ON THE DIFFERENCE BETWEEN THE METHOD OF EXPANSION IN A SERIES WITH RESPECT TO A PARAMETER AND THE METHOD OF SUCCESSIVE APPROXIMATIONS IN SOLVING NONLINEAR INTEGRAL EQUATIONS

B. A. BEL’TYUKOV

In discussing certain methods for the approximate solution of nonlinear integral equations of Hammerstein type, L. V. Ovsyannikov drew our attention to the need to investigate the methods of expansion in a series with respect to the parameter \(\lambda\) and Picard’s successive approximations, with the aim of comparing the rates of their convergence.

In the present work this and certain other questions are studied by the method of explicit majorants.

We take as a basis the generalized integral equation of Hammerstein type

\[ \varphi=\lambda K(\varphi), \tag{1} \]

where

\[ K(\varphi)=\int_a^b K[x,y,\varphi(y)]\,dy; \]

\(\lambda\) is a numerical parameter.

1. Nekrasov’s method. We shall present this method in a form somewhat different from that usually given in the literature (see, for example, [5, 4, 1, 2]), with a view to its actual use for the approximate solution of equation (1).

Suppose that \(\varphi_0(x)\) is a solution of the given equation (1) for the value \(\lambda=\lambda_0\). Putting in this equation

\[ \varphi=\varphi_0+\psi,\qquad \lambda=\lambda_0+\underline{\lambda}, \]

we shall have

\[ \varphi_0+\psi=(\lambda_0+\underline{\lambda})K(\varphi_0+\psi). \]

Subtracting from this

\[ \varphi_0=\lambda_0K(\varphi_0), \]

we may write

\[ \psi=\underline{\lambda}K(\varphi_0+\psi)+\lambda_0[K(\varphi_0+\psi)-A^{(0)}1-A^{(1)}\psi]+\lambda_0A^{(1)}\psi, \tag{2} \]

where \(A^{(0)}\) and \(A^{(1)}\) are linear Fredholm integral operators with kernels

\[ A^{(0)}(x,y)=K[x,y,\varphi_0(y)],\qquad A^{(1)}(x,y)= \left.\frac{\partial K(x,y,z)}{\partial z}\right|_{z=\varphi_0(y)} . \]

Moreover, we shall assume that \(A^{(1)}(x,y)\not\equiv 0\).

Let \(\lambda_0\) not be a characteristic number of the kernel \(A^{(1)}(x,y)\), and let \(\Gamma^{(1)}(x,y,\lambda_0)\) be the resolvent of this kernel. Then, solving (2) as a linear equation with kernel \(A^{(1)}(x,y)\), we arrive at the following equation for \(\psi(x)\):

\[ \psi=\widetilde{K}(\psi,\underline{\lambda}), \tag{3} \]

where

\[ {}''K(\psi,\lambda)=\int_a^b {}''K[x,y,\psi(y),\lambda]\,dy, \]

\[ {}''K(x,y,t,\lambda)=\lambda\,{}'K(x,y,t)+\lambda_0\bigl[{}'K(x,y,t)-{}'K^{(0)}(x,y)-{}'K^{(1)}(x,y)t\bigr], \]

\[ {}'K(x,y,t)=K[x,y,\varphi_0(y)+t]+ \]

\[ +\lambda_0\int_a^b \Gamma^{(1)}(x,\eta,\lambda_0)K[\eta,y,\varphi_0(y)+t]\,d\eta, \]

\[ {}'K^{(0)}(x,y)=A^{(0)}(x,y)+\lambda_0\int_a^b \Gamma^{(1)}(x,\eta,\lambda_0)A^{(0)}(\eta,y)\,d\eta, \]

\[ {}'K^{(1)}(x,y)=A^{(1)}(x,y)+\lambda_0\int_a^b \Gamma^{(1)}(x,\eta,\lambda_0)A^{(1)}(\eta,y)\,dy =\Gamma^{(1)}(x,y,\lambda_0). \]

Thus, finding the solution of equation (1) in a neighborhood of the point \(\lambda_0\) has been reduced to finding the solution of equation (3) in a neighborhood of the point \(\lambda=0\).

Let us show that equation (3) admits a significant simplification if the original equation (1) is an ordinary Hammerstein equation, i.e.

\[ \varphi(x)=\lambda\int_a^b K(x,y)f[y,\varphi(y)]\,dy. \tag{4} \]

In this case

\[ A^{(0)}(x,y)=K(x,y)b^{(0)}(y),\qquad A^{(1)}(x,y)=K(x,y)b^{(1)}(y), \]

where

\[ b^{(0)}(y)=f[y,\varphi_0(y)],\qquad b^{(1)}(y)=\left.\frac{\partial f(y,z)}{\partial z}\right|_{z=\varphi_0(y)}. \]

The function \({}'K(x,y,t)\) will be equal to

\[ {}'K(x,y,t)=\left\{K(x,y)+\lambda_0\int_a^b \Gamma^{(1)}(x,\eta,\lambda_0)K(\eta,y)\,d\eta\right\} f[y,\varphi_0(y)+t], \]

or

\[ {}'K(x,y,t)=\frac{\Gamma^{(1)}(x,y,\lambda_0)}{b^{(1)}(y)}\,f[y,\varphi_0(y)+t], \]

since the expression in braces is essentially the right-hand side of the resolvent equation \(\Gamma^{(1)}(x,y,\lambda_0)\), divided by \(b^{(1)}(y)\). Similarly,

\[ {}'K^{(0)}(x,y)=\frac{\Gamma^{(1)}(x,y,\lambda_0)}{b^{(1)}(y)}\,b^{(0)}(y). \]

As a result, equation (3) assumes the relatively simple form:

\[ \psi(x)=\lambda\int_a^b \frac{\Gamma^{(1)}(x,y,\lambda_0)}{b^{(1)}(y)} \,{}'f[x,y,\psi(y),\lambda]\,dy, \tag{5} \]

where

\[ {}'f(x,y,t,\lambda)=\lambda f[y,\varphi_0(y)+t]+\lambda_0\{f[y,\varphi_0(y)+t]-b^{(0)}(y)-b^{(1)}(y)t\}. \]

We shall assume that for equation (3) the function \({}'K(x,y,t)\) is representable, for the time being formally, by a power series

\[ {}'K(x,y,t)=\sum_{s=0}^{\infty}{}'K^{(s)}(x,y)t^s, \tag{6} \]

whose coefficients

\[ {}'K^{(s)}(x,y)=\left.\frac{1}{s!}\frac{\partial^s{}'K(x,y,z)}{\partial z^s}\right|_{z=0}, \quad s=0,1,\ldots \tag{7} \]

belong to the space \(L_2[a,b]\), and \({}'K^{(0)}(x,y)\ne 0\).

In the case of the ordinary Hammerstein equation, the coefficients (7) have the form

\[ {}'K^{(s)}(x,y)= \frac{\Gamma^{(1)}(x,y,\lambda_0)}{b^{(0)}(y)}\,b^{(s)}(y), \tag{8} \]

where

\[ b^{(s)}(y)= \left.\frac{1}{s!}\frac{\partial^s f(y,z)}{\partial z^s}\right|_{z=\varphi_0(y)} . \]

Let us apply to the solution of equation (3) the method of expansion in a series with respect to the parameter \(\lambda\), presenting it in the form of the proof of the following theorem.

Theorem 1. Under the conditions

\[ \|{}'K^{(0)}1\|_{L_2}\leq B,\qquad \|{}'K^{(s)}\|_{L_2}\leq \frac{B}{r^s},\quad s=1,2,\ldots \]

where \(B\) and \(r\) are certain constants, equation (3) has in the closed disk

\[ |\lambda|\leq \frac{r^*}{4B},\quad (C_\lambda), \tag{9} \]

where

\[ r^*=\frac{r}{1+|\lambda_0|\dfrac{B}{r}} \tag{10} \]

a unique solution representable by the Nekrasov series

\[ \psi^*=\sum_{n=0}^{\infty}\lambda^{n+1}\psi^{(1,n)}, \tag{11} \]

which converges in the mean. The convergence has order

\[ \|\psi^*-S_k\|_{L_2}= \begin{cases} O\!\left(k^{-3/2}\gamma^{k+1}\right), & \text{for } \gamma<1,\\[4pt] O\!\left(k^{-1/2}\right), & \text{for } \gamma=1, \end{cases} \tag{12} \]

where

\[ S_k=\sum_{n=0}^{k-1}\lambda^{n+1}\psi^{(1,n)},\qquad \gamma=|\lambda|\frac{4B}{r^*}. \]

For the proof of the theorem, let us try to satisfy equation (3) by the series (11). Substituting this series into the equation, we use expansion (6), as well as an expansion for \(\psi^s\) of the form

\[ \psi^s=\sum_{n=0}^{\infty}\lambda^{n+s}\psi^{(s,n)},\quad s=2,3,\ldots \tag{13} \]

whose coefficients we determine recursively by putting \(\psi^s=\psi\cdot\psi^{s-1}\), i.e.,

\[ \psi^{(s,n)}=\sum_{p=0}^{n}\psi^{(1,p)}\psi^{(s-1,n-p)}. \tag{14} \]

The use of expansion (14) makes it possible to establish complete recurrent formulas for the coefficients of the series (11), which are needed for the practical implementation of the method.

Using the method of undetermined coefficients, we arrive at the following relations:

\[ \psi^{(1,0)}={}'K^{(0)}1, \]

\[ \psi^{(1,n)}=\sum_{s=1}^{n}{}'K^{(s)}\psi^{(s,n-s)} +\lambda_0\sum_{s=2}^{n+1}{}'K^{(s)}\psi^{(s,n+1-s)*}, \quad n=1,2,\ldots \tag{15} \]

From these relations, with the aid of (14), one can determine step by step the coefficients \(\psi^{(1,n)}\), \(n=0,1,2,\ldots\), and construct the series (11), which formally satisfies equation (3). Let us give expressions for several of the first coefficients:

\[ \psi^{(1,0)}={}'K^{(0)}1, \]

\[ \psi^{(1,1)}={}'K^{(1)}\psi^{(1,0)}+\lambda_0{}'K^{(2)}\psi^{(2,0)}, \]

where

\[ \psi^{(2,0)}=\psi^{(1,0)^2}, \]

\[ \psi^{(1,2)}={}'K^{(1)}\psi^{(1,1)}+{}'K^{(2)}\psi^{(2,0)} +\lambda_0\left({}'K^{(2)}\psi^{(2,1)}+{}'K^{(3)}\psi^{(3,0)}\right), \]

where

\[ \psi^{(2,1)}=2\psi^{(1,0)}\psi^{(1,1)}, \]

\[ \psi^{(3,0)}=\psi^{(1,0)}\psi^{(2,0)}, \]

\[ \psi^{(1,3)}={}'K^{(1)}\psi^{(1,2)}+{}'K^{(2)}\psi^{(2,1)}+{}'K^{(3)}\psi^{(3,0)}+ \]

\[ +\lambda_0\left({}'K^{(2)}\psi^{(2,2)}+{}'K^{(3)}\psi^{(3,1)}+{}'K^{(4)}\psi^{(4,0)}\right), \]

where

\[ \psi^{(2,2)}=2\psi^{(1,0)}\psi^{(1,2)}+\psi^{(1,1)^2}, \]

\[ \psi^{(3,1)}=\psi^{(1,0)}\psi^{(2,1)}+\psi^{(1,1)}\psi^{(2,0)}, \]

\[ \psi^{(4,0)}=\psi^{(1,0)}\psi^{(3,0)}, \]

\[ \psi^{(1,4)}={}'K^{(1)}\psi^{(1,3)}+{}'K^{(2)}\psi^{(2,2)}+{}'K^{(3)}\psi^{(3,1)}+{}'K^{(4)}\psi^{(4,0)}+ \]

\[ +\lambda_0\left({}'K^{(2)}\psi^{(2,3)}+{}'K^{(3)}\psi^{(3,2)}+{}'K^{(4)}\psi^{(4,1)}+{}'K^{(5)}\psi^{(5,0)}\right), \]

\[ \text{* Taking (8) into account, it is easy to refine these formulas for the case of equation (5).} \]

where

\[ \psi^{(2,3)}=2\psi^{(1,0)}\psi^{(1,3)}+2\psi^{(1,1)}\psi^{(1,2)}, \]

\[ \psi^{(3,2)}=\psi^{(1,0)}\psi^{(2,2)}+\psi^{(1,1)}\psi^{(2,1)}+\psi^{(1,2)}\psi^{(2,0)}, \]

\[ \psi^{(4,1)}=\psi^{(1,0)}\psi^{(3,1)}+\psi^{(1,1)}\psi^{(3,0)}, \]

\[ \psi^{(5,0)}=\psi^{(1,0)}\psi^{(4,0)} \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

In order to regard the constructed series (11) as a genuine solution of equation (3), we shall determine the domain of values of \(\lambda\) for which it converges in the mean. We shall show that

\[ \left\|\sum_{n=n_0+1}^{n=n_0+p}\lambda^{n+1}\psi^{(1,n)}\right\|_{L_2}\to 0 \quad \text{as } n_0\to\infty \tag{15} \]

in the circle \(C_\lambda\) (9).

From (15), by virtue of the conditions of Theorem 1, we shall have

\[ \left\|\psi^{(1,0)}\right\|_{L_2}\leq B, \]

\[ \left\|\psi^{(1,n)}\right\|_{L_2} \leq \sum_{s=1}^{n}\frac{B}{r^s}\left\|\psi^{(s,n-s)}\right\|_{L_2} +\lambda_0\sum_{s=2}^{n+1}\frac{B}{r^s}\left\|\psi^{(s,n+1-s)}\right\|_{L_2}, \tag{16} \]

moreover, according to (14),

\[ \left\|\psi^{(s,n)}\right\|_{L_2} \leq \sum_{p=0}^{n}\left\|\psi^{(1,p)}\right\|_{L_2} \left\|\psi^{(s-1,n-p)}\right\|_{L_2}. \tag{17} \]

We introduce into the proof a nonlinear algebraic equation with respect to \(\Phi\):

\[ \Phi=\lambda\frac{B}{1-\dfrac{\Phi}{r}} +\lambda_0\frac{B}{1-\dfrac{\Phi}{r}}\frac{\Phi^2}{r^2}. \tag{18} \]

It is easy to verify that this equation has the solution

\[ \Phi=\frac{r^*}{2}\left(1-\sqrt{1-\lambda\frac{4B}{r^*}}\right), \]

where \(r^*\) is (10). In the circle \(C_\lambda\) it expands into a power series, similar to the series (11), of the form

\[ \Phi=\sum_{n=0}^{\infty}\lambda^{n+1}\Phi^{(1,n)}, \tag{19} \]

where

\[ \Phi^{(1,n)}= \frac{r^*}{2}\, \frac{(2n)!}{2^{2n+1}n!(n+1)!} \left(\frac{4B}{r^*}\right)^{n+1}. \tag{20} \]

We shall now find the coefficients \(\Phi^{(1,n)}\) of the series (19), starting from equation (18), in a way analogous to that by which the coefficients of the series (11) were found.

In the present case the role of the expansions (6) and (13) is played, respectively, by

\[ \frac{B}{1-\dfrac{t}{r}}=\sum_{s=0}^{\infty}\frac{B}{r^s}t^s,\qquad \Phi^s=\sum_{p=0}^{n}\lambda^{\,n+s}\Phi^{(s,n)}, \]

where

\[ \Phi^{(s,n)}=\sum_{p=0}^{n}\Phi^{(1,p)}\Phi^{(s-1,n-p)}. \]

Similarly to (15), we obtain

\[ \Phi^{(1,0)}=B, \]

\[ \Phi^{(1,n)}=\sum_{s=1}^{n}\frac{B}{r^s}\Phi^{(s,n-s)} +\lambda_0\sum_{s=2}^{n+1}\frac{B}{r^s}\Phi^{(s,n+1-s)}. \]

Comparing the last formulas with inequalities (16) for \(n=0,1,2,\ldots\), it is not difficult to see that

\[ \|\psi^{(1,n)}\|_{L_2}\leq \Phi^{(1,n)}, \tag{21} \]

\[ \|\psi^{(s,n)}\|_{L_2}\leq \Phi^{(s,n)},\qquad s=2,3,\ldots \tag{22} \]

Using inequalities (21), we may write:

\[ \left\|\sum_{n=n_0+1}^{n=n_0+p}\lambda^{\,n+1}\psi^{(1,n)}\right\|_{L_2} \leq \sum_{n=n_0+1}^{n=n_0+p}|\lambda|^{\,n+1}\Phi^{(1,n)} < \sum_{n=n_0+1}^{\infty}|\lambda|^{\,n+1}\Phi^{(1,n)}. \]

The last sum is the remainder of the series (19), convergent in the disk \(C_\lambda\), which proves assertion \((15')\). Thus, Nekrasov’s series (11) converges in the mean for \(\lambda\in C_\lambda\). By a well-known theorem of analysis, the sum of the series \(\psi^*(x)\in L_2[a,b]\).

Further, from the very method of obtaining the coefficients \(\psi^{(1,n)}\) it follows that equation (3) has no other solution of the form (11).

Finally, let us estimate the rate of convergence of the series (11). Denoting

\[ S_k=\sum_{n=0}^{k-1}\lambda^{\,n+1}\psi^{(1,n)}, \]

we have, by virtue of (21),

\[ \|\psi^*-S_k\|_{L_2} \leq \sum_{n=k}^{\infty}|\lambda|^{\,n+1}\Phi^{(1,n)}. \tag{23} \]

Let \(\gamma=|\lambda|\dfrac{4B}{r^*}<1\); then, using (20), we derive

\[ \|\psi^*-S_k\|_{L_2} < \frac{r^*}{2}\, \frac{(2k)!}{2^{2k+1}k!(k+1)!}\, \frac{\gamma^{\,k+1}}{1-\gamma}. \]

From this, applying to the factorials the well-known Stirling formula, we easily arrive at the first estimate (12).

Let \(\gamma=1\). Applying Stirling’s formula again, we derive

\[ \|\psi^* - S_k\|_{L_2} < \frac{r^*}{2}\,\frac{1}{2\sqrt{n}}\sum_{n=k}^{\infty} \frac{1}{\sqrt{\pi}(n+1)} < \]

\[ < \frac{r^*}{2}\,\frac{1}{\sqrt{\pi}}\sum_{n=k}^{\infty} \frac{1}{(n+1)^{3/2}} < \frac{r^*}{2}\,\frac{1}{\sqrt{\pi}}\,\frac{1}{2\sqrt{k}}, \]

which confirms the second estimate (12). The theorem is proved.

Analyzing the estimates (12), we see that if the value of \(\lambda\) for which it is required to find an approximate solution of equation (3) lies inside the circle \(C_\lambda\) and is removed from its boundary (\(\gamma\) is small), then the Nekrasov series (11) converges sufficiently rapidly and can be successfully used to obtain an approximate solution of this equation.

A rigorous estimate of the error of the approximate solution can be carried out with the aid of inequality (23), rewriting it as

\[ \|\psi^* - S_k\|_{L_2} \le \frac{r^*}{2} \left( 1-\sqrt{1-\gamma} - \sum_{n=0}^{k-1} \frac{(2n)!}{2^{2n+1} n!(n+1)!}\,\gamma^{n+1} \right). \tag{24} \]

This error estimate is a priori, since it contains no quantities connected with the solution of equation (3).

Theorem 2. If there exists a constant \(A\) such that, together with the conditions of Theorem 1, the inequalities

\[ \|K^{(0)}1\|_M \le A^*, \qquad \left\{ \sup_x \int_a^b |'K^{(s)}(x,y)|^2\,dy \right\}^{1/2} \le \frac{A}{r^s}, \quad s=1,2,\ldots, \tag{25} \]

hold, then the Nekrasov series (11) converges regularly in the circle \(C_\lambda\).

Proof. From relations (15), by virtue of the conditions of the present theorem,

\[ \|\psi^{(1,0)}\|_M=\|K^{(0)}1\|_M \le A, \]

\[ \|\psi^{(1,n)}\|_M = \sum_{s=1}^{n} \left\{ \sup_x \int_a^b |'K^{(s)}(x,y)|^2\,dy \right\}^{1/2} \|\psi^{(s,n-s)}\|_{L_2} + \]

\[ + |\lambda_0| \sum_{s=2}^{n+1} \left\{ \sup_x \int_a^b |'K^{(s)}(x,y)|^2\,dy \right\}^{1/2} \|\psi^{(s,n+1-s)}\|_{L_2} \le \]

\[ \le \sum_{s=1}^{n} \frac{A}{r^s}\, \|\psi^{(s,n-s)}\|_{L_2} + |\lambda_0| \sum_{s=2}^{n+1} \frac{A}{r^s}\, \|\psi^{(s,n+1-s)}\|_{L_2}, \quad n=1,2,\ldots \]

Substituting here inequalities (21) and (22), we obtain

\[ \|\psi^{(1,0)}\|_M \le \frac{A}{B}\,\Phi^{(1,0)}, \]

\(*\) In the present paper the letter \(M\) denotes the space of functions bounded on \([a,b]\), with
\[ \|\psi\|_M=\sup_x |\psi(x)|. \]

\[ \|\psi^{(1,n)}\|_M \leq \frac{A}{B}\left[\sum_{s=1}^{n}\frac{B}{r^s}\Phi^{(s,n-s)}+|\lambda_0|\sum_{s=2}^{n+1}\frac{B}{r^s}\Phi^{(s,n+1-s)}\right]\leq \frac{A}{B}\Phi^{(1,n)}, \]

\[ n=\overline{1,\infty}. \]

As we see, the series (11) is majorized by the series (19), multiplied by \(\dfrac{A}{B}\). Consequently, the series (11) converges regularly, at least in the disk \(C_\lambda\).

The theorem is proved.

Analogously to (24), the estimate

\[ \|\psi^*-S_k\|_M \leq \frac{A}{B}\frac{r^*}{2}\left(1-\sqrt{1-\gamma}-\sum_{n=0}^{k-1}\frac{(2n)!}{2^{2n+1}n!(n+1)!}\gamma^{n+1}\right). \tag{26} \]

holds.

Let, in the particular case, \(\lambda_0=0\) and, consequently, \(\varphi_0=0\). Then, as is known, the Nekrasov—Nazarov method becomes the Neumann method. Equation (3) coincides with the given equation (1). The role of the expansion (6) is now played by

\[ K(x,y,t)=\sum_{s=0}^{\infty}K^{(s)}(x,y)t^s, \tag{27} \]

where

\[ K^{(s)}(x,y)=\frac{1}{s!}\frac{\partial^s K(x,y,z)}{\partial z^s}\bigg|_{z=0}. \]

The series (11) becomes the ordinary Neumann series

\[ \varphi^*=\sum_{n=0}^{\infty}\lambda^{n+1}\varphi^{(1,n)}. \tag{28} \]

The formulas for the coefficients of the series take the form

\[ \varphi^{(1,0)}=K^{(0)}1,\qquad \varphi^{(1,n)}=\sum_{s=1}^{n}K^{(s)}\overline{\varphi}^{(s,n-s)},\qquad n=1,2,\ldots, \tag{29} \]

where

\[ \varphi^{(s,n)}=\sum_{p=1}^{n}\varphi^{(1,p)}\varphi^{(s-1,n-p)}. \]

Further, since in the present case \(r^*=r\), the circle of convergence of the series (28) has the following form:

\[ |\lambda|\leq \frac{r}{4B}\quad (C_\lambda). \tag{30} \]

Let us consider an example of the use of Theorems 1 and 2 in this particular case. Take the equation

\[ \varphi(x)=\lambda\int_{0}^{1} xye^{-\frac{x^2y^2\varphi(y)}{4}}\,dy. \tag{31} \]

Here

\[ K(x,y,t)=xy e^{-\frac{x^2y^2t}{4}}, \qquad K^{(s)}(x,y)=\frac{(xy)^{2s+1}}{4^s s!}, \qquad s=0,1,\ldots . \]

Let us compute

\[ \|K^{(0)}1\|_{L_2}=\frac{1}{2\sqrt{3}}, \qquad \|K^{(s)}\|_{L_2}=\frac{1}{4^s s!(4s+3)}, \qquad s=1,2,\ldots . \]

Consequently, one may put

\[ B=\frac{1}{2\sqrt{3}}, \qquad r=\frac{14}{\sqrt{3}}. \]

Next let us establish the existence of the constant \(A\). We have

\[ \|K^{(0)}1\|_{M}=\frac{1}{2}, \qquad \left\{\sup_x \int_a^b |K^{(s)}(x,y)|^2\,dy\right\}^{\frac12} =\frac{1}{4^s s!\sqrt{4s+3}}, \qquad s=1,2,\ldots . \]

Hence

\[ A=\frac{\sqrt{7}}{2\sqrt{3}} . \]

Thus, equation (31) has a unique solution in the form of the series (28), converging uniformly at least in the disk

\[ |\lambda|\le 7 . \]

We shall now construct an approximate solution of equation (31), restricting ourselves in the Neumann series to three terms

\[ S_3=\lambda\varphi^{(1,0)}+\lambda^2\varphi^{(1,1)}+\lambda^3\varphi^{(1,2)} . \]

We take the value of \(\lambda\) equal to \(\dfrac14\). First, for preliminary orientation, let us estimate the error of this solution using (26). In the present example

\[ \gamma=|\lambda|\frac{4B}{r}=\frac{1}{28}. \]

Substituting all this into (26) (\(k=3\)) and carrying out the usual computations, we find

\[ \|\varphi^*-S_3\|_{M}<\frac{1}{10^6}. \tag{32} \]

The terms of the sum \(S_3\) are easily computed, together with \(\lambda\), from formulas (29). As a result,

\[ S_3(x)=\frac{x}{8}+\frac{113}{71680}x^3+\frac{x^5}{65536}. \]

To judge how much estimate (32) overstates the true error of the obtained solution, let us also write \(S_4(x)\):

\[ S_4(x)=\frac{x}{8}+\left(\frac{113}{71680}+0.0000002\right)x^3+ \]

\[ +\left(\frac{1}{65536}+0.0000003\right)x^5+0.0000001x^7. \]

2. Solution of equation (3) by the method of successive approximations.
The iteration formulas have the form:

\[ \psi_{(n)}=\mathcal{K}(\psi_{(n-1)},\lambda), \qquad n=1,2,3,\ldots \tag{33} \]

Concerning the convergence of the process, the following holds.

Theorem 3. If the conditions of Theorem 1 are satisfied and the value \(\lambda\) lies inside the circle \(C_\lambda\), then, independently of the choice of the initial approximation \(\psi_{(0)}\) from the domain

\[ \|\psi\|_{L_2} \leq \frac{r^*}{2}\left(1-\sqrt{1-\gamma}\right) \tag{34} \]

the process (33) converges on the average to the solution \(\psi^*\) of equation (3), which lies in the domain (34) and is unique in the domain

\[ \|\psi\|_{L_2}<\frac{r^*}{2}. \tag{35} \]

The convergence has order

\[ \|\psi^*-\psi_{(n)}\|_{L_2}=O(\alpha^n), \tag{36} \]

where

\[ \alpha=\frac{\gamma(1+\chi)} {(1+\sqrt{1-\gamma})(1+\chi+\sqrt{1-\gamma})}, \qquad \chi=2|\lambda_0|\frac{B}{r}. \]

The validity of this theorem follows from the well-known Banach theorem (see, for example, [3], Ch. III, § 1). We shall show this. Define the ball \(T\) by inequality (34) and find the Lipschitz constant \(\alpha\). Let \(\psi_1,\psi_2\in T\); then

\[ \|\,''K(\psi_2,\lambda)-''K(\psi_1,\lambda)\|_{L_2} \leq \|\lambda\omega+\lambda_0(\omega-{}'K^{(1)}1)\|_{L_2} \|\psi_2-\psi_1\|_{L_2}, \]

where

\[ \omega=\sum_{s=1}^{\infty}K^{(s)}(\psi_2^{s-1}+\psi_2^{s-2}\psi_1+\cdots+\psi_1^{s-1}). \]

By virtue of the conditions of Theorem 1 and the inequality defining the ball \(T\), we shall have

\[ \begin{aligned} \|\lambda\omega+\lambda_0(\omega-{}'K^{(1)}1)\|_{L_2} &\leq |\lambda|\,\|\omega\|_{L_2} +|\lambda_0|\,\|\omega-{}'K^{(1)}1\|_{L_2}\leq \\ &\leq |\lambda|\sum_{s=1}^{\infty}\frac{B}{r^s}\, s\left[\frac{r^*}{2}\left(1-\sqrt{1-\gamma}\right)\right]^{s-1} +|\lambda_0|\sum_{s=2}^{\infty}\frac{B}{r^s}\times \\ &\quad{}\times s\left[\frac{r^*}{2}\left(1-\sqrt{1-\gamma}\right)\right]^{s-1} = \frac{\gamma(1+\chi)} {(1+\sqrt{1-\gamma})(1+\chi+\sqrt{1-\gamma})}, \end{aligned} \]

where \(\chi=2|\lambda_0|\,B/r\). Consequently, one may take

\[ \alpha=\frac{\gamma(1+\chi)} {(1+\sqrt{1-\gamma})(1+\chi+\sqrt{1-\gamma})}. \]

It is clear that inside the circle \(C_\lambda\) we shall have \(\alpha<1\), since \(\gamma<1\). We note in passing that, within the imposed conditions, the value found for the Lipschitz constant \(\alpha\) cannot be decreased.

Next we shall show that if \(\psi\in T\), then \( ''K(\psi,\lambda)\in T\). We have

\[ \|\,''K(\psi,\lambda)\|_{L_2} \leq |\lambda|\sum_{s=0}^{\infty}\frac{B}{r^s} \left[\frac{r^*}{2}\left(1-\sqrt{1-\gamma}\right)\right]^s + \]

\[ +|\lambda_0|\sum_{s=2}^{\infty}\frac{B}{r^s} \left[\frac{r^*}{2}\left(1-\sqrt{1-\gamma}\right)\right]^s = \frac{r^*}{2}\left(1-\sqrt{1-\gamma}\right), \]

i.e. \(K(\psi,\lambda)\in T\).

As a result we conclude that the process (33) converges in the mean to the solution \(\psi^*\) of equation (3), lying in the ball (34) and unique in this ball. From the possibility of increasing the radius of the ball (34) to one arbitrarily close to the radius of the ball (35), there follows the uniqueness of the solution \(\psi^*\) in the entire ball (35).

Using inequality (1.4) from the above-mentioned Banach theorem [3], we may write

\[ \left\|\psi^*-\psi_{(n)}\right\|_{L_2} \leq \frac{\alpha^n}{1-\alpha} \left\|\psi_{(1)}-\psi_{(0)}\right\|_{L_2}, \tag{37} \]

whence (36) follows. The theorem is proved.

We note in passing that from inequality (37) one can easily obtain a practically convenient estimate of the form

\[ \left\|\psi-\psi_{(n)}\right\|_{L_2} \leq \frac{\alpha}{1-\alpha} \left\|\psi_{(n)}-\psi_{(n-1)}\right\|_{L_2}. \tag{38} \]

For this it is enough to put \(n=1\) in (37), and then take, respectively, \(\psi_{(n-1)}\) and \(\psi_{(n)}\) as \(\psi_{(0)}\) and \(\psi_{(1)}\).

The solution of equation (3) obtained in Theorem 3 obviously coincides with the solution in the form of the Nekrasov series if \(\lambda\) lies inside \(C_\lambda\).

A comparison of the estimates (12) and (36), supplied by Theorems 1 and 3, shows a higher order of convergence of the iterations (33) than of the Nekrasov series.

It is interesting to note that the convergence of the process (33) remains sufficiently fast in a circle that is only slightly smaller than the circle \(C_\lambda\). For example, \(\alpha\leq \frac12\) if the value of \(\lambda\) lies in the circle

\[ |\lambda|\leq \frac{3\chi^2+10\chi+8}{4\chi^2+12\chi+9}\, \frac{r^*}{4B}. \tag{39} \]

In the case when \(\lambda_0=0\) (\(\varphi_0=0\)), the process (33) turns into the process

\[ \varphi_{(n)}=\lambda K(\varphi_{(n-1)}),\quad n=1,\ 2,\ 3,\ldots, \tag{40} \]

which converges to the solution of the original equation (1) inside the circle \(C_\lambda\) (30). This solution, instead of (34), lies in the domain

\[ \|\varphi\|_{L_2}\leq \frac{r}{2}\left(1-\sqrt{1-\gamma}\right), \quad \gamma=|\lambda|\,\frac{4B}{r}, \tag{41} \]

and is unique in the domain

\[ \|\varphi\|_{L_2}<\frac{r}{2}. \]

For example, for equation (31) these domains respectively have the form

\[ \|\varphi\|_{L_2}\leq 0.072825,\quad \|\varphi\|_{L_2}<4.041452 \quad \left(\lambda=-\frac14\right). \]

According to the foregoing, the successive approximations (40) have a higher order of convergence than the Neumann series (28). The disk (39), in which the convergence has order not lower than \(\left(\frac12\right)^n\), is here quite close to the disk \(C_\lambda\):

\[ |\lambda|\leq \frac{8}{9}\frac{4B}{r}. \]

However, it should be said that the quadratures which must be computed here at each step are usually more complicated than the quadratures determining the coefficients of the Neumann series. For illustration, let us turn to example (31). Let us try to solve it by the method of successive approximations, taking \(\varphi_0=\frac1{16}\). Then the first approximation

\[ \varphi_{(1)}(x)=\frac{8}{x}\left(e^{\frac{x^2}{64}}-1\right) \]

is easily computed. But already the second approximation

\[ \varphi_{(2)}(x)=\frac14\int_0^1 x y e^{2x^2y\left(e^{\frac{y^2}{64}}-1\right)}\,dy \]

is difficult to compute.

Theorem 4. If the conditions of Theorems 3 and 2 are satisfied, then, independently of the choice of \(\psi_{(0)}\) from the domain (34), the process (33) converges uniformly to the solution \(\psi^*\) of equation (3) lying in the domain

\[ \|\psi\|_M \leq \frac{A}{B}\frac{r^*}{2}\left(1-\sqrt{1-\gamma}\right). \]

For the sake of brevity we shall not give the proof of the theorem. Instead of estimate (38), the following estimate is valid here:

\[ \|\psi-\psi_{(n)}\|_M \leq \frac{A}{B}\frac{\sigma}{1-\alpha}\|\psi_{(n)}-\psi_{(n-1)}\|_{L_2}. \]

Moreover, thanks to the use of norms from different spaces, this estimate is less restrictive and more accurate than the analogous estimate carried out entirely in the space \(M\).

3. Continuation of the solution beyond the disk \(C_\lambda\). Let us return to the Neumann series (28). As is known, the full disk of convergence of this series has the form

\[ |\lambda|<|\lambda_1^{(1)}|, \tag{42} \]

where \(\lambda_1^{(1)}\) is the first, in modulus, characteristic number of the kernel

\[ A^{(1)}(x,y,\lambda)=\left.\frac{\partial K(x,y,z)}{\partial z}\right|_{z=\varphi^*(y)} \tag{43} \]

(see [4]). It is clear that it is practically very difficult to compute this number \(\lambda_1^{(1)}\). At the same time it is comparatively simple to construct the disk \(C_\lambda\) (30), in which the Neumann series necessarily converges to the solution.

Denote by \(\lambda_0\neq 0\) a point lying in the disk \(C_\lambda\). Let \(\varphi_0\) be the solution of the original equation (1) corresponding to \(\lambda=\lambda_0\). To find \(\varphi_0\)

can be done by means of the Neumann series or by the method of successive approximations.

From general considerations it is clear that any value of \(\lambda\) belonging to the circle \(C_\lambda\) (in particular \(\lambda=\lambda_0\)) will not be a characteristic number of the kernel (43). This can also be proved directly.

For the kernel \(A^{(1)}(x,y,\lambda)\), a circle free of characteristic numbers has the form

\[ |\lambda|<\frac{1}{\|A^{(1)}\|_{L_2}} . \tag{44} \]

Taking into account inequality (41) and expansion (27), we obtain

\[ \|A^{(1)}\|_{L_2} \leq \sum_{s=1}^{\infty} s\|K^{(s)}\|_{L_2}\|\varphi\|_{L_2}^{s-1} \leq \frac{\dfrac{4B}{r}}{\left(1+\sqrt{1-\gamma}\right)^2} \leq \frac{4B}{r}. \]

As we see, the radius of the circle (44) is not smaller than the radius of the circle \(C_\lambda\).

This circumstance is very convenient in the sense that it makes it possible to find \(\Gamma^{(1)}(x,y,\lambda_0)\) from the resolvent equation by means of the ordinary method of successive approximations. The convergence will then be of order \(O(q_0^n)\), where

\[ q_0=\frac{\gamma_0}{(1+\sqrt{1-\gamma_0})^2},\qquad \gamma_0=|\lambda_0|\frac{4B}{r}. \]

Having determined \(\varphi_0(x)\) and \(\Gamma^{(1)}(x,y,\lambda_0)\) and having formed equation (3), we can, by means of Nekrasov’s series or by the method of successive approximations, find the solution of the original equation in the circle with center at the point \(\lambda_0\) and radius

\[ \frac{r^*}{4B}. \]

Under favorable conditions this circle may extend beyond the circle \(C_\lambda\) and possibly even beyond the circle (42). This will make it possible to find the solution of equation (1) for certain values of \(\lambda\) not belonging to \(C_\lambda\), or even to the circle (42).

In accordance with what has been said, a numerical solution of equation (1) can be obtained if, in the formulas derived above, the integrals are replaced by quadrature sums.

References

1. Akhmedov K. T. UМN, 12, no. 4 (76), 1957, pp. 135–153.
2. Vainberg M. M. and Trenogin V. A. UМN, 17, no. 2 (104), 1962, pp. 13–75.
3. Krasnosel’skii M. A. Topological Methods in the Theory of Nonlinear Integral Equations. Moscow, Gostekhizdat, 1956.
4. Nazarov N. N. Nonlinear integral equations of Hammerstein type. Proceedings of SAGU, 5, Mathematics, issue 33, 1941.
5. Nekrasov A. I. Exact Theory of Waves of Steady Form on the Surface of a Heavy Fluid. Moscow, Publishing House of the USSR Academy of Sciences, 1951.

Received by the editors
January 29, 1965

Irkutsk Pedagogical Institute