Abstract Generated abstract
This paper studies convergence of the method of averaging functional corrections for linear Fredholm integral equations of the second kind with square integrable real kernel and inhomogeneous term. It formulates the iterative scheme, estimates successive corrections in the \(L^2\) norm using Minkowski and Cauchy inequalities, and derives an explicit sufficient condition \(\mathcal L^2 < 1\) involving integral characteristics of the kernel and the parameter. Under this condition the approximations converge in \(L^2\) to the unique solution, and the criterion is stated to be less restrictive than a previous one. An example shows convergence of the averaging method in a case where the usual sufficient condition for successive approximations fails and the ordinary successive approximation process diverges.
Full Text
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).