MATHEMATICS

V. E. MIRAKOV

THE PRINCIPLE OF MAJORANTS AND THE METHOD OF TANGENT PARABOLAS FOR NONLINEAR FUNCTIONAL EQUATIONS

(Presented by Academician A. N. Kolmogorov on 15 XI 1956)

Suppose it is required to find an approximate solution of the functional equation

\[ P(x)=0, \tag{1} \]

where \(y=P(x)\) is a nonlinear operation transforming \(x\in X\) into \(y\in Y\). By \(X\) and \(Y\) we mean complete linear spaces, normed in the sense of L. V. Kantorovich by means of linear semi-ordered spaces \(Z\) and \(W\), spaces of type \(B_k\) \((^{1,2})\).

We shall solve equation (1) by the method of successive approximations, in which the iterations are defined by the following equality \((^{3,4})\):

\[ x_{n+1}=x_n+\sum_{k=1}^{m}\frac{(-1)^k}{k!}\Phi^{(k)}(P(x_n))(P(x_n))^k, \tag{2} \]

where

\[ \Phi'(P(x_n))=[P'(x_n)]^{-1}=\Gamma_n,\qquad \Phi''(P(x_n))=-\frac{1}{2}\Gamma_n P''(x_n)\Gamma_n^2 \]

and so on.

Method (2) in the case \(m=1\) gives Newton’s method

\[ x_{n+1}=x_n-\Gamma_nP(x_n), \]

considered in \((^1)\) for spaces of type \(B_k\).

For \(m=2\) we obtain Chebyshev’s method

\[ x_{n+1}=x_n-\Gamma_nP(x_n)-\frac{1}{2}\Gamma_nP''(x_n)[\Gamma_nP(x_n)]^2, \]

considered in \((^{4,5})\) for Banach spaces and in \((^6)\) for spaces of type \(B_k\).

It is natural to call process (2) an analogue of the process of tangent parabolas, since in the case of a real equation the finding of the approximation \(x_{n+1}\) from \(x_n\) is equivalent to finding the point of intersection of the axis of abscissas with the parabola

\[ x=\sum_{k=0}^{m}a_ky^k, \]

with appropriately chosen coefficients \(a_k\), having with the curve \(y=P(x)\) at the point \((x_n,P(x_n))\) contact of order \(m\).

We shall consider a process analogous to (2) for the equation

\[ Q(z)=0, \tag{3} \]

where \(Q\) is an operation from \(Z\) to \(W\).

Theorem 1. If in equations (1) and (3) \(P,Q\) are continuous operations having, respectively on the paths \(x_0,x_1,x_2,\ldots\) and \(z_0,z_1,z_2,\ldots\), weak derivatives in the sense of Gâteaux up to order \(m+1\) inclusive, and the following conditions are fulfilled:

1) there exist \([P'(x_0)]^{-1}=\Gamma_0\) and \([Q'(z_0)]^{-1}=\Delta_0\), where \(\Delta_0\) is \((oo)\)-continuous and \(|\Gamma_0|\le -\Delta_0\);

2) \(|\Gamma_0P(x_0)|\le -\Delta_0Q(z_0)\);

3) \(|\Gamma_0 P^{(k)}(x_0)| \leq -\Delta_0 Q^{(k)}(z_0)\) for \(k=2,3,\ldots,m;\ m>1\) (for \(m=1\), condition 3) is omitted);

4) \(|\Gamma_0 P^{(m+1)}(x)| \leq -\Delta_0 Q^{(m+1)}(z)\) for \(x \leftrightarrow z\) (to the point \(x=x_n+\theta(x_{n+1}-x_n)\) there is assigned the point \(z=z_n+\theta(z_{n+1}-z_n);\ 0\leq\theta\leq 1\)), then from the existence and convergence of the process of tangent parabolas for equation (3) there follows its existence and convergence for equation (1).

Proof. Denote by \(\Delta x_{l,n}\) the general term of the sum (2)

\[ \Delta x_{l,n}=\frac{(-1)^l}{l!}\Phi^{(l)}(P(x_n))(P(x_n))^l . \]

Then, according to (7), we shall have

\[ \Delta x_{1,n}=-\Gamma_n P(x_n); \]

\[ \Delta x_{l,n} = -\Gamma_n \sum_{k=2}^{l} \frac{1}{k!}P^{(k)}(x_n) \sum_{\substack{i_1+\cdots+i_{l-1}=k\\ 1\cdot i_1+\cdots+(l-1)i_{l-1}=l}} \frac{k!}{i_1!\cdots i_{l-1}!} \Delta x_{1,n}^{\,i_1}\cdots \Delta x_{l-1,n}^{\,i_{l-1}} . \tag{4} \]

From the condition of the theorem it follows directly that \(|x_1-x_0|\leq z_1-z_0\). Further, on the basis of Taylor’s formula,

\[ \Gamma_0 P(x_1) = \Gamma_0 \sum_{k=0}^{m} \frac{P^{(k)}(x_0)}{k!}(x_1-x_0)^k + \frac{1}{m!}\Gamma_0 \int_{x_0}^{x_1} P^{(m+1)}(\bar{x})(x_1-\bar{x})^m\,d\bar{x}. \tag{5} \]

Putting \(n=0\) in (4) and substituting in (5), we obtain

\[ |\Gamma_0 P(x_1)| = \left| \Gamma_0 \sum_{k=2}^{m} \frac{P^{(k)}(x_0)}{k!} \sum_{\substack{i_1+\cdots+i_m=k\\ 1\cdot i_1+\cdots+m i_m>m}} \frac{k!}{i_1!\cdots i_m!} \Delta x_{1,0}^{\,i_1}\cdots \Delta x_{m,0}^{\,i_m} + \frac{1}{m!}\Gamma_0 \int_{x_0}^{x_1} P^{(m+1)}(\bar{x})(x_1-\bar{x})^m\,d\bar{x} \right| \leq -\Delta_0 Q(z_1). \tag{6} \]

Since

\[ |I-\Gamma_0 P'(x_1)| = \left| -\Gamma_0 \sum_{k=2}^{m} \frac{P^{(k)}(x_0)}{k!}(x_1-x_0)^k - \frac{1}{m!}\Gamma_0 \int_{x_0}^{x_1} P^{(m+1)}(\bar{x})(x_1-\bar{x})^m\,d\bar{x} \right| \leq \]

\[ \leq I-\Delta_0 Q'(z_1), \]

we conclude analogously to (1) that there exists an operator \(\Gamma_1=[P'(x_1)]^{-1}\), and in this case
\[ |\Gamma_1|\leq -\Delta_1=-[Q'(z_1)]^{-1};\qquad |\Gamma_1P(x_1)|\leq -\Delta_1Q(z_1). \]
It is now easy to verify that if in the conditions of the theorem the points \(x_0\) and \(z_0\) are replaced by \(x_1\) and \(z_1\), then these conditions will still be satisfied, and therefore the analogous process can be continued further.

Let us now consider the case when \(X\) and \(Y\) are Banach spaces.

Theorem 2. If

1) there exists \(\Gamma_0=[P'(x_0)]^{-1}\) and \(\|\Gamma_0\|\leq B\);

2) \(\|\Gamma_0P(x_0)\|\leq\eta\);

3) \(\|P^{(k)}(x_0)\|\leq M_k\) for \(k=2,3,\ldots,m;\ m>1\) (for \(m=1\), condition 3) is omitted);

4) \(\|P^{(m+1)}(x)\|\leq M_{m+1}\) in a domain containing the path \(x_0, x_1, x_2,\ldots\) (it is sufficient that \(\|x-x_0\|\leq z^*\leq 2\eta\)) and the condition is satisfied

\[ h=B\eta\sum_{k=2}^{m+1}\frac{M_k}{(k-2)!}(2\eta)^{k-2}\leq \frac12 \tag{7} \]

for the existence of a positive root \(z^*\) (where \(z^*\) is the least positive root) of the equation

\[ Q(z)=\frac{\eta}{B}-\frac{z}{B}+\sum_{k=2}^{m+1}\frac{M_k z^k}{k!}=0, \tag{8} \]

then the process of tangent parabolas converges to the solution \(x^*\) of equation (1), which exists and lies in the domain

\[ \|x-x_0\|\leq z^* . \tag{9} \]

The rate of convergence is characterized by the inequality

\[ \|x^*-x_n\|\leq z^*-z_n . \tag{10} \]

Remark. It should be noted that the estimates (9) and (10), obtained in Theorem 2, cannot be improved, since they are attained for equation (8).

Moscow
Institute of Physics and Technology

Received
3 IX 1956

CITED LITERATURE

\(^{1}\) L. V. Kantorovich, DAN, 76, No. 1, 17 (1951).
\(^{2}\) L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Semi-Ordered Spaces, M.—L., 1950, p. 455.
\(^{3}\) P. L. Chebyshev, Collected Works, 5, Izd. AN SSSR, 1951, p. 7.
\(^{4}\) M. I. Nechipurenko, Uspekhi matem. nauk, 9, issue 2, 163 (1954).
\(^{5}\) M. A. Mertvetsova, Izv. Kazansk. fil. AN SSSR, ser. phys.-mat. and techn. sciences, issue 8, 154 (1955).
\(^{6}\) V. E. Mirakov, Uspekhi matem. nauk, 11, issue 3, 171 (1956).
\(^{7}\) E. E. Tamm, DAN, 103, No. 5, 769 (1955).