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ON THE CONSTRUCTION OF A ZERO APPROXIMATION IN SOLVING THE CAUCHY PROBLEM FOR A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS BY S. A. CHAPLYGIN’S METHOD
V. S. BLINCHEVSKII
It is assumed in the paper that the right-hand sides of the systems of differential equations under consideration satisfy conditions of the Lipschitz-condition type, and moreover the growth of the Lipschitz constant with an increase in the radius of the domain is bounded in a certain sense.
In Theorems 1–3 the zero approximation mentioned is expressed by two, and in Theorem 4 by one quadrature, but one of the functions is given by its inverse.
The idea of constructing a zero Chaplygin [1] approximation arose in the analysis of Lemma 2 [2].
Under the conditions of the theorems obtained, the solution of the Cauchy problem can be approximated arbitrarily accurately by Chaplygin functions.
Consider the Cauchy problem
\[ x_i(a)=l_i,\qquad i=1,\ldots,n \tag{1} \]
for the system
\[ \dot{x}_i=a_i(t,x_1,\ldots,x_n),\qquad i=1,\ldots,n, \tag{2} \]
where the \(a_i\) are piecewise continuous in \(t\), \(a\leq t\leq b\).
Denote by \(u_i(t)\), \(i=1,\ldots,n\), arbitrary piecewise smooth functions for which \(u_i(a)=l_i\), and by \(\eta_i(t,K)\)
\[ \int_a^t \left[ K\sum_{j=1}^n \int_a^\tau \alpha_j(s)\exp[nK(\tau-s)]\,ds+\alpha_i(\tau) \right]\,d\tau, \]
where
\[ \alpha_j(t)\geq |u'_j(t)-a_j(t,u_1(t),\ldots,u_n(t))|. \]
Lemma. Suppose
\[ |a_i(t,x_1,\ldots,x_n)-a_i(t,y_1,\ldots,y_n)| \leq K\sum_{j=1}^n |x_j-y_j| \]
for \(a\leq t\leq b\), \(u_i-\eta_i\leq x_i\), \(y_i\leq u_i+\eta_i\). Then on \([a,b]\) the functions
\[ \varphi_i(t)=u_i(t)-\eta_i(t),\qquad \psi_i(t)=u_i(t)+\eta_i(t),\qquad i=1,\ldots,n, \]
are respectively lower and upper Chaplygin functions of problem (1) for system (2), i.e.,
\[ \varphi_i(a)=\psi_i(a)=l_i, \]
\[ \varphi'_i\leq a_i(t,s_1,\ldots,s_{i-1},\varphi_i,s_{i+1},\ldots,s_n), \]
\[ \psi'_i\geq a_i(t,s_1,\ldots,s_{i-1},\psi_i,s_{i+1},\ldots,s_n) \]
provided
\[ \varphi_j(t)\leq s_j(t)\leq \psi_j(t),\qquad i,j=1,\ldots,n. \]
Proof. It is not difficult to verify that \(\eta_i(t)\), \(i=1,\ldots,n\), is a solution of the system
\[ \eta_i' = K \sum_{r=1}^{n} \eta_r + \alpha_i(t), \qquad i=1,\ldots,n, \]
with zero initial conditions.
We shall prove that \(\psi_i(t)\) are upper functions:
\[ \begin{aligned} &(u_i+\eta_i)' - a_i(t,s_1,\ldots,s_{i-1},u_i+\eta_i,s_{i+1},\ldots) = u_i' - \\[2mm] &\quad - a_i(t,u_1,\ldots,u_n) + a_i(t,u_1,\ldots,u_n) + K\sum_{1}^{n}\eta_r+\alpha_i(t) - \\[2mm] &\quad - a_i(t,s_1,\ldots,s_{i-1},u_i+\eta_i,s_{i+1},\ldots) \ge u_i' - a_i(t,u_1,\ldots,u_n)+ \\[2mm] &\quad + |u_i' - a_i(t,u_1,\ldots,u_n)| + K\sum_{1}^{n}\eta_r - K\sum_{r\ne i}|s_r-u_r| - K\eta_i \ge \\[2mm] &\quad \ge K\sum_{r\ne i}(\eta_r-|s_r-u_r|)\ge 0 \qquad \text{for } |s_r-u_r|\le \eta_r . \end{aligned} \]
Analogously it is proved that \(\varphi_i(t)\) are lower functions. An obvious consequence of the lemma is
Theorem 1. Suppose
\[ |a_i(t,x_1,\ldots,x_n)-a_i(t,y_1,\ldots,y_n)| \le K\sum_{j=1}^{n}|x_j-y_j| \]
for \(a\le t\le b\), \(-\infty<x_j,y_j<+\infty\), \(i,j=1,\ldots,n\). Then the conclusion of the lemma is valid.
Theorem 2. Let \(r_i(t)>0\) be arbitrary functions continuous on \([a,b]\), and suppose that in the set \(|x_k-u_k(t)|\le r_k(t)\), \(k=1,\ldots,n\), \(a\le t\le b\),
\[ |a_i(t,\xi_1,\ldots,\xi_n)-a_i(t,\eta_1,\ldots,\eta_n)| \le \overline{K}\sum_{j=1}^{n}|\xi_j-\eta_j|. \]
Then, on a sufficiently small interval \([a,c]\), the conclusion of the lemma is valid for \(K=\overline{K}\).
Proof. The functions \(\eta_i(t,\overline{K})\) do not decrease from zero as \(t\) increases.
Let \([a,c]\), \(c\le b\), be the maximal interval on which the inequalities \(\eta_i(t)\le r_i(t)\) hold. Then on \([a,c]\) all the conditions of the lemma are satisfied, and hence also its conclusion (for \(K=\overline{K}\)).
Remark. It is easy to see that for the conditions of Theorem 2 to hold it is sufficient that continuous \(\dfrac{\partial a_i}{\partial x_j}\) exist for \(a\le t\le b\), \(|x_k-u_k(t)|\le r_k(t)\), \(i,j,k=1,\ldots,n\).
Theorem 3. Suppose
\[ 1^\circ.\quad |a_i(t,x_1,\ldots,x_n)-a_i(t,y_1,\ldots,y_n)|\le \]
\[ \le f(r_1,\ldots,r_n)\sum_{j=1}^{n}|x_j-y_j|, \]
where \(r_j=\max\bigl(|x_j-u_j(t)|,\ |y_j-u_j(t)|\bigr)\), for \(a\le t\le b\), \(-\infty<x_j,y_j<+\infty\).
\(2^\circ.\) \(f\) is a nondecreasing function in each of its arguments.
\(3^\circ.\) The equation \(f[\eta_1(b,\bar K),\ldots,\eta_n(b,\bar K)] = K\) is solvable for some \(K=\bar K\). Then, for \(K=\bar K\), the conclusion of the lemma is valid.
Proof. Note that in the set \(a\le t\le b\), \(|x_i-u_i(t)|\le \eta_i(t,\bar K)\), the Lipschitz condition with constant \(\bar K\) is valid. Indeed, using the monotonicity of \(f\) and \(\eta_j\), we conclude that in it
\[ f(r_1,\ldots,r_n)\le f[\eta_1(t,\bar K),\ldots,\eta_n(t,\bar K)]\le \]
\[ \le f[\eta_1(b,\bar K),\ldots,\eta_n(b,\bar K)]=\bar K. \]
All the conditions of the lemma are fulfilled for \(K=\bar K\), and hence its conclusion is also valid.
Remark. If continuous \(\dfrac{\partial a_i}{\partial x_j}\) exist, it is clear that there will always be found a nondecreasing function of its arguments \(f(r_1,\ldots,r_n)\), where \(r_j=|x_j-u_j(t)|\), such that
\[ \left|\frac{\partial a_i}{\partial x_j}\right|\le f(r_1,\ldots,r_n). \]
Hence it is clear that conditions \(1^\circ, 2^\circ\) of Theorem 3 are satisfied.
Theorem 4. Suppose
\[ 1^\circ.\ |a_i(t,x_1,\ldots,x_n)-a_i(t,y_1,\ldots,y_n)|\le \]
\[ \le f\left(\sum_1^n r_k\right)\sum_{j=1}^n |x_j-y_j|, \]
where \(r_k=\max (|x_k-u_k(t)|,\ |y_k-u_k(t)|)\), for \(a\le t\le b\), \(-\infty<x_j,\ y_j<+\infty\).
\(2^\circ.\) \(f(r)\) is a piecewise smooth nondecreasing function for \(0\le r<+\infty\).
\[ 3^\circ.\ \int_0^{+\infty}\frac{dr}{nrf(r)+M}>b-a,\quad \text{where } M>\sum_{i=1}^n |u_i'-a_i(t,u_1,\ldots,u_n)|. \]
Then on \([a,b]\) the Chaplygin functions of problem (1) for system (2) are constructed.
Proof. Denote by \(\gamma(t)\) the solution of the equation \(r'=nrf(r)+M\) with the zero initial condition \(\gamma(a)=0\). It can be continued to \([a,b]\), since by \(2^\circ\) \(\gamma(t)\) is bounded on \([a,b]\).
The function \(\gamma(t)\) can be sought as the inverse function
\[ t=\int_0^\gamma \frac{dr}{nrf(r)+M}+a. \]
Introduce the functions
\[ \xi_i(t)=\int_a^t \left(\gamma(\tau)f[\gamma(\tau)]+|u_i'(\tau)-a_i(\tau,u_1,\ldots,u_n)|\right)d\tau. \]
Note that
\[ \sum_1^n \xi_i\le \gamma,\quad \text{since}\quad \sum_1^n \xi_i(a)=\gamma(a)=0,\quad \sum_1^n \xi_i'\le \gamma'. \]
We shall show that \(u_i+\xi_i,\ u_i-\xi_i\) are, respectively, upper and lower Chaplygin functions:
\[ \begin{aligned} &(u_i+\xi_i)' - a_i(t,s_1,\ldots,s_{i-1},u_i+\xi_i,s_{i+1},\ldots) = \\ &= u_i' - a_i(t,u_1,\ldots,u_n) + a_i(t,u_1,\ldots,u_n) + \gamma f(\gamma) + \\ &\quad + |u_i' - a_i(t,u_1,\ldots,u_n)| - a_i(t,s_1,\ldots,s_{i-1},u_i+\xi_i,s_{i+1},\ldots) \ge \\ &\ge \gamma f(\gamma) - \sum_{1}^{n}\xi_j f\left(\sum_{1}^{n}\xi_j\right) \ge 0 \end{aligned} \]
when \(|u_j-s_j|\le \xi_j\), in accordance with the monotonicity of \(f\). The argument for \(u_i-\xi_i\) is analogous.
Remark. Conditions \(1^\circ, 2^\circ\) will be fulfilled if the continuous \(\dfrac{\partial a_i}{\partial x_j}\) exist. Indeed, set
\[ \varphi\left(\sum_{1}^{n} r_k\right) = \max_{i,j\, |x_k-u_k(t)|<r_k} \left|\frac{\partial a_i}{\partial x_j}\right| \]
and approximate it from above by a smooth nondecreasing function
\[ f\left(\sum_{1}^{n} r_k\right). \]
Then
\[ \left|\frac{\partial a_i}{\partial x_j}\right| \le f\left(\sum_{1}^{n} r_k\right), \quad \text{where } r_k=|x_k-u_k(t)|. \]
In the following theorem the zero Chaplygin approximations obtained are related to the solution of the Cauchy problem.
Theorem 5. Suppose that the conditions of one of Theorems 1–4 are fulfilled. Then the solution of the Cauchy problem can be continued to the interval appearing in the conclusion of the theorem and can be approximated there with arbitrary accuracy by Chaplygin functions.
Proof. The solution is continued to the corresponding interval of the parameter, since it is bounded by the functions of the zero approximation. In the set bounded by these functions the Lipschitz constant will be fixed. As was proved in Theorem 1 [1], under this condition the solution can be approximated with arbitrary accuracy by Chaplygin functions on the same interval.
References
- B. S. Blinnchevskii, Zhurnal vych. matem. i matem. fiziki, 3, No. 6, 1117–1121, 1963.
- L. S. Rakovshchik, DAN SSSR, 117, No. 3, 378–379, 1957.
Received by the editors
November 16, 1964
Moscow Technological Institute
of the Food Industry