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UDC 517.948
MATHEMATICS
A. YANUSHAUSKAS
ON A CLASS OF SINGULAR INTEGRAL EQUATIONS
(Presented by Academician M. A. Lavrent'ev on May 6, 1965)
Let \(R^3\) be three-dimensional Euclidean space; \(S:\{x^2+y^2+z^2=1\}\) the unit sphere of the space \(R^3\). Consider the following singular integral equation:
\[ \begin{aligned} f_1(x,y,z)={}&[\alpha(z)(1-z^2)+z\beta(z)]u_1(x,y,z)+{}\\ &+\int_S \frac{\alpha(z)[x(x-x_0)+y(y-y_0)]+\beta(z)(z-z_0)} {\bigl[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2\bigr]^{3/2}}\, u_1(x_0,y_0,z_0)\,ds, \end{aligned} \tag{1} \]
where \(P=(x,y,z)\) and \(Q=(x_0,y_0,z_0)\) are points of the sphere \(S\); \(\alpha(r)\), \(\beta(r)\), and \(f_1(x,y,z)\) are Hölder-continuous functions, \(\alpha^2+\beta^2\ne 0\), \(\beta(1)\ne 0\).
Using the equation of the sphere \(S\) and taking as independent variables on \(S\) the variables \(z\) and \(\varphi=\operatorname{arctg} y/x\), we reduce equation (1) to the form
\[ \begin{aligned} f(z,\varphi)={}&[\alpha(z)(1-z^2)+z\beta(z)]u(z,\varphi)+{}\\ &+\frac{\alpha(z)z-\beta(z)}{2\sqrt2} \int_{-1}^{+1}\int_0^{2\pi} \frac{(z-z_0)u(z_0,\psi)} {(1-zz_0)^{3/2}[1-\lambda\cos(\psi-\varphi)]^{3/2}}\, d\psi\,dz_0+{}\\ &+\frac{\alpha(z)}{\sqrt2} \int_{-1}^{+1}\int_0^{2\pi} \frac{u(z_0,\psi)} {(1-zz_0)^{1/2}[1-\lambda\cos(\psi-\varphi)]^{1/2}}\, d\psi\,dz_0, \end{aligned} \tag{2} \]
where
\[ \lambda=\bigl[(1-z^2)(1-z_0^2)\bigr]^{1/2}/(1-zz_0). \]
The operator
\[ T(u)=\frac{\alpha(z)}{\sqrt2} \int_{-1}^{+1}\int_0^{2\pi} \frac{u(z_0,\psi)} {(1-zz_0)^{1/2}[1-\lambda\cos(\psi-\varphi)]^{1/2}}\, d\psi\,dz_0 \]
is completely continuous; therefore, first of all, we consider the equation
\[ \begin{aligned} g(z,\varphi)={}&[\alpha(z)(1-z^2)+z\beta(z)]u(z,\varphi)+{}\\ &+\frac{\alpha(z)z-\beta(z)}{2\sqrt2} \int_{-1}^{+1}\int_0^{2\pi} \frac{(z-z_0)u(z_0,\psi)} {(1-zz_0)^{3/2}[1-\lambda\cos(\psi-\varphi)]^{3/2}}\, d\psi\,dz_0 . \end{aligned} \tag{3} \]
We shall seek solutions of equations (2) and (3) in the class of Hölder-continuous functions. Since \(g(z,\varphi)\) and \(u(z,\varphi)\) are Hölder-continuous on the sphere \(S\), they can be represented by uniformly convergent series of the form \((^1)\)
\[ u(z,\varphi)=\sum_{l=0}^{\infty}\bigl[u_l(z)\cos l\varphi+v_l(z)\sin l\varphi\bigr], \tag{4a} \]
\[ g(z,\varphi)=\sum_{l=0}^{\infty}\bigl[f_l(z)\cos l\varphi+h_l(z)\sin l\varphi\bigr]. \tag{4b} \]
Substituting (4a) into (3), we obtain
\[ \begin{gathered} [\alpha(z)(1-z^2)+z\beta(z)]u(z,\varphi) +\frac{\alpha(z)z-\beta(z)}{2\sqrt{2}} \int_{-1}^{+1} \frac{(z-z_0)\sqrt{\pi}}{(1-zz_0)^{3/2}(1-z^2)} \\ {}\times \sum_{l=0}^{\infty} \frac{\Gamma\left(l+\frac{3}{2}\right)}{2^{l-1}\Gamma(l+1)} \lambda^l F\left(l/2+\frac14,\; \frac{l}{2}-\frac14;\; l+1;\; \lambda^2\right) \\ {}\times [u_l(z_0)\cos l\varphi+v_l(z_0)\sin l\varphi]\,dz_0 = g(z,\varphi). \end{gathered} \tag{5} \]
Equation (5) can be rewritten in the following form:
\[ \begin{gathered} g(z,\varphi)=[\alpha(z)(1-z^2)+z\beta(z)]u(z,\varphi)+ \\ {}+\frac{\alpha(z)z-\beta(z)}{2} \sum_{l=0}^{\infty}\int_{-1}^{+1} \frac{\lambda^l(1-zz_0)^{1/2}}{z-z_0} [u_l(z_0)\cos l\varphi+v_l(z_0)\sin l\varphi]\,dz_0 \\ {}+\int_{-1}^{+1}\int_{0}^{2\pi} H(z,z_0,\psi-\varphi)u(z_0,\psi)\,d\psi\,dz_0, \end{gathered} \tag{6} \]
where
\[ H(z,z_0,\psi-\varphi)= \]
\[ = \frac{1}{z-z_0} \sum_{l=0}^{\infty}\lambda^l \left[ \frac{\sqrt{\pi}\,\Gamma\left(l+\frac32\right)} {2^{l-1}\Gamma(l+1)} F\left(l/2+\frac14,\;l/2-\frac14;\;l+1;\;\lambda^2\right)-1 \right]\cos l(\psi-\varphi). \]
The operator
\[ A(u)=\int_{-1}^{+1}\int_{0}^{2\pi} H(z,z_0,\psi-\varphi)u(z_0,\psi)\,d\psi\,dz_0 \]
is completely continuous.
Consider the equation
\[ \begin{gathered} g(z,\varphi)=[\alpha(z)(1-z^2)+z\beta(z)]u(z,\varphi)+ \\ {}+\frac{\alpha(z)z-\beta(z)}{2} \sum_{l=0}^{\infty}\int_{-1}^{+1} \frac{\lambda^l(1-zt)^{1/2}}{z-t} [u_l(t)\cos l\varphi+v_l(t)\sin l\varphi]\,dt. \end{gathered} \tag{7} \]
Separating the variables in equation (7), we obtain
\[ f_l(z)=[\alpha(z)(1-z^2)+z\beta(z)]u_l(z)+ \]
\[ {}+\frac{\alpha(z)z-\beta(z)}{2} \int_{-1}^{+1} \frac{\lambda^l(1-zt)^{1/2}}{z-t}u_l(t)\,dt \tag{8} \]
(an analogous equation is also obtained for \(v_l(z)\)). To equation (8) the theory developed in (2) is applicable.
Introduce the following notation:
\[ Q(z)=-[\alpha(z)z-\beta(z)]\pi i/2,\qquad P(z)=\alpha(z)(1-z^2)+z\beta(z), \]
\[ G(z)=\frac{P(z)-Q(z)(1-z^2)^{1/2}} {P(z)+Q(z)(1-z^2)^{1/2}}, \]
\[ P^*(z)= \frac{P(z)} {\{[P(z)]^2-[Q(z)]^2(1-z^2)\}}, \]
\[ Q^*(z)= \frac{Q(z)(1-z^2)^{1/2}} {\{[P(z)]^2-[Q(z)]^2(1-z^2)\}}. \]
Choose that branch of the logarithm for which \(\ln G(-1)=0\); then we have \(\ln G(+1)=2n\pi i\), where \(2n\pi\) is the increment of the argument of the function \(G(z)\) on the interval \(-1\le z\le 1\), and \(n\) is an integer. Consider the integral operator
\[ K^*(h)=P^*(z)h(z)-\frac{Q^*(z)Z(z)}{\pi i}\int_{-1}^{+1}\frac{h(t)}{Z(t)(t-z)}\,dt, \]
where \(Z(z)=\omega(z)(z-1)^n\); \(\omega(z)\) is a completely determined function depending only on \(P\) and \(Q\); \(\omega(z)\ne 0\) for \(-1\le z\le 1\).
Let \(n\le 0\). From (8) we have
\[ u_l(z)=K^*(f_l)+\mathcal L_l(u), \tag{9} \]
where
\[ \mathcal L_l(u)=\frac{1}{\pi i}\int_{-1}^{+1}K^*\left(Q(r)\frac{\lambda^l(1-zt)^{1/2}-(1-z^2)^{1/2}}{z-t}\right)u(t)\,dt. \]
As \(l\to\infty\), the operators \(\mathcal L_l\) converge to the operator
\[ A(u)=-K^*\left(\frac{Q(z)\sqrt{1-z^2}}{\pi i}\int_{-1}^{+1}\frac{u(t)}{t-z}\,dt\right)+Q^*(z)Q(z)(1-z^2)^{1/2}u(z). \]
The singular integral equation
\[ f(z)=u(z)+A(u) \]
is always solvable and has a unique solution. It follows from this that equation (9), for sufficiently large \(l\), is always solvable and has a unique solution. For the solution of equation (8) we have the representation \((^2)\)
\[ u_l(z)=P(z)f_l(z)+\int_{-1}^{+1}\frac{N_l(z,t)}{t-z}f_l(t)\,dt+\sum_{i=1}^{m}C_i\omega_i(z), \tag{10} \]
where \(N_l(z,t)\) is a completely determined bounded function; \(C_i\) are arbitrary constants; \(\omega_i\) are solutions of the homogeneous equation (8), and, for the validity of representation (10), it is necessary and sufficient that the function \(f_l(z)\) satisfy \(-n+m\) orthogonality conditions
\[ \int_{-1}^{+1} f_l(z)\psi_{li}(z)\,dz=0,\qquad i=0,1,\ldots,-n+m-1, \tag{11} \]
where \(\psi_{li}(z)\) are linearly independent solutions of the adjoint homogeneous equation.
From the convergence of the operators \(\mathcal L_l\) to the operator \(A\), and from the fact that the equation \(f(z)=u(z)+A(u)\) is always solvable and has a unique solution, it follows that the functions \(N_l(z,t)\) are bounded uniformly with respect to \(l\). Consider the series
\[ N(z,t,\psi-\varphi,\tau)=\frac{1}{z-t}\sum_{l=0}^{\infty}N_l(z,t)\tau^l\cos l(\psi-\varphi),\qquad 0\le \tau<1. \]
The solution of equation (7) is written in the following form:
\[ u(z,\varphi)=P(z)g(z,\varphi)+\lim_{\tau\to 1}\int_{-1}^{+1}\int_{0}^{2\pi}N(z,t,\psi-\varphi,\tau)g(t,\psi)\,d\psi\,dt, \tag{12} \]
and, in order that a solution (12) exist, it is necessary and sufficient that the function \(g(z,\varphi)\) satisfy a countable set of conditions
orthogonality
\[ \int_{-1}^{+1}\int_{0}^{2k}\mu_{lj}(z)e^{il\varphi}g(z,\varphi)\,d\varphi\,dz=0, \tag{13} \]
\[ j=0,1,\ldots,-n+m-1;\quad l=0,1,2,\ldots \]
The operator
\[ N(g)=\lim_{\tau\to 1}\int_{-1}^{+1}\int_{0}^{2\pi}N(z,t,\psi-\varphi,\tau)\,g(t,\psi)\,d\psi\,dt \]
is a bounded operator; therefore the following theorem is valid.
Theorem 1. If \(n<0\), then for the solvability of equation (3) it is necessary and sufficient that the function \(g(z,\varphi)\) satisfy a countable set of orthogonality conditions. The homogeneous equation corresponding to equation (3) has no more than a finite number of linearly independent solutions. Equation (3) is normally solvable in the sense of Hausdorff.
The equation
\[ h(z,\varphi)=\bigl[\alpha(z)(1-z^{2})+z\beta(z)\bigr]\,v(z,\varphi)- \]
\[ -\frac{1}{2\sqrt{2}}\int_{-1}^{+1}\int_{0}^{2\pi} \frac{[t\alpha(t)-\beta(t)](z-t)} {(1-tz)^{3/2}[1-\lambda\cos(\psi-\varphi)]^{3/2}}\, v(t,\psi)\,d\psi\,dt \tag{14} \]
will be called the adjoint equation to equation (3). Equation (14) is also normally solvable in the sense of Hausdorff \((^3)\).
If \(n>0\) for equation (3), then for the adjoint equation (14) \(n<0\); therefore, also in the case \(n>0\), equation (3) is normally solvable in the sense of Hausdorff.
Theorem 2. Equation (3) is normally solvable in the sense of Hausdorff. For \(n<0\), for the solvability of equation (3) it is necessary and sufficient to impose on the function \(g(z,\varphi)\) a countable set of orthogonality conditions; the corresponding homogeneous equation has no more than a finite number of linearly independent solutions. For \(n>0\), the homogeneous equation corresponding to equation (3) has infinitely many linearly independent solutions, and for the solvability of equation (3) it is necessary and sufficient to impose on the function \(g(z,\varphi)\) no more than a finite number of orthogonality conditions. For \(n=0\), equation (3) is Fredholm.
Theorem 2 is also valid for the more general integral equation
\[ h(z,\varphi)=\bigl[\alpha(z)(1-z^{2})+z\beta(z)\bigr]\,u(z,\varphi)+T(u)+ \]
\[ +\int_{-1}^{+1}\int_{0}^{2\pi} \frac{[z\alpha(z)-\beta(z)](z-t)} {2\sqrt{2}(1-zt)^{3/2}[1-\lambda\cos(\psi-\varphi)]^{3/2}}\, u(t,\psi)\,d\psi\,dt, \tag{15} \]
where the operator \(T(u)\) is completely continuous. A special case of equation (15) is equation (2).
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
6 V 1965
REFERENCES
- N. K. Bari, Trigonometric Series, Moscow, 1961.
- N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
- F. Hausdorff, Set Theory, Moscow–Leningrad, 1937.