A. Yu. LUCHKA

A SUFFICIENT CONDITION FOR THE CONVERGENCE OF THE METHOD OF AVERAGING FUNCTIONAL CORRECTIONS

(Presented by Academician N. N. Bogolyubov, 13 V 1958)

MATHEMATICS

Consider the linear Fredholm integral equation of the second kind

\[ y(x)=\varphi(x)+\lambda\int_a^b K(x,\xi)y(\xi)\,d\xi \qquad (0<|\lambda|<\infty). \tag{1} \]

Assume that the functions \(\varphi(x)\) and \(K(x,\xi)\) are real and belong to the space \(L^2(a,b)\). Let equation (1) have a unique solution for a certain value of the parameter \(\lambda\).

The method of averaging functional corrections, set forth in \((^{1-4})\), consists in the following: in the first approximation we put

\[ y_1(x)=\varphi(x)+\lambda\alpha_1\int_a^b K(x,\xi)\,d\xi, \tag{2} \]

where

\[ \alpha_1=\frac{1}{h}\int_a^b y_1(x)\,dx \qquad (h=b-a>0). \tag{3} \]

From equalities (2) and (3) we determine \(\alpha_1\):

\[ \alpha_1=\frac{1}{D(\lambda)}\int_a^b \varphi(x)\,dx, \tag{4} \]

where

\[ D(\lambda)=h-\lambda\int_a^b\int_a^b K(x,\xi)\,d\xi\,dx. \]

In the \(n\)-th approximation we put

\[ y_n(x)=\varphi(x)+\lambda\int_a^b K(x,\xi)\bigl(y_{n-1}(\xi)+\alpha_n\bigr)\,d\xi, \tag{5} \]

where

\[ \alpha_n=\frac{1}{h}\int_a^b \delta_n(x)\,dx; \tag{6} \]

\[ \delta_n(x)=y_n(x)-y_{n-1}(x) \qquad (n=2,3,\ldots). \tag{7} \]

From equalities (5), (6), and (7) we obtain

\[ \delta_n(x)=\lambda\int_a^b K(x,\xi)\bigl(\delta_{n-1}(\xi)-\alpha_{n-1}\bigr)\,d\xi +\lambda\alpha_n\int_a^b K(x,\xi)\,d\xi; \tag{8} \]

\[ \alpha_n=\frac{\lambda}{D(\lambda)} \int_a^b\int_a^b K(x,\xi)\bigl(\delta_{n-1}(\xi)-\alpha_{n-1}\bigr)\,d\xi\,dx \qquad (n=2,3,\ldots). \tag{9} \]

Let \(D(\lambda)\ne0\). Then from the assumptions concerning the functions \(\varphi(x)\) and \(K(x,\xi)\) it follows that all the functions \(y_n(x)\), and hence also all the functions \(\delta_n(x)\), belong to the space \(L^2(a,b)\).

The functions \(\delta_n(x)\) and \(\alpha_n\) can be represented in the form

\[ \delta_n(x)=\lambda\int_a^b \bigl(K(x,\xi)-M(x)\bigr) \bigl(\delta_{n-1}(\xi)-\lambda h\alpha_{n-1}t\bigr)\,d\xi +\lambda\alpha_n h M(x); \tag{8'} \]

\[ \alpha_n=\frac{\lambda}{D(\lambda)} \int_a^b\int_a^b \bigl(K(x,\xi)-M(x)\bigr) \bigl(\delta_{n-1}(\xi)-\lambda h\alpha_{n-1}t\bigr)\,d\xi\,dx, \tag{9'} \]

where

\[ M(x)=\frac{1}{h}\int_a^b K(x,\xi)\,d\xi, \]

\(t\) is an arbitrary parameter.

Subtracting \(\lambda h\alpha_n t\) from both sides of equality \((8')\), squaring the result obtained, and integrating with respect to \(x\), we then obtain

\[ \Phi_n(t)=\int_a^b\bigl(\delta_n(x)-\lambda h\alpha_n t\bigr)^2\,dx= \]

\[ =\lambda^2\int_a^b \left\{ \int_a^b \bigl(K(x,\xi)-M(x)\bigr) \bigl(\delta_{n-1}(\xi)-\lambda h\alpha_{n-1}t\bigr)\,d\xi +\alpha_n h\bigl(M(x)-t\bigr) \right\}^2 dx . \]

Using Minkowski’s inequality, we obtain:

\[ \{\Phi_n(t)\}^{1/2}\le |\lambda| \left\{ \int_a^b \left[ \int_a^b \bigl(K(x,\xi)-M(x)\bigr) \bigl(\delta_{n-1}(\xi)-\lambda h\alpha_{n-1}t\bigr)\,d\xi \right]^2 dx \right\}^{1/2} + \]

\[ +|\lambda|h \left\{ \alpha_n^2\int_a^b \bigl(M(x)-t\bigr)^2\,dx \right\}^{1/2}. \]

Applying the Cauchy–Bunyakovsky inequality, we finally obtain the inequality

\[ \{\Phi_n(t)\}^{1/2}\le |\lambda| \left\{ \int_a^b\int_a^b \bigl(K(x,\xi)-M(x)\bigr)^2\,d\xi\,dx \right\}^{1/2} \cdot \{\Phi_{n-1}(t)\}^{1/2} + \]

\[ +|\lambda|h \left\{ \alpha_n^2\int_a^b \bigl(M(x)-t\bigr)^2\,dx \right\}^{1/2}. \tag{10} \]

From \((9')\) we have:

\[ \alpha_n^2 \le \frac{\lambda^2 h}{D^2(\lambda)}\,\Phi_{n-1}(t) \int_a^b\int_a^b \bigl(K(x,\xi)-M(x)\bigr)^2\,d\xi\,dx . \tag{11} \]

On the basis of (11), from (10) we obtain the relation

\[ \{\Phi_n(t)\}^{1/2}\leq |\lambda|\,\{\Phi_{n-1}(t)\}^{1/2} \left\{\int_a^b\int_a^b (K(x,\xi)-M(x))^2\,d\xi\,dx\right\}^{1/2}\times \]

\[ \times\left\{1+\frac{h^{3/2}|\lambda|}{|D(\lambda)|} \left[\int_a^b(M(x)-t)^2\,dx\right]^{1/2}\right\}. \]

Put

\[ t=K=\frac1{h^2}\int_a^b\int_a^b K(x,\xi)\,d\xi\,dx; \]

then

\[ \Phi_n(K)\leq \mathcal L^2\Phi_{n-1}(K), \tag{12} \]

where

\[ \mathcal L^2=\lambda^2\int_a^b\int_a^b (K(x,\xi)-M(x))^2\,d\xi\,dx \left\{1+\frac{h^{3/2}|\lambda|}{|D(\lambda)|} \left[\int_a^b(M(x)-K)^2\,dx\right]^{1/2}\right\}^2, \]

or

\[ \mathcal L^2=\lambda^2(B^2-hM^2) \left\{1+\frac{h^{3/2}|\lambda|}{|D(\lambda)|} [M^2-hK^2]^{1/2}\right\}^2; \]

\[ B^2=\int_a^b\int_a^b K^2(x,\xi)\,d\xi\,dx; \tag{13} \]

\[ M^2=\frac1{h^2}\int_a^b\left(\int_a^b K(x,\xi)\,d\xi\right)^2\,dx. \]

Since the functions \(\varphi(x)\), \(K(x,\xi)\) belong to the space \(L^2(a,b)\), it follows from equalities (2) and (4) \((D(\lambda)\ne0)\) that

\[ \Phi_1(K)\leq C \qquad (\delta_1(x)=y_1(x)). \tag{14} \]

Let \(\mathcal L^2<1\); then from (12) and (14) it follows that, as \(n\to\infty\), \(\Phi_n(K)\to0\); consequently, by (11), \(\alpha_n^2\to0\), and hence also \(\alpha_n\to0\).

From

\[ \Phi_n(K)=\int_a^b(\delta_n(x)-\lambda hK\alpha_n)^2\,dx \]

it follows that, as \(n\to\infty\),

\[ \int_a^b \delta_n^2(x)\,dx\to0; \]

i.e. the sequence of functions \(y_n(x)\) converges in itself. Since the space \(L^2(a,b)\) is complete, it follows that the sequence of functions \(y_n(x)\) converges, as \(n\to\infty\), to a function \(Y(x)\) belonging to the space \(L^2(a,b)\). It is evident that the function \(Y(x)\) is a solution of equation (1).

The derived condition \(\mathcal L^2<1\) is less restrictive than the condition given in paper \((^2)\).

In the cases \(K(x,\xi)\equiv C\) and \(K(x,\xi)\equiv K(x)\), \(B^2-hM^2=0\), \(\mathcal L^2=0\), \(\alpha_2=0\); the first approximation gives the exact solution.

If

\[ \int_a^b K(x,\xi)\,d\xi=0, \]

then \(\mathcal L^2=\lambda^2B^2\). In this case the method of averaging functional corrections degenerates into the method of successive approximations. For \(\lambda^2B^2<1\), as is known, the convergence of the method of successive approximations has been proved.

Example. Consider the simple equation

\[ y(x)=-20.2\sqrt{x}+3\int_{0}^{1}\sqrt{x}\,(\xi+10)y(\xi)\,d\xi, \]

which has the obvious solution \(y(x)=\sqrt{x}\).

For this example we have: \(B^{2}=\dfrac{331}{6}\); \(\quad M^{2}=\dfrac{441}{8}\); \(\quad K^{2}=49\); \(\quad D(\lambda)=1-\)

\[ -3\cdot 7=-20;\qquad \mathscr{L}^{2}=\frac{3}{8}\left(1+\frac{21}{80}\sqrt{2}\right)^{2}<1. \]

It should be noted that the usually employed sufficient condition for convergence of the method of successive approximations is not satisfied in the present case, since \(\lambda^{2}B^{2}=\dfrac{331}{6}\cdot 9>1\). The ordinary process of successive approximations diverges in this example. However, by the method of averaging functional corrections this equation is solved. The first and second approximations have the form

\[ y_{1}(x)=\sqrt{x}+0.01\sqrt{x}; \]

\[ y_{2}(x)=\sqrt{x}-0.0001\sqrt{x}. \]

\[ y_{1}(x)-y_{2}(x)=0.0001\sqrt{x}, \]

i.e., the relative error is \(0.01\%\).

Institute of Mathematics
Academy of Sciences of the Ukrainian SSR

Received
9 V 1958

REFERENCES

\(^{1}\) Yu. D. Sokolov, Reports of the Academy of Sciences of the Ukrainian SSR, No. 2 (1955).
\(^{2}\) Yu. D. Sokolov, Ukrainian Mathematical Journal, 9, No. 1 (1957).
\(^{3}\) Yu. D. Sokolov, Ukrainian Mathematical Journal, 9, No. 4 (1957).
\(^{4}\) Yu. D. Sokolov, Ukrainian Mathematical Journal, 10, No. 2 (1958).