MATHEMATICS

L. S. RAKOVSHCHIK

ON ONE CONDITION FOR THE UNRESTRICTED APPLICABILITY OF S. A. CHAPLYGIN’S THEOREM ON INEQUALITIES TO SYSTEMS OF FIRST-ORDER DIFFERENTIAL EQUATIONS

(Presented by Academician S. L. Sobolev on 31 V 1957)

For a single first-order differential equation there is Chaplygin’s theorem on inequalities:

If a function (u=u(x)) is such that (u' - f(x,u)>0) ((<0)) on the segment ([x_0,x_1]) and (u(x_0)=y_0), then for the solution (y=y(x)) of the equation (y'=f(x,y)), (y(x_0)=y_0), the inequality (y(x)u(x))) holds on ((x_0,x_1]). (1).

This theorem does not extend directly to systems of differential equations. This may be seen from the following example:
(y'_1-y_1+y_2-2x-1=0,\quad y'_2-y_2+y_1-2x+1=0,\quad y_1(0)=0,\quad y_2(0)=1.)
The solution of the system is (y_1=x^2,\ y_2=1+x^2). As comparison functions let us take
(u_1=\dfrac{x}{5}) and (u_2=1+3x). We have
(u'_1-u_1+u_2-2x-1=\dfrac{1+4x}{5}>0,)
(u'_2-u_2+u_1-2x+1=3-\dfrac{24}{5}x>0) for (x<{}^5/_8). But to the right of the point (x={}^1/_5),
(u_1-y_1=({}^1/_5-x)x<0), despite the fulfillment of the differential inequalities.

Thus, if we have a system of equations

[
y'_i=f_i(x,y_1,y_2,\ldots,y_n)
\tag{1}
]

with initial conditions

[
y_i(x_0)=y_{i0},
\tag{2}
]

then, in order that from the relations (u'i-f_i(x,u_1,u_2,\ldots,u_n)>0) ((<0)), (u_i(x_0)=y), it should follow that (u_i(x)>y_i(x)) ((u_i(x)<y_i(x))), it is necessary to impose additional conditions either on the right-hand sides of the equations of the system (3), or on the comparison functions (u_1,u_2,\ldots,u_n).

In the present paper one of the possible conditions imposed on the comparison functions is indicated.

First we record several auxiliary propositions. In doing so we shall suppose that the functions (f_i(x,y_1,y_2,\ldots,y_n)) are continuous in some domain (D) of variation of their arguments and satisfy in this domain the Lipschitz condition with constant (K) with respect to the arguments (y_1,y_2,\ldots,y_n). In addition, we assume that for (x\in[x_0,x_1]) the values of all introduced comparison functions do not leave the domain (D).

Lemma 1. If for the functions (u_i=u_i(x)), (\vartheta_i=\vartheta_i(x)) ((i=1,2,\ldots,n)) the conditions are satisfied:

1) (u_i(x_0)=\vartheta_i(x_0)=y_{i0});

2) (u'_i-f_i(x,s_1,s_2,\ldots,s_n)\le 0,\ \vartheta'_i-f(x,s_1,s_2,\ldots,s_n)\ge 0) for (x\in[x_0,x_1]) and any (s_i) lying between (u_i(x)) and (\vartheta_i(x)),

then on the entire segment ([x_0,x_1]) (u_i\le y_i\le \vartheta_i), where (y_1,y_2,\ldots,y_n) is the solution of system (1) satisfying the initial conditions (2).

Lemma 2. Let, for the functions (u_i=u_i(x)), (i=1,2,\ldots,n), the initial conditions (2) be satisfied; let (\eta_1,\eta_2,\ldots,\eta_n) be a solution of the system

[
\eta_i' - K \sum_{r=1}^{n} \eta_r
=
\left|u_i' - f_i(x,u_1,\ldots,u_n)\right|,
\tag{3}
]

satisfying zero initial conditions. Then on ([x_0,x_1]) the values of the functions (u_i+\eta_i) ((u_i-\eta_i)) are not less than (not greater than) the corresponding values of the functions (y_1,y_2,\ldots,y_n) giving a solution of system (1) for the same initial conditions, i.e. on ([x_0,x_1])

[
u_i-\eta_i \leq y_i \leq u_i+\eta_i .
]

For the proof, let us note that (\eta_i(x)\geq 0) on ([x_0,x_1]) (this can be verified by using the results of the work ((^3))). Consequently,

[
u_i-\eta_i \leq u_i \leq u_i+\eta_i .
]

Consider the difference ((u_i+\eta_i)'-f_i(x,s_1,s_2,\ldots,s_n)), where, for the given (x), (s_r) assumes arbitrary values lying in the segment ([u_r(x)-\eta_r(x),\,u_r(x)+\eta_r(x)]).

We have:

[
\begin{aligned}
&(u_i+\eta_i)' - f_i(x,s_1,s_2,\ldots,s_n) =\
&= u_i' - f_i(x,u_1,u_2,\ldots,u_n)
+ \left|u_i' - f_i(x,u_1,u_2,\ldots,u_n)\right|
+ \eta_i' - K\sum_{r=1}^{n}\eta_r -\
&\quad - \left|u_i' - f_i(x,u_1,u_2,\ldots,u_n)\right|
+ f_i(x,u_1,u_2,\ldots,u_n)
- f_i(x,s_1,s_2,\ldots,s_n) +\
&\quad + K\sum_{r=1}^{n}\eta_r
\geq
K\sum_{r=1}^{n}{\eta_r-|u_r-s_r|}\geq 0,
\end{aligned}
]

since (|u_r-s_r|\leq \eta_r). Similarly we prove that

[
(u_i-\eta_i)' - f_i(x,s_1,s_2,\ldots,s_n)\leq 0
]

for the same (s_1,s_2,\ldots,s_n). It remains to apply Lemma 1.

Lemma 3. If

[
\eta_i' - K\sum_{r=1}^{n}\eta_r=\varepsilon_1(x), \qquad
\sigma_i' - K\sum_{r=1}^{n}\sigma_r=\varepsilon_2(x),
]

(\sigma_i(x_0)=\eta_i(x_0)), (i=1,2,\ldots,n), and (\varepsilon_1(x)\leq \varepsilon_2(x)) on ([x_0,x_1]), then on the same interval (\eta_i(x)\leq \sigma_i(x)).

The required conclusion is obtained by applying the result of the work ((^3)).

Theorem. Let the functions (u_1,u_2,\ldots,u_n) satisfy the initial conditions (2) and the inequalities

[
u_i' - f_i(x,u_1,u_2,\ldots,u_n)\geq 0 \;(\leq 0)
]

for (x\in[x_0,x_1]). Let (\vartheta_i=u_i+\eta_i) ((\vartheta_i=u_i-\eta_i)), where (\eta_1,\eta_2,\ldots,\eta_n) is a solution of the system

[
\eta_i' - K\sum_{r=1}^{n}\eta_r
=
\left|u_i' - f_i(x,u_1,\ldots,u_n)\right|,
]

vanishing for (x=x_0), and let (y_1,y_2,\ldots,y_n) be a solution of system (1) satisfying conditions (2). Then, if

[
\left|\vartheta_i' - f_i(x,\vartheta_1,\vartheta_2,\ldots,\vartheta_n)\right|
\leq
\left|u_i' - f_i(x,u_1,u_2,\ldots,u_n)\right|,
]

then

[
u_i\geq y_i \quad (u_i\leq y_i)
]

on the entire segment ([x_0,x_1]).

By virtue of Lemma 1, the values of the functions (\vartheta_i=u_i+\eta_i) ((\vartheta_i=u_i-\eta_i)) are not less than (not greater than) the values of the functions (y_i).

Further, if

[
\sigma_i' - K\sum_{r=1}^{n}\sigma_r
=
\left|\vartheta_i' - f_i(x,\vartheta_1,\vartheta_2,\ldots,\vartheta_n)\right|,
\qquad
\sigma_i(x_0)=0,
]

then the functions (\vartheta_i-\sigma_i) ((\vartheta_i+\sigma_i)) are not greater than (not less than) the functions (y_i), i.e.

[
\vartheta_i-\sigma_i\leq y_i
\quad
(\vartheta_i+\sigma_i\geq y_i).
]

By Lemma 3 and the hypotheses of the theorem, (\sigma_i\leq \eta_i); therefore

[
u_i\leq u_i+\eta_i-\sigma_i
=
\vartheta_i-\sigma_i
\leq y_i
\quad
\left(
u_i\geq u_i-\eta_i+\sigma_i
=
\vartheta_i+\sigma_i
\geq y_i
\right).
]

Novosibirsk Electrotechnical Institute

Received
25 X 1957

REFERENCES

1. S. A. Chaplygin, Foundations of a New Method of Approximate Integration of a First-Order Differential Equation, Collected Works, 1, 1948.
2. S. A. Chaplygin, Solution of a System of Two Differential Equations, Collected Works, 2, 1948.
3. A. O. Gel’fond, Izv. AN SSSR, OMEN, No. 6 (1938).
4. N. V. Azbelev, DAN, 99, No. 4, 493 (1954).