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.

See also the references in Differential inequality.

References

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

Comments

On page 123 of [a1] Chaplygin's theorem is formulated as a problem.

References

[a1]	I.G. Petrovskii, "Ordinary differential equations" , Prentice-Hall (1966) (Translated from Russian)