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UDC 517.948.32
A GENERAL THEORY OF INTEGRAL EQUATIONS WITH A POWER KERNEL
F. V. CHUMAKOV
Integral equations with a Cauchy kernel, whose distinctive feature is that the kernel becomes infinite of first order when the arguments coincide, have been well studied [3], and a general theory has been constructed for them. Its core consists of three Noether theorems. The simplest of these equations—the characteristic one—is solved in closed form. In doing so, the method of analytic continuation into the plane of the complex variable is used. An analytic function is introduced in the form of a Cauchy-type integral whose density is the desired solution of the integral equation. The formulas of Yu. V. Sokhotskii for its limiting values make it possible to reduce the characteristic Cauchy integral equation to a Riemann boundary-value problem for the analytic function introduced.
Here we shall consider integral equations with a power kernel. By the nature of the kernel (its becoming infinite of order less than one), they seem to be Fredholm-type equations. But the presence of an additional feature of the kernel (a change in the form of the kernel when passing across the main diagonal of the domain of definition) and the fact that they are equations of the first kind make these equations special. As K. D. Sakalyuk [6] showed, the simplest types of these equations reduce to the Riemann problem for analytic functions. This relates these equations to special integral equations of Cauchy type. In the present paper a general theory of linear integral equations with a power kernel will be set forth. In studying such equations we shall make extensive use of the theory of the Riemann problem and of integral equations with a Cauchy kernel.
§ 1. BASIC DEFINITIONS AND NOTATION
1.1. Integral equation with a power kernel. On the segment \([\alpha,\beta]\) of the real axis in the plane of the complex variable \(z\), we shall consider an equation of the form
\[ M\varphi = A(x)\int_{\alpha}^{x}\frac{\varphi(t)}{(x-t)^{\mu}}\,dt + B(x)\int_{x}^{\beta}\frac{\varphi(t)}{(t-x)^{\mu}}\,dt + \]
\[ + \int_{\alpha}^{\beta} m(x,t)\varphi(t)\,dt = f(x), \tag{1.1} \]
where \(0<\mu<1\); \(A(x)\), \(B(x)\), \(f(x)\), \(m(x,t)\) are given functions, real on the interval of integration. The class of these functions will be defined
later. In what follows, equation (1.1) will be called the complete generalized Abel integral equation with external coefficients. Here \(M\) denotes the symbol of the operation performed on the function \(\varphi(x)\) in the left-hand side of the equation under consideration. The operator defined by the expression
\[ M^0\varphi \equiv A(x)\int_\alpha^x \frac{\varphi(t)}{(x-t)^\mu}\,dt + B(x)\int_x^\beta \frac{\varphi(t)}{(t-x)^\mu}\,dt, \tag{1.2} \]
will be called characteristic, and the operator
\[ m\varphi \equiv \int_\alpha^\beta m(x,t)\varphi(t)\,dt \tag{1.3} \]
regular. The equation
\[ M^0\varphi \equiv A(x)\int_\alpha^x \frac{\varphi(t)}{(x-t)^\mu}\,dt + B(x)\int_x^\beta \frac{\varphi(t)}{(t-x)^\mu}\,dt = f(x) \tag{1.4} \]
will be called the characteristic equation with external coefficients.
We write the original integral equation (1.1) in operator form
\[ M\varphi \equiv M^0\varphi + m\varphi = f. \tag{1.5} \]
As will be shown below, the operator \(M^0\) plays, in the theory of integral equations with a power kernel, the same role as the characteristic operator in the general theory of singular integral equations with Cauchy kernel.
1.2. The adjoint (transposed) equation. The equation obtained from the complete homogeneous Abel integral equation with external coefficients \(M\varphi=0\) by transposition (interchange) of the variables in the kernel will be called the transposed, or adjoint, equation to (1.1). To construct it, we transform, using the symbol
\[ \operatorname{sign}(x-t)= \begin{cases} -1, & t>x,\\ 1, & t<x, \end{cases} \]
the characteristic part of equation (1.1) to the new form
\[ M^0\varphi \equiv \frac{A(x)+B(x)}{2}\int_\alpha^\beta \frac{\varphi(t)\,dt}{|t-x|^\mu} + \frac{B(x)-A(x)}{2}\int_\alpha^\beta \frac{\varphi(t)}{|t-x|^\mu}\operatorname{sign}(t-x)\,dt. \tag{1.6} \]
Taking (1.6) into account and interchanging the variables in the kernel of \(M\varphi=0\), we arrive at the adjoint equation
\[ M'\omega \equiv M^{0'}\omega + m'\omega \equiv \frac{1}{2}\int_{\alpha}^{\beta}\frac{A(t)+B(t)}{|t-x|^\mu}\,\omega(t)\,dt + \frac{1}{2}\int_{\alpha}^{\beta}\frac{B(t)-A(t)}{|t-x|^\mu}\,\operatorname{sign}(x-t)\omega(t)\,dt + \int_{\alpha}^{\beta}m(t,x)\omega(t)\,dt=0. \tag{1.7} \]
We write the integral equation obtained in (1.7) in a form analogous to equation (1.1), substituting the value of the symbol \(\operatorname{sign}(x-t)\):
\[ M'\omega \equiv \int_{\alpha}^{x} B(t)\frac{\omega(t)}{(x-t)^\mu}\,dt + \int_{x}^{\beta} A(t)\frac{\omega(t)}{(t-x)^\mu}\,dt + \int_{\alpha}^{\beta} m(t,x)\omega(t)\,dt=0. \tag{1.8} \]
The equation
\[ M'\omega = \int_{\alpha}^{x} B(t)\frac{\omega(t)}{(x-t)^\mu}\,dt + \int_{x}^{\beta} A(t)\frac{\omega(t)}{(t-x)^\mu}\,dt + \int_{\alpha}^{\beta} m(t,x)\omega(t)\,dt=h(x) \tag{1.9} \]
will be called the complete generalized Abel integral equation with internal coefficients. The role of the characteristic operator for \(M'\) will be played by the expression
\[ M^{0'}\omega = \int_{\alpha}^{x} B(t)\frac{\omega(t)\,dt}{(x-t)^\mu} + \int_{x}^{\beta} A(t)\frac{\omega(t)}{(t-x)^\mu}\,dt. \tag{1.10} \]
The integral equation
\[ M^{0'}\omega \equiv \int_{\alpha}^{x} B(t)\frac{\omega(t)}{(x-t)^\mu}\,dt + \int_{x}^{\beta} A(t)\frac{\omega(t)}{(t-x)^\mu}\,dt = h(x), \tag{1.11} \]
adjoint (transposed) to the characteristic equation with external coefficients, will be called characteristic with internal coefficients. We further note that the equation adjoint to the characteristic equation with external coefficients (1.2) is obtained by interchanging the functions \(A(x)\), \(B(x)\) and placing them under the integral sign. In what follows it will be shown that, in considering questions of solvability and the number of linearly independent solutions, there is a close connection between equations (1.1) and (1.9). For complete equations with a power kernel, questions of regularization will also be investigated, i.e., their reduction to Fredholm equations.
§ 2. Auxiliary Relations
Let, in the plane of the complex variable \(z\), cut along the segment \([\alpha,\beta]\) of the real axis, there be given an analytic function
\[ \Phi(z)=[(z-\alpha)(\beta-z)]^{\frac{\mu-1}{2}} \int\limits_{\alpha}^{\beta}\frac{\varphi(t)}{(t-z)^{\mu}}\,dt . \tag{2.1} \]
We shall establish the connection between the function \(\varphi(x)\), which is the density of the integral representation (2.1), and the function \(\psi(x)\)—the density of an integral of Cauchy type
\[ \Phi(z)=\frac{1}{2\pi i}\int\limits_{\alpha}^{\beta}\frac{\psi(t)\,dt}{t-z}, \tag{2.2} \]
which gives a representation of one and the same function \(\Phi(z)\).
The analogues of Yu. V. Sokhotskii’s formulas for the function \(\Phi(z)\) given by the representation (2.1) have the following form [6]:
\[ \Phi^{+}(x)=\frac{1}{R(x)} \left[ e^{\mu\pi i}\int\limits_{\alpha}^{x}\frac{\varphi(t)\,dt}{(x-t)^{\mu}} + \int\limits_{x}^{\beta}\frac{\varphi(t)\,dt}{(t-x)^{\mu}} \right], \tag{2.3} \]
\[ \Phi^{-}(x)=-\frac{1}{R(x)} \left[ \int\limits_{\alpha}^{x}\frac{\varphi(t)\,dt}{(x-t)^{\mu}} + e^{\mu\pi i}\int\limits_{x}^{\beta}\frac{\varphi(t)}{(t-x)^{\mu}}\,dt \right], \tag{2.3'} \]
where
\[ R(x)=[(x-\alpha)(\beta-x)]^{\frac{1-\mu}{2}} . \]
From the equality of the boundary values of the function \(\Phi(z)\), represented by the two expressions (2.1), (2.2), there follow the following identities:
\[ \frac{1}{2}\psi(x)+\frac{1}{2\pi i}\int\limits_{\alpha}^{\beta}\frac{\psi(t)\,dt}{t-x} = \frac{1}{R(x)} \left[ e^{\mu\pi i}\int\limits_{\alpha}^{x}\frac{\varphi(t)\,dt}{(x-t)^{\mu}} + \int\limits_{x}^{\beta}\frac{\varphi(t)\,dt}{(t-x)^{\mu}} \right], \]
\[ -\frac{1}{2}\psi(x)+\frac{1}{2\pi i}\int\limits_{\alpha}^{\beta}\frac{\psi(t)\,dt}{t-x} = -\frac{1}{R(x)} \left[ \int\limits_{\alpha}^{x}\frac{\varphi(t)\,dt}{(x-t)^{\mu}} + e^{\mu\pi i}\int\limits_{x}^{\beta}\frac{\varphi(t)\,dt}{(t-x)^{\mu}} \right]. \]
From the last relations there follows the connection between the functions \(\varphi(x)\) and \(\psi(x)\), expressed by the two formulas
\[ \psi(x)=\frac{1+e^{\mu\pi i}}{R(x)} \int\limits_{\alpha}^{\beta}\frac{\varphi(t)}{|t-x|^{\mu}}\,dt \equiv T\varphi, \tag{2.4} \]
\[ \frac{1}{\pi i}\int\limits_{\alpha}^{\beta}\frac{\psi(t)\,dt}{t-x} = \frac{1-e^{\mu\pi i}}{R(x)} \int\limits_{\alpha}^{\beta}\frac{\varphi(t)}{|t-x|^{\mu}}\operatorname{sign}(t-x)\,dt . \tag{2.5} \]
Substituting into formula (2.5) the value of \(\psi(x)\) from (2.4) and performing a rearrangement of the order of integration in the double integral, which can be done by virtue of the nonsingularity of one of the integrals, we obtain an identity from which the equality follows
\[ \operatorname{sign}(\tau - x) = \frac{R(x)|\tau - x|^\mu}{\pi \operatorname{tg}\frac{\mu\pi}{2}} \int_\alpha^\beta \frac{dt}{R(t)|t-\tau|^\mu(t-x)} . \tag{2.6} \]
Relation (2.6), establishing a connection between the symbol \(\operatorname{sign}(\tau-x)\) and an integral of Cauchy type, can also be proved by a direct calculation using the theory of residues for the integral appearing on the right-hand side of (2.6).
For the subsequent exposition we shall need certain relations following from the solution of Abel’s equation [6]:
\[ \int_\alpha^x \frac{\varphi(t)}{(x-t)^\mu}\,dt = g(x). \tag{2.7} \]
If the right-hand side of equation (2.7) can be represented in the form
\[ g(x)=(x-\alpha)^{\varepsilon_1}(\beta-x)^{\varepsilon_2}g^*(x), \tag{2.8} \]
where \(\varepsilon_1,\varepsilon_2\) are certain positive numbers, and the function \(g^*(x)\) has a derivative satisfying the Hölder condition on the interval \([\alpha,\beta]\), then its solution is unique and belongs to the class of integrable functions
\[ \varphi(x)= \frac{\varphi^*(x)} {(x-\alpha)^{1-\mu-\varepsilon_1}(\beta-x)^{1-\mu-\varepsilon_2}}, \tag{2.9} \]
where \(\varphi^*(x)\) is a Hölder function.
§ 3. CHARACTERISTIC EQUATION WITH EXTERNAL COEFFICIENTS
K. D. Sakalyuk [4] gave a solution of the equation
\[ M^0\varphi \equiv A(x)\int_\alpha^x \frac{\varphi(t)}{(x-t)^\mu}\,dt + B(x)\int_x^\beta \frac{\varphi(t)}{(t-x)^\mu}\,dt = f(x) \tag{3.1} \]
in closed form by reducing it to a Riemann problem for a function given by the integral (2.1), followed by the inversion of one of Abel’s integrals, determined by formulas (2.3), (2.3′). We shall give another solution of this equation, based on reducing it to an equivalent equation with a Cauchy kernel. We shall assume that the coefficients of the equation under consideration do not vanish simultaneously and have derivatives \(A'(x)\), \(B'(x)\) satisfying the Hölder condition. The solution of (3.1) is sought in the class of functions (2.9). The class of the free term will be indicated below.
3.1. Integral equation with Cauchy kernel, equivalent to (3.1). Let us write equation (3.1) by means of the symbol \(\operatorname{sign}(x-t)\)
\[ M^0\varphi \equiv \frac{A(x)+B(x)}{2} \int_\alpha^\beta \frac{\varphi(t)}{|t-x|^\mu}\,dt + \]
\[ + \frac{B(x)-A(x)}{2}\int_{\alpha}^{\beta}\frac{\varphi(t)}{|t-x|^\mu}\operatorname{sign}(t-x)\,dt=f(x). \tag{3.2} \]
Substituting into (3.2) the expressions for the integrals from the auxiliary formulas (2.4), (2.5), we arrive at an equation with a Cauchy kernel of the following form:
\[ K^0\psi \equiv M^0T^{-1}\psi = \frac{A(x)+B(x)}{2(1+e^{\mu\pi i})}\,R(x)\psi(x)+ \]
\[ +\frac{B(x)-A(x)}{2(1-e^{\mu\pi i})}\, \frac{R(x)}{\pi i}\int_{\alpha}^{\beta}\frac{\psi(t)\,dt}{t-x} =f(x). \tag{3.3} \]
The connection between the functions \(\varphi(x)\) and \(\psi(x)\) is expressed by formula (2.4), which, with respect to \(\varphi(x)\), in turn represents a certain particular case of an integral equation with a power kernel (3.1). The mutual one-to-one correspondence between these functions follows from the fact that the solution of equation (2.4) reduces to the solution of a jump problem for analytic functions, which is solved uniquely, followed by the inversion of the Abel integral, which is also performed uniquely. Consequently, between the solutions of equation (3.1) and the equation with Cauchy kernel (3.3) there is established a mutual one-to-one correspondence. Let us establish the class of functions in which the solution of the special equation (3.3) is sought.
K. D. Sakalyuk [6] showed that the Abel integral
\[ \Phi(x)=\int_{\alpha}^{x}\frac{\varphi(t)}{(x-t)^\mu}\,dt \]
takes any function of the form
\[ \varphi(x)=\frac{\varphi^*(x)}{(x-\alpha)^\gamma} \qquad (\gamma<1,\ \varphi^*(x)\in H(\lambda)) \]
into the function
\[ \Phi(x)=\frac{\Phi^*(x)}{(x-\alpha)^{\mu+\gamma-1}}, \quad \text{where } \Phi^*(x)\in H(\lambda). \]
The behavior of the Abel integral at the point \(\beta\) is analogous. Hence it follows that the function \(\psi(x)\) at the points \(\alpha,\beta\) may tend to infinity of order not exceeding, respectively, the numbers
\[ \frac{1-\mu}{2}-\varepsilon_1,\qquad \frac{1-\mu}{2}-\varepsilon_2. \]
The latter circumstance means that, in the case when \(\varphi(x)\) belongs to the class (2.9), the function \(\psi(x)\), defined by formula (2.4), belongs to the class
\[ \psi(x)= \frac{\psi^*(x)} {(x-\alpha)^{\frac{1-\mu}{2}-\varepsilon_1} (\beta-x)^{\frac{1-\mu}{2}-\varepsilon_2}}, \tag{3.4} \]
where \(\psi^*(x)\) satisfies a Hölder condition on \([\alpha,\beta]\) with the same exponent as \(\varphi^*(x)\). Let us find what sufficient conditions must be imposed on the function \(\psi(x)\) in order to ensure that the solution \(\varphi(x)\) of equation (2.4) belongs to the class (2.9). It is known [6] that equation (2.4) reduces to inversion of the Abel integral
\[ \int_{\alpha}^{x}\frac{\varphi(t)\,dt}{(x-t)^\mu} = \frac{R(x)}{2(1+e^{\mu\pi i})} \left[ \psi(x) - \frac{\operatorname{ctg}\frac{\mu\pi}{2}}{\pi} \int_{\alpha}^{\beta}\frac{\psi(t)}{t-x}\,dt \right], \tag{3.5} \]
where \(\psi(x)\) is the solution of (3.3). In order that the function \(\varphi(x)\), defined from (3.5), belong to the class (2.9), it is sufficient to require of the right-hand side of (3.5) that it be representable in the form (2.8). It is easy to see that this requirement will be fulfilled if \(\psi(x)\) is of class (3.4). In this case \([\psi^{*}(x)]'\) satisfies the Hölder condition. The latter circumstance will force us to impose certain conditions on the function \(f(x)\). The restrictions imposed on \(f(x)\) will follow from the following arguments.
3.2. Solution of equation (3.1). In what follows we shall use formulas given in the book of F. D. Gakhov [2, § 47]. Introducing the function (2.2) and using Yu. V. Sokhotskii’s formulas for its limiting values, we reduce the solution of (3.3) to the Riemann problem
\[ \Phi^{+}(x)= \frac{e^{\mu\pi i}B(x)-A(x)} {e^{\mu\pi i}A(x)-B(x)}\,\Phi^{-}(x) + \frac{e^{2\mu\pi i}-1} {e^{\mu\pi i}A(x)-B(x)}\,\frac{f(x)}{R(x)} . \tag{3.6} \]
In view of the fact that \(\psi(x)\) belongs to the class (3.4), the solution of problem (3.6) must be sought in the class of functions having, in neighborhoods of the ends of the segment \([\alpha,\beta]\) and of infinity, respectively, the estimates
\[ \Phi(z)= \begin{cases} O\!\left[(z-\alpha)^{\frac{\mu-1}{2}+\varepsilon_{1}}\right],\\[4pt] O\!\left[(\beta-z)^{\frac{\mu-1}{2}+\varepsilon_{2}}\right],\\[4pt] O\!\left(\dfrac{1}{z}\right). \end{cases} \tag{3.7} \]
From the formula for the solution of the Riemann problem (3.6) it follows that the order of the function \(\Phi(z)\) at the ends of the integration segment, generally speaking, coincides with the order of the canonical function \(X(z)\) for an arbitrary choice of the function \(f(x)\). Hence it follows that the solution of problem (3.6) will satisfy the conditions (3.7) if one requires that the order of the canonical function \(X(z)\) at the points \(\alpha,\beta\) be less than \((1-\mu)/2\).
The solution of equation (3.3) is given by the formula
\[ \psi(x)=\Phi^{+}(x)-\Phi^{-}(x) = \frac{1}{2}\left[1+\frac{1}{G(x)}\right]g(x)+ \]
\[ {}+X^{+}(x)\left[1-\frac{1}{G(x)}\right] \left[ \frac{1}{2\pi i}\int_{\alpha}^{\beta} \frac{g(t)}{X^{+}(t)}\,\frac{dt}{t-x} - P_{\varkappa-1}(x) \right], \tag{3.8} \]
where
\[ G(x)= \frac{e^{\mu\pi i}B(x)-A(x)} {e^{\mu\pi i}A(x)-B(x)}; \qquad g(x)= \frac{e^{2\mu\pi i}-1} {e^{\mu\pi i}A(x)-B(x)}\,\frac{f(x)}{R(x)}; \]
\(\varkappa\) is the index of the Riemann problem (3.6) in the class of sought functions. The quantity
\[ \varkappa=\operatorname{Ind} \frac{e^{\mu\pi i}B(x)-A(x)} {e^{\mu\pi i}A(x)-B(x)} \]
will henceforth be called the index of the characteristic equation (3.1) with external coefficients. Let us determine the conditions that must be imposed on the function \(f(x)\). Substitute the value of \(\psi(x)\) from (3.8) into equation (3.5). In solving the latter, one has to take the derivative of the right-hand side, which contains a Cauchy-type integral in whose density \(f(x)\) enters. It is known that the derivative of a singular integral exists if its density has a derivative pro—
a derivative satisfying the Hölder condition and vanishes at the ends of the contour of integration. This circumstance is precisely what requires that \(f(x)\) belong to the class
\[ f(x)=(x-\alpha)^{n_1}(\beta-x)^{n_2} f^{*}(x), \tag{3.9} \]
where
\[ n_1>\frac{1-\mu}{2}-p_1;\qquad n_2>\frac{1-\mu}{2}-p_2; \]
\([f^{*}(x)]'\) is a Hölder function; \(p_1,p_2\) are the orders of the canonical function at the points \(\alpha,\beta\), respectively.
If the index \(\varkappa\) in the class of sought solutions is negative, then there arise \(|\varkappa|\) solvability conditions
\[ \int_{\alpha}^{\beta} \frac{f(t)}{R(t)X^{+}(t)\left[e^{\mu\pi i}A(t)-B(t)\right]} \, t^{k-1}\,dt=0,\qquad k=1,2,\ldots,|\varkappa|. \tag{3.10} \]
We write the solution (3.3) in the following form:
\[ \psi(x)=\widetilde{\psi}(x)+\sum_{k=1}^{\varkappa} c_k\psi_k(x), \tag{3.11} \]
where
\[ \widetilde{\psi}(x)= \frac{1}{2}\left[1+\frac{1}{G(x)}\right]g(x) +X^{+}(x)\left[1-\frac{1}{G(x)}\right]\times \]
\[ \times \frac{1}{2\pi i}\int_{\alpha}^{\beta} \frac{g(t)}{X^{+}(t)}\frac{dt}{t-x}; \qquad \psi_k(x)=X^{+}(x)\left[1-\frac{1}{G(x)}\right]x^{k-1}. \]
From (2.4) it follows that
\[ \varphi(x)=\widetilde{\varphi}(x)+\sum_{k=1}^{\varkappa} c_k\varphi_k(x), \tag{3.12} \]
where \(\widetilde{\varphi}(x)=T^{-1}\psi(x)\) is a particular solution of the nonhomogeneous equation (3.1), and \(\psi_k(x)=T^{-1}\psi_k(x)\) form a complete system of linearly independent solutions of the equation \(M^{0}\varphi=0\). In the case \(\varkappa\leqslant0\), the solution of (3.1) is given by formula (3.12), in which the sum is absent.
Let us formulate the results.
Theorem. Suppose that the coefficients \(A(x)\), \(B(x)\) of the characteristic equation (3.1) with outer coefficients do not vanish simultaneously and have Hölder derivatives on the segment \([\alpha,\beta]\), and that the free term \(f(x)\) belongs to the class of functions (3.9). Then
- If \(\varkappa>0\), the equation \(M^{0}\varphi=0\) has \(\varkappa\) linearly independent solutions
\[ \varphi_k(x)=T^{-1}\psi_k(x),\qquad k=1,2,\ldots,\varkappa \]
from the class (2.9).
-
If \(\varkappa\leqslant0\), then \(M^{0}\varphi=0\) is unsolvable.
-
If \(\varkappa\geqslant0\), then the equation \(M^{0}\varphi=f\) is unconditionally solvable in the class (2.9), and its general solution depends on \(\varkappa\) arbitrary constants.
-
If \(\varkappa<0\), then \(M^{0}\varphi=f\) is solvable in the class (2.9) only when the \(|\varkappa|\) solvability conditions (3.10) are satisfied.
§ 4. CHARACTERISTIC EQUATION (ADJOINT) WITH INTERNAL COEFFICIENTS
Consider the equation
\[ M^{0'}\omega \equiv \int_{\alpha}^{x} B(t)\frac{\omega(t)}{(x-t)^\mu}\,dt +\int_{x}^{\beta} A(t)\frac{\omega(t)}{(t-x)^\mu}\,dt=h(x). \tag{4.1} \]
We shall set forth the scheme for solving equation (4.1). The conditions imposed on the coefficients and the free term will be indicated below.
4.1. Integral equation with Cauchy kernel equivalent to (4.1). Write equation (4.1) in the form
\[ M^{0'}\omega \equiv \frac{1}{2}\int_{\alpha}^{\beta} \frac{A(t)+B(t)}{|t-x|^\mu}\,\omega(t)\,dt +\frac{1}{2}\int_{\alpha}^{\beta} \frac{B(t)-A(t)}{|t-x|^\mu}\, \operatorname{sign}(x-t)\omega(t)\,dt=h(x). \tag{4.2} \]
Substituting in (4.2) the expression for \(\operatorname{sign}(x-t)\) through a Cauchy-type integral (2.6), and interchanging the order of integration in the double integral, which can be done because one of the integrals is nonsingular, we obtain
\[ M^{0'}\omega \equiv \int_{\alpha}^{\beta} \frac{dt}{R(t)|t-x|^\mu} \left[ \frac{A(t)-B(t)}{2}\,R(t)\omega(t)- \right. \]
\[ \left. -\frac{\operatorname{ctg}\dfrac{\mu\pi}{2}}{\pi} \int_{\alpha}^{\beta} \frac{B(\tau)-A(\tau)}{2}\, \frac{R(\tau)\omega(\tau)}{\tau-t}\,d\tau \right] =h(x). \tag{4.3} \]
Introduce the operator obtained from (2.4) by interchanging the variables in the kernel,
\[ T'\delta \equiv (1+e^{\mu\pi i}) \int_{\alpha}^{\beta} \frac{\delta(t)}{R(t)|t-x|^\mu}\,dt. \tag{4.4} \]
It is known [2] that the solution of the equation \(T'\delta=p\) reduces to the inversion of the Abel integral, which is carried out uniquely. The homogeneous equation \(T'\delta=0\) has no nontrivial solutions. Consequently, the operator \(T'\) has no eigenfunctions. Acting on equation (4.3) from the left with the operator \((T')^{-1}\), we arrive at the equivalent equation with Cauchy kernel
\[ K^{0'}\omega \equiv (T')^{-1}M^{0'}\omega \equiv \frac{A(x)+B(x)}{2(1+e^{\mu\pi i})}\,R(x)\omega(x)- \]
\[ -\frac{\operatorname{ctg}\dfrac{\mu\pi}{2}} {(1+e^{\mu\pi i})\pi} \int_{\alpha}^{\beta} \frac{B(\tau)-A(\tau)}{2}\, \frac{R(\tau)\omega(\tau)}{\tau-x}\,d\tau =(T')^{-1}h. \tag{4.5} \]
It is easy to see that the equation with Cauchy kernel (4.5) is adjoint to (3.3),
4.2. Solution of equation (4.1).
Introducing the analytic function
\[ \Omega(z)=\frac{1}{2\pi i}\int_{\alpha}^{\beta} \frac{A(t)-B(t)}{2}\,\frac{R(t)\omega(t)}{t-z}\,dt \tag{4.6} \]
and proceeding as usual, we arrive at the Riemann boundary-value problem
\[ \Omega^{+}(x)= \frac{e^{\mu\pi i}A(x)-B(x)} {e^{\mu\pi i}B(x)-A(x)}\,\Omega^{-}(x)+ \]
\[ +\frac{e^{\mu\pi i}-1}{2}\, \frac{A(x)-B(x)} {e^{\mu\pi i}B(x)-A(x)}\,(T')^{-1}h(x). \tag{4.7} \]
Next note that the coefficient of the obtained Riemann problem (4.7) is inverse to the coefficient of problem (3.6). If \(X(z)\) is the canonical function of (3.6), then it is easy to see that
\[ X'(z)=\frac{1}{X(z)} \]
is the canonical function of problem (4.7) in the adjoint class. Obviously, the index of (4.7) in the adjoint class is equal to the index of (3.6) with the opposite sign. By the formula
\[ \omega(x)=\frac{2}{R(x)}\, \frac{\Omega^{+}(x)-\Omega^{-}(x)} {A(x)-B(x)}. \tag{4.8} \]
we find the solution of equation (4.1), which we write in the form
\[ \omega(x)=\widetilde{\omega}(x)+\sum_{k=1}^{\chi} d_k\omega_k(x), \tag{4.9} \]
where \(\widetilde{\omega}(x)\) is a particular solution of \(M^{0'}\omega=h\), and the functions
\[ \omega_k(x)= \frac{x^k} {R(x)X^{+}(x)\,[e^{\mu\pi i}A(x)-B(x)]}, \qquad k=0,1,\ldots,|\chi|-1 \]
form a complete system of linearly independent solutions of the homogeneous equation \(M^{0'}\omega=0\). If \(-\chi\leqslant 0\), then the sum must be absent in (4.9); moreover, if \(\chi<0\), then the solution exists when \(\chi\) solvability conditions are fulfilled:
\[ \int_{\alpha}^{\beta} \frac{A(t)-B(t)} {e^{\mu\pi i}B(t)-A(t)} \,(T')^{-1}h(t)X^{+}(t)t^{j-1}\,dt=0, \quad j=1,2,\ldots,\chi . \tag{4.10} \]
Let us note here also that the solvability conditions (3.10) of the characteristic equation with external coefficients (3.1) may be represented in the usual form
\[ \int_{\alpha}^{\beta} f(t)\omega_k(t)\,dt=0, \quad k=0,1,2,\ldots,|\chi|-1 \tag{4.11} \]
through the solutions of the adjoint homogeneous equation.
We shall clarify the conditions imposed on the coefficients and the free term of equation (4.1). Since we seek the solution of the Riemann problem (4.7) in the adjoint class, it is easy to see that the solution of (4.1) must be sought in the class of functions (2.9). Obviously, the coefficients \(A(x)\), \(B(x)\) must satisfy the Hölder condition and \(A(x)-B(x)\ne0\). The representability of the function
\(h(x)\) in the form (3.9), under the condition \(n_1>\dfrac{1-\mu}{2}\), \(n_2>\dfrac{1-\mu}{2}\), ensures that the solution \(\omega(x)\) belongs to the required class (2.9). Thus, equations (3.1) and (4.1) possess the properties of the characteristic equation with Cauchy kernel and of its adjoint.
§ 5. THE COMPLETE GENERALIZED ABEL INTEGRAL EQUATION WITH EXTERNAL COEFFICIENTS
It is not possible to obtain the solution of the complete equation with an arbitrary regular part in closed form. We shall investigate the regularization of these equations, i.e., their reduction to equations of Fredholm type. Here there arises the problem of establishing the relation between the solutions of the original and the regular equation (questions of equivalent regularization). We shall also establish Noether theorems for the complete equation.
Consider the equation
\[ M\varphi \equiv M^0\varphi + m\varphi = f, \tag{5.1} \]
where \(M^0\), \(m\) are given by formulas (1.2), (1.3).
5.1. A special integral equation with Cauchy kernel equivalent to (5.1). Let
\[ \psi(x)=\frac{1+e^{\mu\pi i}}{R(x)} \int_\alpha^\beta \frac{\varphi(t)}{|t-x|^\mu}\,dt \equiv T\varphi . \tag{5.2} \]
Equation (5.2) is uniquely solvable [2, § 53], and its solution is given by the formula
\[ \varphi=T^{-1}\psi . \tag{5.3} \]
Substituting into the original equation (5.1), instead of the function \(\varphi\), its expression (5.3) in terms of \(\psi\), we arrive at an integral equation of the following form:
\[ MT^{-1}\psi \equiv M^0T^{-1}\psi + mT^{-1}\psi = f . \tag{5.4} \]
Introduce the notation \(MT^{-1}\equiv K\), \(M^0T^{-1}\equiv K^0\), \(mT^{-1}\equiv k\). In the new notation equation (5.4) is written as follows:
\[ K\psi \equiv K^0\psi + k\psi = f . \tag{5.5} \]
In considering the characteristic equation (3.1) with external coefficients it was shown that the expression \(K^0=M^0T^{-1}\) is a special Cauchy integral operator. Its explicit expression is given by formula (3.3). We shall seek the regular part of equation (5.5) in the form
\[ k\psi=\int_\alpha^\beta K(x,t)\psi(t)\,dt . \tag{5.6} \]
We shall find the expression \(K(x,t)\) in terms of the known function \(m(x,t)\). Clearly,
\[ k\psi=\int_\alpha^\beta K(x,t)\psi(t)\,dt = \int_\alpha^\beta m(x,t)T^{-1}\psi(t)\,dt . \]
Substituting in the last relation the expression \(\psi\) (5.2) and interchanging the order of integration in the double integral (the latter
one may do this owing to the nonsingularity of the integrals), we obtain an identity from which the relation follows
\[ \int_{\alpha}^{\beta} \frac{K(x,t)}{R(t)|t-\tau|^\mu}\,dt = \frac{m(x,\tau)}{1+e^{i\pi i}} . \tag{5.7} \]
The latter may be regarded as an equation for determining the kernel of the regular part of the Cauchy equation (5.5). We write equation (5.7) for determining \(K(x,t)\) by means of the operator \(T\) in the following form:
\[ T'_t K(x,t)=m(x,\tau). \tag{5.8} \]
Here \(x\) is a parameter. The solution of (5.8) is given by the formula
\[ K(x,t)=(T')_t^{-1}m(x,t). \tag{5.9} \]
From the properties of the operator \(T\) (§ 3) there follows the equivalence of the resulting integral equation with Cauchy kernel (5.5) with respect to the new function \(\psi(x)\) and the original equation (5.1) with power kernel.
5.2. The adjoint equation. Reduction to an equivalent equation with Cauchy kernel. In subsection 1.2 an expression was given for the transposed operator \(M'\) (1.9). The equation corresponding to this operator will be written in the form
\[ M'\omega \equiv M^{0'}\omega + m'\omega = h, \tag{5.10} \]
where the operator \(M^{0'}\) is given by formula (1.10). Acting on equation (5.10) by the operator \((T')^{-1}\) from the left, we arrive at the following expression:
\[ (T')^{-1}M'\omega \equiv (T')^{-1}M^{0'}\omega + (T')^{-1}m'\omega = (T')^{-1}h. \tag{5.11} \]
In § 4 it was established that the operator \((T')^{-1}M^{0'}\) is a Cauchy operator. Moreover, the equality holds
\[ K^{0'} \equiv (T')^{-1}M^{0'}, \tag{5.12} \]
where \(K^0\) is from (5.4). Arguing analogously to subsection 5.1, it is easy to show that
\[ (T')^{-1}m' \equiv k', \tag{5.13} \]
where \(k\) is from (5.6). Consequently, equation (5.10), adjoint to the complete equation (5.1) with external coefficients, is reduced to the equivalent (by virtue of the properties of the operator \(T\)) equation with Cauchy kernel
\[ K'\omega \equiv K^{0'}\omega + k'\omega = (T')^{-1}h, \tag{5.14} \]
where the operator \(K'\) is adjoint to the operator \(K\). Thus, the complete generalized Abel integral equation with external coefficients (5.1) and its adjoint (the complete generalized Abel integral equation (5.10) with internal coefficients) are reduced respectively to equivalent complete equations with Cauchy kernel (5.5), (5.14). Moreover, (5.14) is adjoint to (5.5). The theory of complete equations with Cauchy kernel is well developed. This theory is set forth, for example, in the book of F. D. Gakhov [2]. Questions of regularization are considered and Noether theorems are established. We shall transfer the results concerning complete equations with Cauchy kernel to complete equations with power kernel. We seek the solution of (5.1) in the class (2.9). In order that \(\varphi(x)\), determined by formula (5.3), should belong to the class (2.9), it is sufficient
seek a solution of (5.5) in the class (3.4), where \([\psi^*(x)]'\) is a Hölder function. We assume that \(A'(x)\), \(B'(x)\) satisfy the Hölder condition, and that \(f(x)\) belongs to the class (3.9). The conditions imposed on the kernel \(m(x,t)\) will be specified later.
5.3. Regularization of (5.1) by a solution of the characteristic equation.
Let us write the complete equation (5.5) in the form
\[ K^0 \psi = f - k\psi \tag{5.15} \]
and solve it as a characteristic equation, regarding the right-hand side as a known function. The solution \(\psi(x)\) of (5.15) will belong to the desired class if the right-hand side of (5.15) admits the representation (3.9). Let us clarify the conditions imposed on the kernel \(m(x,t)\) of the original equation. Suppose that for \(m(x,t)\) the representation
\[ m(x,t)=(x-\alpha)^{n_1}(\beta-x)^{n_2}m^*(x,t), \tag{5.16} \]
is valid, where \(n_1, n_2\) are from (3.9); \(m^*(x,t)\) has a Hölder derivative with respect to \(x\) and may, at the ends of the interval of integration, tend to infinity with respect to \(t\) of order \(<\mu\). From formula (5.9) there follows a representation for the kernel of the regular part of equation (5.5)
\[ K(x,t)=(x-\alpha)^{n_1}(\beta-x)^{n_2}K^*(x,t), \tag{5.17} \]
where \(K^*(x,t)\) tends to infinity of order less than \((1+\mu)/2\) at the points \(\alpha,\beta\) with respect to \(t\), and has a Hölder derivative \(K_x'(x,t)\). Solving, as usual, the characteristic equation (5.15), we arrive at the following equation:
\[ \psi(x)+\int_{\alpha}^{\beta} K_1(x,t)\psi(t)\,dt=f_1(x), \tag{5.18} \]
where
\[ \begin{aligned} K_1(x,t) &=\frac{e^{2\mu\pi i}-1}{2} \left[1+\frac{1}{G(x)}\right] \frac{K(x,t)}{R(x)\,[e^{\mu\pi i}A(x)-B(x)]} \\ &\quad +\frac{e^{2\mu\pi i}-1}{2}\, X^+(x)\left[1-\frac{1}{G(x)}\right]\times \\ &\quad\times \frac{1}{\pi i}\int_{\alpha}^{\beta} \frac{K(\xi,t)\,d\xi} {X^+(\xi)R(\xi)[e^{\mu\pi i}A(\xi)-B(\xi)](\xi-x)}, \end{aligned} \]
\[ \begin{aligned} f_1(x) &=\frac{1}{2}\left[1+\frac{1}{G(x)}\right]g(x) +X^+(x)\left[1-\frac{1}{G(x)}\right]\times \\ &\quad\times \left[ \frac{1}{2\pi i}\int_{\alpha}^{\beta} \frac{g(t)}{X^+(t)}\,\frac{dt}{t-x} - P_{\chi-1}(x) \right], \end{aligned} \]
\[ g(x)= -\frac{e^{2\mu\pi i}-1}{e^{\mu\pi i}A(x)-B(x)} \,\frac{f(x)}{R(x)}, \qquad G(x)= \frac{e^{\mu\pi i}B(x)-A(x)} {e^{\mu\pi i}A(x)-B(x)} . \]
The kernel of equation (5.18), \(K_1(x,t)\), and the free term \(f_1(x)\) may tend to infinity of integrable order at the ends \([\alpha,\beta]\). Therefore this
the equation is not Fredholm in the direct sense. However, carrying out an investigation analogously to (2, § 48), one can show that it possesses all the properties of equations with a Fredholm kernel. In the case \(\chi<0\) (\(\chi\) is the index of (5.15) in the class sought), the functional relations must be satisfied that include both the solvability conditions for (5.15) and the conditions for equivalence of the resulting Fredholm equation (5.18) and the equation (5.15) under consideration. Let us also note that the solvability conditions for (5.15) are at the same time the solvability conditions for (5.1). The equation with Cauchy kernel (5.15) is equivalent to the Fredholm equation (5.18) and the functional equalities
\[ \int_{\alpha}^{\beta}\delta_k(t)\psi(t)\,dt=f_k,\qquad k=1,2,\ldots,|\chi|, \tag{5.19} \]
where
\[ f_k=\int_{\alpha}^{\beta} \frac{f(t)}{R(t)X^{+}(t)\left[e^{\mu\pi i}A(t)-B(t)\right]}\,t^{k-1}\,dt; \tag{5.19'} \]
\[ \delta_k(t)=\int_{\alpha}^{\beta} \frac{K(\tau,t)}{R(\tau)X^{+}(\tau)\left[e^{\mu\pi i}A(\tau)-B(\tau)\right]}\,\tau^{k-1}\,d\tau. \tag{5.19''} \]
Following I. N. Vekua [1], one can replace the Fredholm equation (5.18) and the functional relations (5.19) by a certain Fredholm integral equation and by conditions imposed only on the function \(f(x)\). If among the \(|\chi|\) functions \(\delta_1(x),\ldots,\delta_{-\chi}(x)\) there are \(h\) linearly independent ones, then from (5.19) one can extract \(|\chi|-h=\nu\) conditions imposed on the function \(f(x)\) (equivalence conditions),
\[ \int_{\alpha}^{\beta} f(t)\theta_i(t)\,dt=0,\qquad i=1,2,\ldots,\nu, \tag{5.20} \]
where \(\theta_i(x)\) are known functions. The remaining conditions (5.19) make it possible to construct a Fredholm equation for some new function, which, together with (5.20), is equivalent to the equation (5.18) and the functional relations (5.19). Thus, the original integral equation (5.1) is reduced to a Fredholm equation and to conditions imposed on the function \(f(x)\). If the conditions (5.20) are satisfied, then the constructed Fredholm equation is equivalent to (5.1) in the sense that they are simultaneously unsolvable or solvable.
5.4. Other methods of regularization. Equivalent regularization.
It is known [2] that the operator \(N\), playing the role of the resolving operator in solving the characteristic equation with Cauchy kernel
\[ K^{0}\psi=f, \tag{5.21} \]
is regularizing. The expression for the operator \(N\) is obtained from the formula for the solution of (5.21), i.e.
\[ \psi=Nf+\sum_{k=1}^{\chi}c_k\psi_k, \]
where \(\psi_k\) are known functions. In the theory of singular equations with Cauchy kernel, on the basis of the properties of the operator \(N\), the following fact is proved: if the index
of the operator \(K\) in the given class \(\varkappa \geqslant 0\), then, applying the operator \(N\) on the left to equation (5.5), we arrive at the equivalent Fredholm equation
\[ NK\psi = Nf. \tag{5.22} \]
With respect to the original equation (5.1) with a power kernel, we shall have
\[ NK\psi \equiv NMT^{-1}\psi = NM^{0}T^{-1}\psi + NmT^{-1}\psi = Nf. \tag{5.23} \]
In the present case, acting on the operator \(M\) successively by two operators, on the right \((T^{-1})\) and on the left \((N)\), leads to an equivalent equation of Fredholm type with respect to the new function \(\psi\). If, however, the index \(\varkappa \leqslant 0\), then we apply to (5.5) regularization on the right, carrying out a substitution of the form
\[ \psi = N\delta. \tag{5.24} \]
We shall have
\[ KN\delta = f. \tag{5.25} \]
Equivalence must be understood in the sense that both equations (5.25), (5.5) are either solvable or unsolvable simultaneously. In the case of solvability, to each solution of (5.25) there corresponds a solution of the original equation (5.5), \(\psi = N\delta\), and, conversely, to each solution \(\psi\) of equation (5.5) there corresponds a solution of (5.25) by the formula
\[ \delta = K^{0}\psi + \sum_{k=1}^{\varkappa} c_k\delta_k, \tag{5.26} \]
where \(\delta_k\) are eigenfunctions of the operator \(N\). For equation (5.1) this means the following:
\[ MT^{-1}N\delta \equiv M^{0}T^{-1}N\delta + mT^{-1}N\delta = f. \tag{5.27} \]
Consequently, acting on the right on equation (5.1) by the operator \(T^{-1}N\), we arrive at an equivalent Fredholm equation. We shall not give the explicit expression of the operator \(T^{-1}N\) because of the cumbersomeness of the formulas. To each solution \(\delta\) of the regularized equation there corresponds a solution of (5.1)
\[ \varphi = T^{-1}N\delta \tag{5.28} \]
and, conversely, to each solution \(\varphi\) of (5.1) there corresponds a solution of the regularized equation (5.27)
\[ \delta = K^{0}\psi + \sum_{k=1}^{\varkappa} c_k\delta_k = \]
\[ = M^{0}T^{-1}T\varphi + \sum_{k=1}^{\varkappa} c_k\delta_k = M^{0}\varphi + \sum_{k=1}^{\varkappa} c_k\hat{\delta}_k. \tag{5.29} \]
Similarly, one can consider the complete generalized Abel integral equation with internal coefficients.
Let us formulate the results.
- The number of linearly independent solutions of the equation
\[ M\varphi = 0 \]
is finite.
- A necessary and sufficient condition for the solvability of the equation \(M\varphi=f\) in the class (2.9) is the fulfillment of the conditions
\[ \int_{\alpha}^{\beta} f(t)\omega_k(t)\,dt=0, \]
where the functions \(\omega_k(t)\) form a complete system of linearly independent solutions of the adjoint equation
\[ M'\omega=0 \]
in the adjoint class.
- Let \(n\) be the number of solutions of \(M\varphi=0\) in the class (2.9), and let \(n'\) be the number of solutions of \(M'\omega=0\) in the adjoint class. Then, if the index of the equation \(M\varphi=0\) in the sought class is \(\varkappa\), then
\[ n-n'=\varkappa. \]
In conclusion I consider it a pleasant duty to express my sincere gratitude to Professor F. D. Gakhov for his constant attention and interest in the work.
References
- Vekua I. N. Integral equations with a special kernel of Cauchy type. Proceedings of the Tbilisi Mathematical Institute of the Academy of Sciences of the Georgian SSR, vol. X, 1941.
- Gakhov F. D. Boundary-value problems. Fizmatgiz, Moscow, 1963.
- Muskhelishvili N. I. Singular integral equations. Fizmatgiz, Moscow, 1962.
- Sakalyuk K. D. DAN SSSR, 131, No. 4, 1960.
- Sakalyuk K. D. Scientific Notes of Kishinev University, 1, 1962.
- Sakalyuk K. D. Candidate dissertation. Minsk, BSU, 1963.
Received by the editors
June 14, 1965
Belorussian State University
named after V. I. Lenin