In mathematical theory of differential equations the Chaplygin's theorem (Chaplygin's method) states about existence and uniqueness of the solution to an initial value problem for the first order explicit ordinary differential equation. This theorem was stated by Sergey Chaplygin. [1] It is one of many comparison theorems.
Consider an initial value problem: differential equation
in ,
with an initial condition
.
For the initial value problem described above the upper boundary solution and the lower boundary solution are the functions and respectively, both of which are smooth in and continous in , such as the following inequalities are true:
Given the aforementioned initial value problem and respective upper boundary solution and lower boundary solution for . If the right part
holds,
then in there exists one and only one solution for the given initial value problem and moreover for all
.
Inside inequalities within both of definitions of the upper boundary solution and the lower boundary solution signs of inequalities (all at once) can be altered to unstrict. As a result, inequalities sings at Chaplygin's theorem concusion would change to unstrict by and respectively. In particular, any of , could be chosen.
If is already known to be an existent soltion for the initial value problem in , the Lipschitz condition requirement can be omitted entirely for proving the resulting inequality. There exists applications for this method while researching whether the solution is stable or not ( [2] , pp. 7–9). This is often called "Differential inequality method" in literature [4] [5] and, for example, Grönwall's inequality can be proven using this technique. [5]
Chaplygin's theorem answers the question about existence and uniqueness of the solution in and the constant from the Lipschitz condition is, generally speaking, dependent on : . If for both functions and retain their smoothness and for a set is bounded, the theorem holds for all .
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