Chaplygin's theorem

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.

Important definitions

Consider an initial value problem: differential equation

${\displaystyle y'\left(t\right)=f\left(t,y\left(t\right)\right)}$ in ${\displaystyle t\in \left[t_{0};\alpha \right]}$, ${\displaystyle \alpha >t_{0}}$

with an initial condition

${\displaystyle y\left(t_{0}\right)=y_{0}}$.

For the initial value problem described above the upper boundary solution and the lower boundary solution are the functions ${\displaystyle {\overline {z}}\left(t\right)}$ and ${\displaystyle {\underline {z}}\left(t\right)}$ respectively, both of which are smooth in ${\displaystyle t\in \left(t_{0};\alpha \right]}$ and continuous in ${\displaystyle t\in \left[t_{0};\alpha \right]}$, such as the following inequalities are true:

1. ${\displaystyle {\underline {z}}\left(t_{0}\right)<y\left(t_{0}\right)<{\overline {z}}\left(t_{0}\right)}$;
2. ${\displaystyle {\underline {z}}'\left(t\right)<f(t,{\underline {z}}\left(t\right))}$ and ${\displaystyle {\overline {z}}\ '\left(t\right)>f(t,{\overline {z}}\left(t\right))}$ for ${\displaystyle t\in \left(t_{0};\alpha \right]}$.

Statement

Source:^[2][3]

Given the aforementioned initial value problem and respective upper boundary solution ${\displaystyle {\overline {z}}\left(t\right)}$ and lower boundary solution ${\displaystyle {\underline {z}}\left(t\right)}$ for ${\displaystyle t\in \left[t_{0};\alpha \right]}$. If the right part ${\displaystyle f\left(t,y\left(t\right)\right)}$

1. is continuous in ${\displaystyle t\in \left[t_{0};\alpha \right]}$, ${\displaystyle y\left(t\right)\in \left[{\underline {z}}\left(t\right);{\overline {z}}\left(t\right)\right]}$;
2. satisfies the Lipschitz condition over variable ${\displaystyle y}$ between functions ${\displaystyle {\overline {z}}\left(t\right)}$ and ${\displaystyle {\underline {z}}\left(t\right)}$: there exists constant ${\displaystyle K>0}$ such as for every ${\displaystyle t\in \left[t_{0};\alpha \right]}$, ${\displaystyle y_{1}\left(t\right)\in \left[{\underline {z}}\left(t\right);{\overline {z}}\left(t\right)\right]}$, ${\displaystyle y_{2}\left(t\right)\in \left[{\underline {z}}\left(t\right);{\overline {z}}\left(t\right)\right]}$ the inequality

${\displaystyle \left\vert f\left(t,y_{1}\left(t\right)\right)-f\left(t,y_{2}\left(t\right)\right)\right\vert \leq K\left\vert y_{1}\left(t\right)-y_{2}\left(t\right)\right\vert }$ holds,

then in ${\displaystyle t\in \left[t_{0};\alpha \right]}$ there exists one and only one solution ${\displaystyle y\left(t\right)}$ for the given initial value problem and moreover for all ${\displaystyle t\in \left[t_{0};\alpha \right]}$

${\displaystyle {\underline {z}}\left(t\right)<y\left(t\right)<{\overline {z}}\left(t\right)}$.

Remarks

Source:^[2]

Weakening inequalities

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 ${\displaystyle {\overline {z}}\left(t\right)}$ and ${\displaystyle {\underline {z}}\left(t\right)}$ respectively. In particular, any of ${\displaystyle {\overline {z}}\left(t\right)=y\left(t\right)}$, ${\displaystyle {\underline {z}}\left(t\right)=y\left(t\right)}$ could be chosen.

Proving inequality only

If ${\displaystyle y\left(t\right)}$ is already known to be an existent solution for the initial value problem in ${\displaystyle t\in \left[t_{0};\alpha \right]}$, 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]

Continuation of the solution towards positive infinity

Chaplygin's theorem answers the question about existence and uniqueness of the solution in ${\displaystyle t\in \left[t_{0};\alpha \right]}$ and the constant ${\displaystyle K}$ from the Lipschitz condition is, generally speaking, dependent on ${\displaystyle \alpha }$: ${\displaystyle K=K\left(\alpha \right)}$. If for ${\displaystyle t\in \left[t_{0};+\infty \right)}$ both functions ${\displaystyle {\overline {z}}\left(t\right)}$ and ${\displaystyle {\underline {z}}\left(t\right)}$ retain their smoothness and for ${\displaystyle \alpha \in \left(t_{0};+\infty \right)}$ a set ${\displaystyle \left\{K\left(\alpha \right)\right\}}$ is bounded, the theorem holds for all ${\displaystyle t\in \left[t_{0};+\infty \right)}$.