ON THE PROBLEM OF APPROXIMATE INTEGRATION OF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS BY THE METHOD OF S. A. CHAPLYGIN

A. N. VITYUK

The problem of approximate integration of the Cauchy problem for a system of ordinary differential equations, connected with the use of differential and integral inequalities, has been treated in works by many authors.

The multipoint boundary-value problem for the equation \(y^{(n)}=f(x,y,y',\ldots,y^{(n-1)})\) was considered in [1], and for a system of ordinary differential equations in [2]*.

Below a method is proposed for constructing two-sided approximations to the solution of the system

\[ y_i'=f_i[y]\equiv f_i(x,y_1,y_2,\ldots,y_n), \qquad (i=1,2,\ldots,n) \tag{1} \]

under the initial conditions

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

where \(x_i\in [a,b]\), \(y_{i0}\) are arbitrary constants**.

Suppose that the functions \(f_i\) and their partial derivatives \(\partial f_i/\partial y_k\) \((k=1,2,\ldots,n)\) are continuous in the domain \(D[a\le x\le b,\ |y_i-y_{i0}|\le c_i]\).

Let the constants \(m_{ik}, M_{ik}\) \((k=1,2,\ldots,n)\) be such that in the domain \(D\)

\[ m_{ik}\le \frac{\partial f_i}{\partial y_k}, \qquad M_{ik}\ge \left|\frac{\partial f_i}{\partial y_k}\right|, \qquad (k=1,2,\ldots,n;\ k\ne i), \]

\[ m_{ii}<\frac{\partial f_i}{\partial y_i}, \qquad M_{ii}>\left|\frac{\partial f_i}{\partial y_i}\right|. \]

Lemma. Let the functions \(u_i^{(0)}(x)\), \(v_i^{(0)}(x)\) of class \(C_1[a,b]\) satisfy the conditions:

1)

\[ u_i^{(0)}(x_i)=v_i^{(0)}(x_i)=y_{i0}; \]

2) for \(a\le x\le b\):

a)

\[ \left|u_i^{(0)}-y_{i0}\right|\le c_i, \qquad \left|v_i^{(0)}-y_{i0}\right|\le c_i; \tag{3} \]

b)

\[ u_i^{(0)\prime}-f_i[u^{(0)}] +\sum_{\substack{k=1\\ k\ne i}}^{n} M_{ik}\left|v_k^{(0)}-u_k^{(0)}\right| =\psi_i^{(0)}(x)<0, \]

* In this work there is an inaccuracy; to remove it one must additionally assume that \(K(n+1)(b-a)<1\) (see the notation of work [2]).

** In what follows we assume that the index \(i\) takes the values \(1,2,\ldots,n\).

\[ v_i^{(0)\prime}-f_i[v^{(0)}]-\sum_{\substack{k=1\\ k\ne i}}^{n} M_{ik}\left|v_k^{(0)}-u_k^{(0)}\right| =\varphi_i^{(0)}(x)>0 . \tag{4} \]

Then

\[ \begin{aligned} v_i^{(0)}(x)&<u_i^{(0)}(x) &&\text{for } a\leq x<x_i,\\ u_i^{(0)}(x)&<v_i^{(0)}(x) &&\text{for } x_i<x\leq b . \end{aligned} \tag{5} \]

Proof. Subtracting (3) from (4) and denoting \(\gamma_i=v_i^{(0)}-u_i^{(0)}\), we obtain

\[ \gamma_i^{(0)\prime} -\frac{\widetilde{\partial f_i}}{\partial y_i}\gamma_i^{(0)} = \sum_{\substack{k=1\\ k\ne i}}^{n} \left[ 2M_{ik}\left|v_k^{(0)}-u_k^{(0)}\right| + \frac{\widetilde{\partial f_i}}{\partial y_k} \left(v_k^{(0)}-u_k^{(0)}\right) \right] +\varphi_i^{(0)}(x)-\psi_i^{(0)}(x). \]

Analogously to the proof of the first part of the theorem of § 3 in [3], using the relation

\[ \gamma_i^{(0)}(x)= e^{\left[\int_{x_i}^{x}\frac{\widetilde{\partial f_i}}{\partial y_i}\,dt\right]} \int_{x_i}^{x} \left\{ \sum_{\substack{k=1\\ k\ne i}}^{n} \left[ 2M_{ik}\left|\gamma_k^{(0)}\right| + \frac{\widetilde{\partial f_i}}{\partial y_k}\gamma_k^{(0)} \right] + \varphi_i^{(0)}(t)-\psi_i^{(0)}(t) \right\} e^{\left[-\int_{x_i}^{t}\frac{\widetilde{\partial f_i}}{\partial y_i}\,d\tau\right]} \,dt, \]

it is easy to prove the validity of the inequalities (5).

Below we use the following notation:

\[ M=\max_i M_{ii},\qquad \gamma_i^{(p)}=v_i^{(p)}-u_i^{(p)},\qquad \sum_{i,p}=\sum_{\substack{k=1\\ k\ne i}}^{n}M_{ik}\left|\gamma_k^{(p)}\right|, \]

\[ \varphi_i^{(p)}(x)=v_i^{(p)\prime}-f_i[v^{(p)}]-\sum_{i,p}, \]

\[ \psi_i^{(p)}(x)=u_i^{(p)\prime}-f_i[u^{(p)}]+\sum_{i,p} \qquad (p=0,1,2,\ldots). \]

Define \(2n\) functions by setting
\(v_i^{(1)}(x)=v_i^{(0)}(x)-\sigma_i^{(1)}(x)\),
\(u_i^{(1)}(x)=u_i^{(0)}(x)+\delta_i^{(1)}(x)\), where
\(\sigma_i^{(1)}\) and \(\delta_i^{(1)}\) are solutions of the differential equations

\[ \sigma_i^{(1)\prime}\mp M\sigma_i^{(1)}-\varphi_i^{(0)}(x)=0, \tag{6} \]

\[ \delta_i^{(1)\prime}\mp M\delta_i^{(1)}+\psi_i^{(0)}(x)=0 \tag{7} \]

under the initial conditions
\(\sigma_i^{(1)}(x_i)=\delta_i^{(1)}(x_i)=0\), where the upper signs are taken for \(x\in [a,x_i]\), and the lower signs for \(x\in [x_i,b]\).

From Chaplygin’s theorem (see [4]) on differential inequalities for the equation \(y'=f(x,y)\) it follows that \(\sigma_i^{(1)}(x)>0\), \(\delta_i^{(1)}(x)>0\) for \(x_i<x\leq b\); \(\sigma_i^{(1)}(x)<0\), \(\delta_i^{(1)}(x)<0\) for \(a\leq x<x_i\).

Since for \(a\leq x\leq b\)

\[ \gamma_i^{(0)'} \mp M\gamma_i^{(0)}-\varphi_i^{(0)}(x) = \bigl[f_i[u^{(0)}]-u_i^{(0)'}\bigr] + \left(\frac{\partial \widetilde f_i}{\partial y_i}\mp M\right)\gamma_i^{(0)} + \sum_{\substack{k=1\\ k\ne i}}^{n} \left( M_{ik}\,|\gamma_k^{(0)}| + \frac{\partial \widetilde f_i}{\partial y_k}\gamma_k^{(0)} \right)>0, \]

\[ \gamma_i^{(0)'} \mp M\gamma_i^{(0)}+\psi_i^{(0)}(x)<0, \]

then from Chaplygin’s theorem it follows that for \(a\leq x\leq b\)

\[ |\sigma_i^{(1)}(x)|\leq |\gamma_i^{(0)}(x)|,\qquad |\delta_i^{(1)}(x)|\leq |\gamma_i^{(0)}(x)|, \]

with equality only for \(x=x_i\). Consequently,

\[ \begin{aligned} &v_i^{(0)}<u_i^{(1)}<u_i^{(0)},\qquad v_i^{(0)}<v_i^{(1)}<v_i^{(0)} &&\text{for } a\leq x<x_i;\\ &u_i^{(0)}<u_i^{(1)}<v_i^{(0)},\qquad u_i^{(0)}<v_i^{(1)}<v_i^{(0)} &&\text{for } x_i<x\leq b. \end{aligned} \tag{8} \]

Adding (7) and (6), we find the equation for \(\gamma_i^{(1)}(x)\)

\[ \gamma_i^{(1)'} \mp M\gamma_i^{(1)} = \gamma_i^{(0)'} \mp M\gamma_i^{(0)} -\varphi_i^{(0)}(x)+\psi_i^{(0)}(x), \qquad \gamma_i^{(1)}(x_i)=0. \]

Since for \(a\leq x\leq b\)

\[ \gamma_i^{(0)'} \mp M\gamma_i^{(0)} -\varphi_i^{(0)}(x)+\psi_i^{(0)}(x) = \left(\frac{\partial \widetilde f_1}{\partial y_i}\mp M\right)\gamma_i^{(0)} + \sum_{i,p} + \sum_{\substack{k=1\\ k\ne i}}^{n} \left[ M_{ik}|\gamma_k^{(0)}| + \frac{\partial \widetilde f_i}{\partial y_k}\gamma_k^{(0)} \right]\geq 0^{*}, \]

then, by Chaplygin’s theorem,

\[ \begin{aligned} \gamma_i^{(1)}(x)&<0 &&\text{for } a\leq x<x_i,\\ \gamma_i^{(1)}(x)&>0 &&\text{for } x_i<x\leq b. \end{aligned} \tag{9} \]

From (8) and (9) it follows that

\[ \begin{aligned} &v_i^{(0)}<v_i^{(1)}<u_i^{(1)}<u_i^{(0)} &&\text{for } a\leq x<x_i,\\ &u_i^{(0)}<u_i^{(1)}<v_i^{(1)}<v_i^{(0)} &&\text{for } x_i<x\leq b. \end{aligned} \tag{10} \]

Further,

\[ v_i^{(1)'}-f_i[v^{(1)}] = v_i^{(0)'}-\delta_i^{(1)'}-f_i[v^{(1)}] = v_i^{(0)'}\mp M\delta_i^{(1)}-v_i^{(0)'} + \]

\[ + f_i[v^{(0)}] + \sum_{i,0} - f_i[v^{(1)}] = \left(\frac{\partial \widetilde f_i}{\partial y_i}\mp M\right)\delta_i^{(1)} + \]

* Equality in relations of this type is possible, generally speaking, only for \(x=x_i\). For example, when in system (1) there is an equation \(y_i=f_i(x,y_i)\), etc.

\[ + \sum_{\substack{k=1\\ k \ne i}}^{n} \left( M_{ik}|\gamma_k^{(0)}| + \frac{\partial \widetilde f_i}{\partial y_k}\sigma_k^{(1)} \right). \tag{11} \]

We shall show that for \(a \le x \le b\)

\[ M_{ik}|\gamma_k^{(0)}| + \frac{\partial \widetilde f_i}{\partial y_k}\sigma_k^{(1)} \ge M_{ik}|\gamma_k^{(1)}| \quad (k=1,2,\ldots,n;\ k \ne i). \tag{12} \]

Indeed, for \(a \le x \le x_k\)

\[ M_{ik}|\gamma_k^{(0)}| + \frac{\partial \widetilde f_i}{\partial y_k}\sigma_k^{(1)} = M_{ik}(u_k^{(0)}-v_k^{(0)})+ \frac{\partial \widetilde f_i}{\partial y_k}(v_k^{(0)}-v_k^{(1)})= \]

\[ = \left(M_{ik}-\frac{\partial \widetilde f_i}{\partial y_k}\right) (v_k^{(1)}-v_k^{(0)}) + M_{ik}(u_k^{(0)}-v_k^{(1)}) \ge M_{ik}(u_k^{(1)}-v_k^{(1)}), \]

and for \(x_k \le x \le b\)

\[ M_{ik}|\gamma_k^{(0)}| + \frac{\partial \widetilde f_i}{\partial y_k}\sigma_k^{(1)} = M_{ik}(v_k^{(0)}-u_k^{(0)})+ \frac{\partial \widetilde f_i}{\partial y_k}(v_k^{(0)}-v_k^{(1)})= \]

\[ = \left(M_{ik}+\frac{\partial \widetilde f_i}{\partial y_k}\right) (v_k^{(0)}-v_k^{(1)}) + M_{ik}(v_k^{(1)}-u_k^{(0)}) \ge M_{ik}(v_k^{(1)}-u_k^{(1)}). \]

From (11), taking (12) into account, we obtain that \(\varphi_i^{(1)}(x) \ge 0\) for \(a \le x \le b\). Analogously, we prove that \(\psi_i^{(1)}(x) \le 0\) for \(a \le x \le b\). Next, the construction of the functions
\(v_i^{(p)}(x)=v_i^{(p-1)}(x)-\sigma_i^{(p)}(x)\),
\(u_i^{(p)}(x)=u_i^{(p-1)}(x)+\delta_i^{(p)}(x)\)
\((p=2,3,\ldots)\) is continued by means of the equations

\[ \sigma_i^{(p)\prime} \mp M\sigma_i^{(p)} - \varphi_i^{(p-1)}(x)=0, \tag{13} \]

\[ \delta_i^{(p)\prime} \mp M\delta_i^{(p)} + \psi_i^{(p-1)}(x)=0, \tag{14} \]

with initial conditions
\(\sigma_i^{(p)}(x_i)=\delta_i^{(p)}(x_i)=0\), where the upper signs are taken for
\(x \in [a,x_i]\), and the lower signs for \(x \in [x_i,b]\). In a manner similar to that above for the case \(p=1\), we verify that

\[ |\sigma_i^{(p)}| \le |\gamma_i^{(p-1)}|,\quad |\delta_i^{(p)}| \le |\gamma_i^{(p-1)}| \quad \text{for } a \le x \le b; \]

\[ v_i^{(p-1)} < v_i^{(p)} < u_i^{(p)} < u_i^{(p-1)} \quad \text{for } a \le x < x_i; \]

\[ u_i^{(p-1)} < u_i^{(p)} < v_i^{(p)} < v_i^{(p-1)} \quad \text{for } x_i < x \le b; \tag{15} \]

\[ \varphi_i^{(p)}(x) \ge 0,\quad \psi_i^{(p)}(x) \le 0 \quad \text{for } a \le x \le b. \]

Theorem. If

\[ b-a < 1/(3n-1)K, \tag{16} \]

where

\[ K=\max_{i,k}(M_{ik}) \quad (i,k=1,2,\ldots,n), \]

then the sequences
\(\{v_i^{(p)}\}\), \(\{u_i^{(p)}\}\) \((p=1,2,\ldots)\) constructed by means of equations (13), (14) converge uniformly on the interval \([a,b]\) to the solution of problem (1), (2), and moreover

$$ v_i^{(p)} < y_i < u_i^{(p)} \quad \text{for} \quad a \leq x < x_i, $$

$$ u_i^{(p)} < y_i < v_i^{(p)} \quad \text{for} \quad x_i < x \leq b . \tag{17} $$

Proof. We shall prove first that the sequences
$\{|\gamma_i^{(p)}(x)|\}$ $(p=1,2,\ldots)$ converge uniformly to zero on the interval $[a,b]$.

Adding (14) and (13), we find an equation for $\gamma_i^{(p)}(x)$, whose solution, under the initial condition $\gamma_i^{(p)}(x_i)=0$, we write in the form

$$ \gamma_i^{(p)}(x)= \int_{x_i}^{x} e^{\pm M(x-t)} \left[ \left( \frac{\partial \widetilde f_i}{\partial y_i}\mp M \right)\gamma_i^{(p-1)} +\Sigma_{i,p-1} + \sum_{\substack{k=1\\ k\ne i}}^{n} \left( M_{ik}|\gamma_k^{(p-1)}| + \frac{\partial \widetilde f_i}{\partial y_k}\gamma_k^{(p-1)} \right) \right]dt . \tag{18} $$

Let

$$ \gamma=\max_i \max_{[a,b]} |\gamma_i^{(0)}(x)|, $$

then from (18), by induction, we easily obtain the estimates

$$ |\gamma_i^{(p)}(x)|\leq [(3n-1)K(b-a)]^p\gamma \quad (p=1,2,\ldots), $$

from which, by virtue of (16), it follows that the sequences $\{|\gamma_i^{(p)}(x)|\}$ converge uniformly to zero on the interval $[a,b]$, and hence, by virtue of (15), so do the sequences $\{|\sigma_i^{(p)}(x)|\}$, $\{|\delta_i^{(p)}(x)|\}$. Thus,

$$ \lim_{p\to\infty} v_i^{(p)}(x)=\lim_{p\to\infty} u_i^{(p)}(x)=z_i(x). $$

Passing to the limit as $p\to\infty$ in the expression

$$ v_i^{(p)}(x)=y_{i0}+ \int_{x_i}^{x} \left[ f_i[v^{(p)}] + \left( \frac{\partial \widetilde f_i}{\partial y_i}\mp M \right)\sigma_i^{(p)} + \sum_{\substack{k=1\\ k\ne i}}^{n} \left( M_{ik}|\gamma_k^{(p-1)}| + \frac{\partial \widetilde f_i}{\partial y_k}\sigma_k^{(p)} \right) \right]dt $$

we conclude that $z_i(x)$ is a solution of problem (1), (2).

The validity of inequalities (17) is obvious.

Remark 1. The conditions satisfied by the functions $f_i$, together with condition (16), guarantee (see [5]) the uniqueness of the solution of problem (1), (2).

Remark 2. In the case of the Cauchy problem for system (1), condition b) of the lemma may be replaced by the following:

$$ v_i^{(0)\prime}-f_i[v^{(0)}] - \sum_{\substack{k=1\\ k\ne i}}^{n} (|m_{ik}|-m_{ik})(v_k^{(0)}-u_k^{(0)}) = \varphi_i^{(0)}(x)>0, \tag{19} $$

$$ u_i^{(0)\prime}-f_i[u^{(0)}] + \sum_{\substack{k=1\\ k\ne i}}^{n} (|m_{ik}|-m_{ik})(v_k^{(0)}-u_k^{(0)}) = \psi_i^{(0)}(x)<0. \tag{20} $$

Instead of equations (13), (14), in this case we shall have

\[ \sigma_i^{(p)\prime} - m\sigma_i^{(p)} - \varphi_i^{(p-1)}(x)=0,\qquad \delta_i^{(p)\prime} - m\delta_i^{(p)} + \psi_i^{(p-1)}(x)=0, \tag{21} \]

where \(m=\min_i m_{ii}\), and the initial conditions are
\(\sigma_i^{(p)}(a)=\delta_i^{(p)}(a)=0\).

If the functions \(v_i^{(0)}(x)\) and \(u_i^{(0)}(x)\) are such that relations (19) and (20) hold when the term \((|m_{ii}|-m_{ii})(v_i^{(0)}-u_i^{(0)})\) is included in the sum, then the functions \(v_i^{(p)}(x)\) and \(u_i^{(p)}(x)\) can be determined as follows:

\[ v_i^{(p)}(x)=v_i^{(p-1)}(x)-\int_a^x \varphi_i^{(p-1)}(t)\,dt, \]

\[ u_i^{(p)}(x)=u_i^{(p-1)}(x)-\int_a^x \psi_i^{(p-1)}(t)\,dt, \]

where

\[ \varphi_i^{(p)}(x)=v_i^{(p)\prime}-f_i[v^{(p)}]-\sum_{k=1}^{n}(|m_{ik}|-m_{ik})(v_k^{(p)}-u_k^{(p)}), \]

\[ \psi_i^{(p)}(x)=u_i^{(p)\prime}-f_i[u^{(p)}]+\sum_{k=1}^{n}(|m_{ik}|-m_{ik})(v_k^{(p)}-u_k^{(p)})\qquad (p=1,2,\ldots). \]

References

1. Chichkin E. S. Izv. vuzov, Matematika, No. 2, 1962, pp. 170–179.
2. Blinchevskii V. S. Zhurnal vychislitel’noi matem. i matem. fiz., 3, No. 6, 1963, pp. 1117–1121.
3. Babkin B. N. Matem. sbornik, 46, No. 4, 1958, pp. 389–397.
4. Chaplygin S. A. Collected Works, vol. 1, Moscow–Leningrad, OGIZ, 1948.
5. Naimark A. B. DAN SSSR, 67, No. 6, 1949, pp. 969–972.

Received by the editors
20 February 1965

Odessa State University
named after I. I. Mechnikov