}$.

The theorem was obtained by S.A. Chaplygin in 1919.

#### References

 [1] Ya.D. Mamedov, S. Ashirov, S. Atdaev, "Theorems on inequalities" , Ashkhabad (1980) (In Russian)