UDC 517.948.32

DEGENERATE CASES OF SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNEL

F. D. GAKHOV

1°. Let \(L\) be a smooth closed contour. A general singular integral equation with Cauchy kernel can be written in the form

\[ K\varphi \equiv a(t)\varphi(t)+\frac{b(t)}{\pi i}\int_L \frac{\varphi(\tau)}{\tau-t}\,d\tau+\int_L k(t,\tau)\varphi(\tau)\,d\tau=f(t). \tag{A} \]

The sum of the first two terms on the left-hand side is called the characteristic part, and the last term the regular part of the equation. If \(a(t)\pm b(t)\) nowhere vanish on the contour \(L\), then equation (A) is called normally solvable. If this expression can vanish at a finite number of points, then the so-called exceptional (singular) cases occur. If

\[ a(t)\pm b(t)\equiv 0, \tag{B} \]

then such a case will be called degenerate. The study of the properties of equation (A) in this case is the subject of the present investigation.

Let us recall some known facts from the theory of singular integral equations (see, for example, [1], Ch. III).

The theory of singular integral equations of class (A) is based on the possibility of their regularization, i.e., reduction to a Fredholm integral equation. In the normal case the regularization process can be carried out in three ways. The first two are based on composition from the left or from the right of the operator under study with its regularizer. The third method (Carleman–Vekua) consists in using the formula for the explicit solution of the characteristic equation. In the normal case all three methods of regularization were used; in exceptional cases only the last was applied. The regularized equation may turn out not to be equivalent to the original singular equation, and under composition from the left and regularization, extraneous solutions may be introduced into the solution of the characteristic equation, while under composition from the right solutions may be lost.

The properties of equation (A) in the normal case are characterized by the following three assertions, usually called Noether theorems.

1. The number of linearly independent solutions of the corresponding homogeneous equation is finite.

2. For the solvability of the nonhomogeneous equation it is necessary and sufficient that the condition

\[ \int_L f(t)\psi(t)\,dt=0 \]

be satisfied, where \(\psi(t)\) is the general solution of the homogeneous equation transposed (adjoint) to the given one.

1. The difference between the numbers of linearly independent solutions of the given homogeneous equation and of its transpose is equal to the index of the equation

\[ \varkappa=\operatorname{Ind}\frac{a(t)-b(t)}{a(t)+b(t)}. \]

In exceptional cases, the first Noether theorem remains valid. The other two theorems, if one remains in the class of functions integrable in the usual sense, become invalid. However, if solutions having polar singularities of sufficiently high order are regarded as admissible, with integration understood in the sense of generalized functions, then all three Noether theorems can be preserved also in exceptional cases. This was shown by L. A. Chikin [4]. In this connection, the integral of a function having a polar singularity was understood as the finite part in the sense of Hadamard.

In the degenerate case investigated by us, the question of regularization and the validity of the Noether theorems will be considered. The results obtained differ essentially from those formulated above for the normal and exceptional cases.

\(2^\circ\). We begin by deriving several auxiliary relations. Let the smooth closed contour \(L\) divide the plane into two domains, the interior \(D^{+}\) and the exterior \(D^{-}\). Consider the Cauchy-type integral

\[ \frac{1}{2\pi i}\int_L \frac{\varphi(\tau)}{\tau-z}\,d\tau = \begin{cases} \varphi^{+}(z), & z\in D^{+},\\ \varphi^{-}(z), & z\in D^{-}, \end{cases} \tag{1} \]

whose density \(\varphi(\tau)\) is a function satisfying a Hölder condition. Denote by

\[ S\varphi \equiv \frac{1}{\pi i}\int_L \frac{\varphi(\tau)}{\tau-t}\,d\tau,\qquad t\in L \tag{2} \]

the singular integral, understood in the sense of the principal value. For the limiting values on the contour of the piecewise analytic function \(\varphi^{\pm}(z)\) defined by the integral (1), the Sokhotski formulas hold:

\[ \varphi^{\pm}(t)=\pm \frac{1}{2}\varphi(t)+\frac{1}{2}S\varphi \tag{3} \]

or

\[ \varphi(t)=\varphi^{+}(t)-\varphi^{-}(t),\qquad S\varphi=\varphi^{+}(t)+\varphi^{-}(t). \tag{4} \]

Let \(J\) be the identity operator \((J\varphi\equiv\varphi)\). Introduce the operators \(M^{\pm}\) by the formulas

\[ M^{\pm}\varphi \equiv \frac{1}{2}(\pm J+S)\varphi. \tag{5} \]

Then the Sokhotski formulas (3) can be written in the form

\[ \varphi^{\pm}(t)=M^{\pm}\varphi. \]

For a function of two variables \(k(t,\tau)\), satisfying the Hölder condition in both variables, we shall use the notation

\[ M_t^{\pm}k = \pm \frac{1}{2}k(t,\tau) + \frac{1}{2\pi i}\int_L \frac{k(t_1,\tau)}{t_1-t}\,dt_1 = \frac{1}{2}(\pm J+S_t)k = k_t^{\pm}(t,\tau). \tag{6} \]

The same operations, performed with respect to the second variable \(\tau\), will be denoted by the subscript \(\tau\).

By the formulas for interchanging the order of integration in a double singular integral, for a closed contour the operator \(S\) satisfies the condition

\[ S^2=J. \tag{7} \]

On the basis of the last formula, as well as the obvious relations \(J^2=J,\ SJ=JS=S\), it is easy to derive the following properties of the operators \(M^{\pm}\):

\[ M^{+}M^{+}=M^{+},\quad M^{-}M^{-}=-M^{-},\quad M^{+}M^{-}=M^{-}M^{+}\equiv 0. \tag{8} \]

\(3^\circ\). Let us proceed to the consideration of equation \((A)\) under condition \((B)\). We shall assume that \(a(t)\) nowhere vanishes.* By division by \(\pm a(t)\) we can reduce equation \((A)\) to the form

\[ K\varphi=-\frac{1}{2}\varphi(t)+\frac{1}{2\pi i}\int_L\frac{\varphi(\tau)}{\tau-t}\,d\tau+\int_L k(t,\tau)\varphi(\tau)\,d\tau=f(t). \tag{9} \]

Introducing the short notation \(k\varphi\equiv \int_L k(t,\tau)\varphi(\tau)d\tau\), the last equation may be written in the form

\[ K\varphi\equiv M^{\pm}\varphi+k\varphi=f. \tag{9'} \]

We transform equation (9) into a system of two equations equivalent to it. Apply to (9), from the left, the operators \(M^{\pm}\), \(M^{\mp}\). Taking into account formulas (8), (6), and the admissibility of interchanging the order of integration in the repeated integral, where only one integral is singular, we obtain the equations

\[ M^{\pm}K\varphi=M^{\pm}\varphi+k_t^{\pm}\varphi=f^{\pm}, \tag{10} \]

\[ M^{\mp}K\varphi=k_t^{\mp}\varphi=f^{\mp}. \tag{11} \]

As is known ([1], Sec. 22.4), composition from the left cannot lead to a loss of solutions; therefore every solution of the original equation (9) satisfies both equations (10), (11). Subtracting termwise equality (11) from equality (10) and using the first of formulas (4), we arrive at the original equation (9). Consequently, every function satisfying both equations (10), (11) also satisfies equation (9). Thus the singular equation (9) is equivalent to the two equations (10), (11), one of which is also singular, but with a simpler kernel and right-hand side, while the other is a Fredholm equation of the first kind.

\(4^\circ\). Let us consider the question of regularization. The most general regularizer for the operator \(K\) may be taken in the form

\[ \widetilde K\psi=-\frac{1}{2}u(t)\psi(t)+\frac{u(t)}{\pi i}\int \frac{\psi(\tau)}{\tau-t}\,d\tau+\int \widetilde k(t,\tau)\psi(\tau)d\tau, \]

where \(u(t)\), \(\widetilde k(t,\tau)\) are arbitrary functions satisfying the Hölder condition ([1], Sec. 22.2). Setting \(u(t)\equiv 1\), \(\widetilde k(t,\tau)\equiv 0\), we obtain the simplest regularizing operator in the form \(\widetilde K\psi=M^{\mp}\psi\). Applying it to the equa—

* Allowing zeros of \(a(t)\) at isolated points would lead to an equation whose kernel has fixed polar singularities. This is a special question which we do not touch upon.

to (9′) on the left and proceeding as in the preceding item, we again arrive at a Fredholm equation of the first kind

\[ M^{\mp}K\varphi \equiv K^{\mp}\psi \equiv \int k_t^{\mp}(t,\tau)\varphi(\tau)\,d\tau=f^{\mp}(t). \]

Thus, regularization on the left reduces the special integral equation in the degenerate case to a Fredholm equation of the first kind. The same result would obviously also be obtained by regularization on the right. This is the first essential difference between the degenerate case and the normal and exceptional cases studied earlier. The regularized equation, generally speaking, will not be equivalent to the original one. From the study in the preceding item it follows that the original equation (9) is satisfied only by those solutions of the regularized equation (11) which simultaneously satisfy the special equation (10). We shall not here go further into the investigation of questions of equivalence.

5°. We shall now use the method of regularization by solving the characteristic equation. In the degenerate case under consideration all the arguments could easily have been carried out directly, without resorting to the general theory. But in order to give the results greater generality we shall follow the general theory. The characteristic equation corresponding to the complete equation under investigation will have the form

\[ K^0\varphi \equiv M^{\pm}\varphi=h. \tag{12} \]

Applying the operator \(M^{\mp}\) to both sides, we obtain that, for the solvability of (12), the condition

\[ M^{\mp}h \equiv h^{\mp}(t)=0 \tag{13} \]

must be satisfied. That is, on the basis of (4), equation (12) must have the form

\[ K^0\varphi \equiv M^{\pm}\varphi=h^{\pm}. \tag{14} \]

Suppose that this condition is fulfilled. Introducing, on the basis of the general theory, for the solution of (14) an analytic function defined by a Cauchy-type integral

\[ \Phi(z)=\frac{1}{2\pi i}\int \frac{\varphi(\tau)}{\tau-z}\,d\tau, \]

we obtain the Riemann boundary-value problem corresponding to the characteristic equation (14) in the form

\[ \Phi^{\pm}(t)=h^{\pm}(t). \tag{15} \]

(The coefficient of the problem \(G=\dfrac{a-b}{a+b}\) here is identically equal to zero or infinity.) The boundary condition (15) allows one to determine only one of the functions: \(\Phi^+(z)\) or \(\Phi^-(z)\); the other remains undetermined. On the basis of the general theory, the solution of the characteristic equation (14) is represented in the form

\[ \varphi(t)=h^{\pm}(t)-\Phi^{\mp}(t), \tag{16} \]

where \(\Phi^{\mp}(t)\) are the boundary values of an arbitrary function analytic in the domain \(D^{\mp}\). Consequently, the solution of the characteristic equation is determined up to the boundary value of an arbitrary function analytic in the domain \(D^{\mp}\). The latter is the general solution of the corresponding homogeneous \((h(t)\equiv0)\) equation.

From this one can already conclude that Noether’s first theorem for the degenerate case is, generally speaking, false. The number of eigenfunctions of the homogeneous equation may turn out to be infinite.

6°. Let us return to the complete equation (9). Moving in it the regular term \(k\varphi\) to the right and putting in (16) and (13) \(h=f-k\varphi\), we arrive at the pair of equations
\(\varphi(t)+k_t^+\varphi=f^+-\Phi^+\), \(k_t^-\varphi=f^-\), or, in expanded form,

\[ \varphi(t)+\int k_t^+(t,\tau)\varphi(\tau)\,d\tau = f^+(t)-\Phi^+(t), \tag{17} \]

\[ \int k_t^-(t,\tau)\varphi(\tau)\,d\tau = f^-(t). \tag{18} \]

From the manner in which equations (17), (18) are obtained from equation (9), it follows that its solutions satisfy (17), (18). If the operator \(M^{\pm}\) is applied to equation (17) and (18) is subtracted from the resulting equation, then we arrive at equation (9). Consequently, the singular integral equation (9) is equivalent to the totality of two equations; of these, equation (17) is a Fredholm equation of the second kind. It contains on the right-hand side the boundary value of an arbitrary function analytic in the domain \(D^+\). Equation (18) is a Fredholm equation of the first kind.

The first part of equation (17) contains a countable set of arbitrary parameters. All of them will, evidently, enter into its general solution. To select the solutions of the original equation, the general solution thus obtained for equation (17) must be substituted into equation (18). Equation (18) includes equalities of two types. Some of them will be equations with respect to the arbitrary parameters contained in the solution of equation (17), while others will not contain these parameters and will be conditions for the solvability of the system of equations (17), (18). By virtue of the equivalence, they will also be conditions for the solvability of the original singular equation (9). Thus equation (18) represents the totality of the conditions for equivalence of the regularized equation (17) to the original equation (9), allowing one to discard extraneous solutions introduced by the regularization process, and of the solvability conditions for the original and the regularized equations. The same occurs in the normal case studied by N. I. Vekua [3], with only the difference that, in the latter case, the number of arbitrary parameters contained in the solution, and the number of equivalence and solvability conditions, can be only finite.

With respect to the number of linearly independent solutions of the homogeneous equation and of its transposed equation, three possible cases are conceivable: 1) both numbers are finite; 2) both numbers are infinite; 3) one of the numbers is finite, the other infinite.

The first case is realized in the normal and exceptional cases. It is not difficult to show (this will be done later) that it also occurs in the degenerate case under consideration. The second possibility, which has no place in the normal and exceptional cases, is also realized here. This can be seen in the simplest case of the characteristic equation. The question of whether the last possibility can be realized remains open. We shall return to this question later.

7°. To make the arguments of the preceding point more definite, let us pass to the consideration of a concrete case, when the contour of integration is the unit circle. In what follows we shall everywhere assume that all functions occurring can be represented by Fourier series.

Let us take, in the singular integral equation given on the circle \(|t|=1\), for definiteness, the operator

\[ K\varphi \equiv M^+\varphi + k\varphi = \frac{1}{2}\varphi(t)+\frac{1}{\pi i}\int \frac{\varphi(\tau)}{\tau-t}\,d\tau +\int k(t,\tau)\varphi(\tau)\,d\tau=f(t), \tag{9''} \]

\[ k(t,\tau)=\sum_{n=-\infty}^{\infty} a_n(\tau)t^n, \qquad f(t)=\sum_{n=-\infty}^{\infty} f_n t^n . \]

Then, evidently,

\[ k_t^+(t,\tau)=\sum_{n=0}^{\infty} a_n(\tau)t^n, \qquad f^+(t)=\sum_{n=0}^{\infty} f_n t^n, \qquad k_t^-(t,\tau)= -\sum_{n=1}^{-\infty} a_n(\tau)t^n, \qquad f^-(t)=-\sum_{n=-1}^{-\infty} f_n t^n . \]

The solvability-equivalence conditions (18) in our case will take the form

\[ \sum_{k=-\infty}^{-1} t^n\left[\int a_n(\tau)\varphi(\tau)\,d\tau-f_n\right]=0 \]

or

\[ \int a_n(\tau)\varphi(\tau)\,d\tau=f_n \qquad (k=-1,-2,\ldots). \tag{19} \]

Suppose that the \(a_n(\tau)\) are connected by the linear relations

\[ A_\nu(a)\equiv a_{\nu_1}a_1+a_{\nu_2}a_2+\ldots \equiv 0 \qquad (\nu=1,2,\ldots). \]

Applying the operations \(A_\nu\) to both sides of the equalities (19), we obtain

\[ A_\nu(f_k)=a_{\nu_1}f_{-1}+a_{\nu_2}f_{-2}+\ldots=0 \qquad (\nu=1,2,\ldots). \tag{20} \]

The latter equalities will be pure solvability conditions. Taking into account that

\[ 2\pi i f_{-1}=\int f(t)t^{n-1}\,dt \]

and denoting \(\sum a_{\nu_n}t^{n-1}=h_\nu(t)\), we may write the solvability conditions (20) in the form of functionals of the right-hand side of the original equations

\[ \int f(t)h_\nu(t)\,dt=0 \qquad (\nu=1,2,\ldots). \tag{21} \]

Suppose that we have succeeded in excluding from the conditions (19) all linearly dependent \(a_n(\tau)\). Then the relations

\[ \int \beta_j(\tau)\varphi(\tau)\,d\tau=F_j \qquad (j=1,2,\ldots), \tag{22} \]

where the \(\beta_j(\tau)\) are linearly independent, will constitute the conditions for equivalence of the original singular equation \((9'')\) to the regularized equation

\[ \varphi(t)+\int k_t^+(t,\tau)\varphi(\tau)\,d\tau = f^+(t)-\Phi^-(t). \tag{17'} \]

They will give equations for the arbitrary parameters \(c_n\) entering into the function appearing on the right-hand side,

\[ \Phi^-(t)=\sum_{k=1}^{\infty} c_n t^{-n}. \]

Let us compose these equations.

We investigate the solutions of the homogeneous equation corresponding to \((17')\) and of its transposed equation. Apply to both sides of the homogeneous equation

\[ \varphi(t)+\int k_t^+(t,\tau)\varphi(\tau)\,d\tau=0 \tag{17^0} \]

the operator \(M^-\). We obtain \(M^-\varphi \equiv 0\). Consequently, the solutions of equation \((17^0)\) are functions of type \((+)\) (boundary values of functions analytic in \(D^+\)). Denote by \(\varphi_1^+(t),\ldots,\varphi_m^+(t)\) a complete system of linearly independent solutions of this equation. Before passing to the transposed equation, transform \((17^0)\). Represent the kernel in the form \(k_t^+=k^{++}-k^{+-}\). Here the second index indicates the type of the function with respect to the second variable \(\tau^*\). By Cauchy’s theorem, \(\int k^{++}(t,\tau)\varphi^+(\tau)\,d\tau=0\). Consequently, equation \((17^0)\) may be replaced by the equivalent equation

\[ \varphi(t)-\int k^{+-}(t,\tau)\varphi(\tau)\,d\tau=0. \tag{17^{00}} \]

It is not difficult to see that the kernel transposed to \(k^{+-}\) will have type \(k^{-+}\). But an equation with such a kernel will have as a solution a function of type \((-)\). Denote by \(\psi_1^-(t),\ldots,\psi_m^-(t)\) a complete system of linearly independent solutions of the equation transposed to \((17^0)\). It is not difficult to show that the generalized resolvent of equation \((17')\) will be a function of type \((+)\) in the variable \(t\). Denote it by \(R_t^+(t,\tau)\). (We note this fact as having independent significance, although here it is not used.) Put

\[ f_1^+(t)=f^+(t)+\int R_t^+(t,\tau)f^+(\tau)\,d\tau, \]

\[ \Phi_1(t)=-\Phi^-(t)-\int R_t^+(t,\tau)\Phi^-(\tau)\,d\tau = \]

\[ = \sum_{n=1}^{\infty} c_k\left[-t^{-n}-\int R_t^+(t,\tau)\tau^{-n}\,d\tau\right] = \sum_{n=1}^{\infty} c_n X_n(t). \]

Then the general solution of equation \((17')\) can be written in the form

\[ \varphi(t)=f_1^+(t)+\Phi_1(t)+\sum_{k=1}^{m} b_k\varphi_k^+(t). \tag{23} \]

Substitute from here the value of \(\varphi(t)\) into the equivalence conditions \((22)\). The latter will give a system of equations for the arbitrary constants \(c_n, b_n\):

\[ \sum_{k=0}^{\infty} c_n \int X_n(\tau)\beta_j(\tau)\,d\tau +\sum_{n=1}^{m} b_k \int \varphi_n^+(\tau)\beta_j(\tau)\,d\tau = F_j - \]

\[ -\int f_1^+(\tau)\beta_j(\tau)\,d\tau. \tag{24} \]

In addition, one must add the solvability conditions for equation \((17')\), which follow from the assumed existence of eigenfunctions \(\psi_i^-(t)\). They are written in the form

\[ \int [f^+(t)-\Phi^-(t)]\psi_n^-(t)\,dt=0 \]

\[ \text{*) A kernel of general form can be represented in the form } k(t,\tau)=k^{++}-k^{+-}-k^{-+}+k^{--}. \]

or, on the basis of Cauchy’s theorem,

\[ \int f^{+}(t)\psi^{-}(t)\,dt=0. \tag{25} \]

The last equations contain no arbitrary parameters \(c_n\); therefore they are pure solvability conditions.

Let us briefly summarize the results obtained. The solution of the given singular integral equation \((9'')\) must begin with checking the solvability conditions (21) and (25). If the answer is positive, the Fredholm equation of the second kind (17) is solved. Its solution, expressed by formula (23), will depend on a countable set of parameters \(c_n\) entering into the function \(\Phi_1(t)\). The solution obtained must be substituted into the equivalence conditions (24). Their number (finite or infinite) will be equal to the number of linearly independent functions of \(\tau\) contained in the bilinear expansion of the kernel \(K_{\tau}^{+}(t,\tau)\). In the case of a finite number of linearly independent functions \(\beta_j(\tau)\) (the kernel \(k_{\tau}^{+}(t,\tau)\) is degenerate), the system (22) will be unconditionally solvable, and the solution of the degenerate singular integral equation \((9'')\) will depend linearly on a countable set of arbitrary parameters. If the number of \(\beta_j(\tau)\) is infinite, then in the general case there are not enough data to judge the rank, and consequently the solvability and number of solutions, of the infinite system (22). In each individual case these questions must be decided by studying the corresponding infinite matrix.

8°. In conclusion, let us consider the question of the basic properties of the singular integral equation expressed by Noether’s theorems. As was already indicated at the end of § 5°, the first Noether theorem for the degenerate case is, generally speaking, false. The number of solutions may turn out to be infinite. This occurs in the case of the simplest characteristic equation. The same certainly occurs in the case of solvable complete equations whose regular kernels are degenerate. In this case only a finite number of parameters will be determined from the system (22), and the solution will depend on an infinite number of parameters. The number of solvability conditions (21) will also be infinite. It is not difficult to construct examples of complete equations in which the number of conditions (22) will be infinite, while the number of parameters remaining arbitrary, as well as the number of solvability conditions, will also be infinite. One such example will be considered below.

Let us now consider the question of the place of the second Noether theorem on the solvability conditions for an inhomogeneous equation. It is not difficult to see that the necessity of the condition expressed by this theorem remains in force also for the degenerate case under consideration. Indeed, the necessity of this condition is a simple consequence of the identity

\[ \int \psi K\varphi\,dt=\int \varphi K'\psi\,dt. \]

The derivation of the last identity is based on the possibility of interchanging the order of integration in the repeated integral. This property obviously remains in force also for the degenerate case considered by us. Hence it follows that the condition

\[ \int f(t)\psi(t)\,dt=0 \tag{26} \]

remains, in the degenerate case as well, a necessary condition of solvability. However, generally speaking, it ceases to be sufficient. It turns out that here solvability requires the fulfillment of additional condi-

of a functional character. To obtain them, let us consider the solvability-equivalence condition (18). Let the kernel \(k(t,\tau)\) belong to the class \(L_2\), and let the solution be sought in the same class. Assume also, for simplicity, that the kernel is symmetric. Then from the Poincaré solvability conditions for an equation of the first kind it follows that, for condition (18) to be fulfilled, it is necessary that its right-hand side \(f^{\pm}(t)\) belong to a narrower class than \(L_2\); namely, the condition \(\sum |\lambda_k f_k|^2 < \infty\) must be satisfied, where \(\lambda_k\) are the eigenvalues of the kernel \(k^{\mp}(t,\tau)\).

To make the result more transparent, let us explain it by an example.

\(9^\circ\). Let an equation be given on the unit circle:

\[ \lambda \varphi(t) + \frac{1}{\pi i}\int \frac{\varphi(\tau)}{\tau - t}\,d\tau + \frac{1}{2\pi i}\int \sum_{n=-\infty}^{\infty} c_k \frac{t^n}{\tau^{n+1}}\varphi(\tau)\,d\tau = f(t). \tag{27} \]

For the time being we shall regard the constant coefficient \(\lambda\) as arbitrary. The degenerate cases will correspond to the values \(\lambda = \pm 1\). We shall consider all functions entering the equation as belonging to the class \(L_2\). Consequently, the expansions

\[ f(t)=\sum_{n=-\infty}^{\infty} f_n t^n,\qquad \varphi(t)=\sum_{n=-\infty}^{\infty} \varphi_n t^n,\qquad \sum_{n=-\infty}^{\infty} c_n \frac{t^n}{\tau^{n+1}} \tag{28} \]

will be regarded as convergent in the mean-square sense, and the coefficients as satisfying the conditions

\[ \sum |c_n|^2 < \infty,\qquad \sum |f_n|^2 < \infty,\qquad \sum |\varphi_n|^2 < \infty . \tag{29} \]

Substitute the expressions (28) into equation (27) and compute the corresponding integrals:

\[ S\varphi = \frac{1}{\pi i}\int \frac{\varphi(\tau)}{\tau - t}\,d\tau = \varphi^+(t)+\varphi^-(t) = \sum_{n=0}^{\infty} c_n t^n - \sum_{n=-1}^{-\infty} c_n t^n = \sum_{n=-\infty}^{\infty} c_n \operatorname{sign} n\, t^n, \]

where

\[ \operatorname{sign} n = \begin{cases} 1, & \text{for } n \ge 0,\\ -1, & \text{for } n < 0, \end{cases} \]

\[ \frac{1}{2\pi i}\int \sum c_k \frac{t^k}{\tau^{k+1}}\varphi(\tau)\,d\tau = \frac{1}{2\pi i}\sum_{n=-\infty}^{\infty}\varphi_n \sum_{k=-\infty}^{\infty} c_k t^k \int \tau^{-k-1}\tau^n\,d\tau = \sum_{n=-\infty}^{\infty} \varphi_n c_n t^n . \]

Substituting \(\varphi\), \(f\), and the values of the last integrals into equation (27), and equating the coefficients of like powers of \(t\), we obtain, for determining the coefficients \(\varphi_n\), the infinite system of equalities

\[ \varphi_n(\lambda + \operatorname{sign} n + c_n)=f_n\quad (n=\ldots,-1,\,+2,\ldots). \tag{30} \]

In the case of the homogeneous equation \((f(t)=0)\), the last system will also be homogeneous. Those values of the index \(n\) for which the equality

\[ \lambda + \operatorname{sign} n + c_n = 0, \tag{31} \]

holds correspond to nontrivial solutions of the homogeneous equation.

there will correspond linearly independent solutions \(t^n\) of the homogeneous equation (27). For the solvability of the nonhomogeneous equation the condition \(f_n=0\) must be satisfied. In integral form it takes the form

\[ \int f(t)t^{-n-1}\,dt=0. \tag{32} \]

Suppose that all solvability conditions are fulfilled. Then from the equalities (30) the coefficients \(\varphi_n\) can be determined:

\[ \varphi_n=\frac{f_n}{\lambda+\operatorname{sign} n+c_n}. \tag{33} \]

For the solution to belong to the class \(L_2\), it is necessary and sufficient that the condition

\[ \sum \frac{|f_n|^2}{|\lambda+\operatorname{sign} n+c_n|^2}<\infty \tag{34} \]

be satisfied.

If \(\lambda\ne \pm 1\), then as \(n\to\infty\), \(|\lambda+\operatorname{sign} n+c_n|\to \lambda+\operatorname{sign} n\ne 0\); therefore condition (34) is equivalent to the condition \(\sum |f_n|^2<\infty\), which, by virtue of the membership of \(f(t)\) in the class \(L_2\), is certainly satisfied. Thus, conditions (32) will be necessary and sufficient conditions for the solvability of equation (27).

Now let the degenerate case occur. Put, for definiteness, \(\lambda=1\). Then

\[ \lambda+\operatorname{sign} n+c_n= \begin{cases} 2+c_n, & n\ge 0,\\ c_n, & n<0. \end{cases} \]

Condition (34) in this case takes the form

\[ \sum_{n=0}^{\infty}\frac{|f_n|^2}{|2+c_n|^2} + \sum_{n=-1}^{-\infty}\left|\frac{f_n}{c_n}\right|^2 <\infty. \tag{35} \]

Since \(c_n\to 0\) as \(n\to\infty\), the coefficients \(f_n\) for negative \(n\) must decrease faster than follows from the conditions (29) for \(f(t)\) to belong to the class \(L_2\). Equation (27) in the exceptional case \(\lambda=1\) is solvable in the class \(L_2\) only if the component \(f^{-}(t)\) of the right-hand side of the equation belongs to a narrower class than \(L_2\). For \(\lambda=-1\), a similar condition must be satisfied by the component \(f^{+}(t)\). These conditions could be obtained from consideration of the equivalence-solvability condition (18), which in our case \((\lambda=1)\) will have the form

\[ \frac{1}{2\pi i}\int \sum_{n=1}^{\infty} c_k\,\frac{\tau^{n+1}}{t^n}\,\varphi(\tau)d\tau = f^{-}(t). \]

It is not difficult to establish that the solvability conditions (32) coincide with the usual solvability conditions following from Noether’s second theorem. Indeed, for \(\lambda=1\) the homogeneous transposed equation here is written in the form

\[ \psi(t)-\frac{1}{\pi i}\int \frac{\psi(\tau)}{\tau-t}\,d\tau + \frac{1}{2\pi i}\int \sum c_n\,\frac{\tau^n}{t^{n+1}}\,\varphi(\tau)d\tau =0. \]

To determine the coefficients \(\psi_n\) of the expansion \(\psi(t)=\sum \psi_n t^n\), we obtain the equations
\[ \psi_n(1-\operatorname{sign} n+c_{-n-1})=0. \]
It is not difficult to see that those values of the coefficients \(c_n\) for which conditions (31) are satisfied will correspond to eigenfunctions \(t^{-n-1}\). Consequently, the solvability conditions (32) can be represented in the form (26). But they are not sufficient. For solvability, the functional conditions (35) must additionally be satisfied.

As was already pointed out in item \(5^\circ\), the coefficient of the Riemann problem corresponding to the degenerate case is identically equal to zero or infinity. Consequently, the concept of an index here loses its meaning, and therefore Noether’s third theorem in the degenerate case likewise loses its meaning.

\(10^\circ\). From the example considered it is clear that, by an appropriate choice of the coefficients \(c_n\) of the regular kernel \(k(t,\tau)\), one can arrange that the corresponding homogeneous equation have either a finite or an infinite number of linearly independent solutions. The corresponding transposed equation will have the same number of solutions. Thus, if one disregards the possibility of an infinite number of solutions, the situation here is the same as in Fredholm theory. In all the other examples we have considered, the situation is the same. Whether other cases can occur is not yet clear. Of serious interest is the question of whether the third of the possibilities indicated in item \(6^\circ\) can be realized, when one of the numbers of solutions of two transposed equations is finite and the other infinite. I. Ts. Gokhberg and M. G. Krein, in their survey article [6], while investigating the general properties of singular operators, hypothetically assume such a possibility (in their terminology, a semi-infinite characteristic), but do not allow the presence of an infinite number of solutions for both equations transposed to one another. Precisely the latter is certainly realized in the degenerate case of a singular integral equation, but it remains open whether the case of a semi-infinite characteristic admitted by the authors mentioned can occur.

\(11^\circ\). A similar investigation could have been carried out under more general assumptions concerning the functions and contours than has been done here. We restricted ourselves to the classical case of a smooth contour and coefficients continuous in the sense of Hölder only so as not to complicate the reasoning with considerations not related to the essence of the question under consideration.

In conclusion, let us note that some of the results established here were briefly formulated in the author’s abstract [2], and that the example considered was borrowed from the work of E. I. Zverovich and G. S. Litvinchuk [5], where it is used for the same purpose as here.

References

1. Gakhov F. D. Boundary Value Problems. Fizmatgiz, Moscow, 1963.

2. Gakhov F. D. Degenerate cases of a singular integral equation. Abstracts of Rostov University for 1959, 37–38, 1960.

3. Vekua I. N. Integral equations with a special kernel of Cauchy type. Transactions of the Tbilisi Mathematical Institute of the Academy of Sciences of the Georgian SSR, vol. X, 1941, pp. 45–72.

4. Muskhelishvili N. I. Singular Integral Equations. Moscow: GITTL, 113, no. 10, 57–105, 1952.

5. Zverovich E. I., Litvinchuk G. S. Izv. AN SSSR, Ser. Mat., 28, no. 5, 1003–1036, 1964.

6. Gokhberg I. Ts., Krein M. G. UMN, vol. XII, issue 2, 43–118, 1957.

Received by the editors
April 19, 1965

Belorussian State University
named after V. I. Lenin