E. I. KIM

SOLUTION OF A CLASS OF SINGULAR INTEGRAL EQUATIONS WITH A CONTOUR INTEGRAL

(Presented by Academician S. L. Sobolev, 5 X 1956)

§ 1.

Consider singular integral equations of the form

\[ \psi(s,t)=\lambda\int_0^t d\tau \int_C K_0(r_{pp_1}^2,t-\tau)\psi(s_1,\tau)\,ds_1+f(s,t)\qquad (t>0), \tag{1} \]

where \(r_{pp_1}\) is the distance between the points \(p\) and \(p_1\) with coordinates \(s\) and \(s_1\) in the arc-coordinate system,

\[ K_0(r_{pp_1}^2,t-\tau)= \frac{1}{(t-\tau)^{3/2}} \int_0^\infty \rho(z) \left[ 1-\frac{r_{pp_1}^2}{2a^2(z)(t-\tau)} \right] \exp\left[ -\frac{r_{pp_1}^2}{4a^2(z)(t-\tau)} \right]dz, \tag{2} \]

\[ \rho(z)=(z^2+a_1^2)^{-3/2}(z^2+a_2^2)^{-1/2},\qquad a^2(z)=a_2^2(z^2+a_1^2)/(z^2+a_2^2). \tag{3} \]

In this equation the function \(f(s,t)\) has a derivative of bounded variation, and

\[ t^\sigma |f(s,t)|,\; V(t^\sigma \partial f(s,t)/\partial s)\leq M \qquad (0\leq \sigma<1). \tag{4} \]

In the present paper we shall show that equation (1), for arbitrary \(\lambda\), has no solution in the class of functions satisfying the inequality

\[ |\psi(s_1,t)-\psi(s_2,t)|\leq Mt^{-\sigma}|s_1-s_2|^\alpha \tag{5} \]

(\(\sigma\) and \(\alpha\) independently satisfy the inequalities \(0\leq\sigma<1\), \(0<\alpha\leq1\)), and exists only for \(\lambda<\lambda_0\), where \(\lambda_0\) is a completely definite number. Consequently, on the basis of Abel’s theorem, the method of successive approximations cannot be applied to equation (1), since this method does not give a complete solution.

§ 2.

We investigate equation (1). In view of the periodicity of the functions \(r_{pp_1}\) and \(f(s,t)\) with respect to \(s\), with period \(2l\) (the length of the arc of the curve \(C\)), the required function \(\psi(s,t)\) must be periodic with period \(2l\). Therefore we represent the solution in the form of a Fourier series, if it exists. If the Fourier series of the function \(\psi(s,t)\) diverges, then a solution of equation (1) cannot exist in the class of functions satisfying inequality (5). Let

\[ \psi(s,t)=\frac{\psi_0(s,t)}{2} +\sum_{k=1}^{\infty} \left( \psi_k^{(1)}(t)\cos\frac{k\pi s}{l} +\psi_k^{(2)}(t)\sin\frac{k\pi s}{l} \right). \tag{6} \]

On the basis of (4), the function \(f(s,t)\) is also expanded in a Fourier series

\[ f(s,t)=\frac{f_0(t)}{2} +\sum_{k=1}^{\infty} \left( f_k^{(1)}(t)\cos\frac{k\pi s}{l} +f_k^{(2)}(t)\sin\frac{k\pi s}{l} \right), \tag{7} \]

moreover

\[ \left| f_k^{(i)}(t) \right| \leq M / k^2 t^\sigma \quad (i=1,2). \tag{8} \]

It is easy to show that, if the curve \(C\) is sufficiently smooth, then

\[ r_{pp_1}^2=(s-s_1)^2\varphi(s,s_1), \tag{9} \]

where \(\varphi(s,s_1)\) is a known function depending on the form of the curve \(C\). Multiply the function \(\frac{1}{l}\cos \frac{n\pi s}{l}\) by both sides of equation (1) and integrate along the contour \(C\). Then

\[ \psi_n^{(1)}(t)=\lambda \int_0^t d\tau \int_C \left\{\frac{1}{l}\int_C K_0\bigl((s-s_1)^2\varphi(s,s_1),t-\tau\bigr)\cos\frac{n\pi s}{l}\,ds\right\} \psi(s_1,\tau)\,ds_1+f_n^{(1)}(t). \]

Making the substitution \(s-s_1=s'\), we obtain

\[ \begin{aligned} \psi_n^{(1)}(t)= {}& \lambda \int_0^t d\tau \int_C \left\{\frac{1}{l}\int_{-l}^{+l} K_0\bigl(s'^2\varphi(s'+s_1,s_1),t-\tau\bigr) \cos\frac{n\pi s'}{l}\,ds'\right\} \psi(s_1,\tau)\cos\frac{n\pi s_1}{l}\,ds_1 \\ &+\lambda \int_0^t d\tau \int_C \left\{\frac{1}{l}\int_{-l}^{+l} K_0\bigl(s'^2\varphi(s'+s_1,s_1),t-\tau\bigr) \sin\frac{n\pi s'}{l}\,ds'\right\} \psi(s_1,\tau)\sin\frac{n\pi s_1}{l}\,ds_1 \\ &+f_n^{(1)}(t). \end{aligned} \tag{10} \]

The inner integrals are not computed in explicit form. Therefore we shall separate the principal parts from these integrals, and estimate the remainder. As a result we shall have

\[ \psi_n^{(1)}(t)= \lambda \int_0^t \left\{4\sqrt{\pi}\lambda_n\int_0^\infty \rho(z)a^3(z)\exp\left[-\lambda_n^2 a^2(z)(t-\tau)\right]\,dz\right\} \psi_n^{(1)}(\tau)\,d\tau +\int_0^t d\tau\int_C \Phi_n^{(1)}(s_1,t-\tau)\psi(s_1,\tau)\,ds_1 +f_n^{(1)}(t). \tag{11} \]

Entirely analogously, we obtain:

\[ \psi_n^{(2)}(t)= \lambda \int_0^t \left\{4\sqrt{\pi}\lambda_n\int_0^\infty \rho(z)a^3(z)\exp\left[-\lambda_n^2 a^2(z)(t-\tau)\right]\,dz\right\} \psi_n^{(2)}(\tau)\,d\tau +\int_0^t d\tau\int_C \Phi_n^{(2)}(s_1,t-\tau)\psi(s_1,\tau)\,ds_1 +f_n^{(2)}(t), \tag{12} \]

where \(\lambda_n=n\pi/l\), and \(\Phi_n^{(i)}(s_1,t-\tau)\) satisfy the inequality

\[ \left|\Phi_n^{(i)}(s_1,t-\tau)\right| \leq M n^{-3/2}(t-\tau)^{-3/4} \quad (i=1,2). \tag{13} \]

Equations (11) and (12) shall be called the equations associated with the singular equation (1).

§ 3. We now consider the integral equation

\[ \psi_n(t)=\lambda\int_0^t K_n^0(t-\tau)\psi_n(\tau)\,d\tau+f_n(t), \tag{14} \]

where

\[ K_n^{(0)}(t-\tau)=4\sqrt{\pi}\lambda_n \int_0^\infty \rho(z)a^3(z)\exp\left[-\lambda_n^2 a^2(z)(t-\tau)\right]\,dz. \tag{15} \]

Equation (14) is obtained from equations (11) and (12) by dropping the second terms. We shall call it the characteristic equation of the adjoint equations.

Applying the operational method to equation (14), we obtain the solution in explicit form

\[ \psi_n(t)=f_n(t)+\lambda\int_0^t \Gamma_n(t-\tau;\lambda)f_n(\tau)\,d\tau, \tag{16} \]

where

\[ \Gamma_n(t-\tau;\lambda)= \frac{2\pi^{3/2}(|\nu|-\nu)}{(\nu^2-1)(a_1^2\nu^2-a_2^2)} \lambda_n^2 \exp[-\lambda_n^2 d_0(t-\tau)] + \]

\[ +\,4\sqrt{\pi}\,a_2^3\mu^2\lambda_n \int_0^\infty \frac{z^2}{(z^2+a_2^2)^2(\nu^2 z^2+a_2^2)} \exp[-\lambda_n^2 d^2(z)(t-\tau)]\,dz, \tag{17} \]

\[ d_0=(a_1^2\nu^2-a_2^2)/(\nu^2-1),\qquad \nu=\mu-1,\qquad \mu=(a_1^2-a_2^2)/2\pi^{-3/2}\lambda . \tag{18} \]

It is obvious that our solution is meaningful if \(\nu\ne \pm 1\), \(\nu\ne \pm a_2/a_1\), but by direct verification one can establish that for \(\nu=1\), \(\nu=\pm a_2/a_1\) the solution can be obtained from (16) by a limiting passage; if, however, \(\nu=-1\), then it follows from (18) that \(\lambda=\infty\). Therefore the latter case is excluded from consideration.

It is easy to verify that the resolvent \(\Gamma_n(t-\tau;\lambda)\) satisfies the equations:

\[ \Gamma_n(t-\tau;\lambda) =K_n^0(t-\tau)+\lambda\int_\tau^t \Gamma_n(t-t_1;\lambda)K_n^0(t_1-\tau)\,dt_1, \tag{19} \]

\[ \Gamma_n(t-\tau;\lambda) =K_n^0(t-\tau)+\lambda\int_0^t K_n^0(t-t_1)\Gamma_n(t_1-\tau;\lambda)\,dt_1; \tag{20} \]

For what follows we introduce the operators

\[ B_n\psi(t)=\psi(t)-\lambda\int_0^t K_n^0(t-\tau)\psi(\tau)\,d\tau, \tag{21} \]

\[ B^{-1}\psi(t)=\psi(t)+\lambda\int_0^t \Gamma_n(t-\tau;\lambda)\psi(\tau)\,d\tau. \tag{22} \]

On the basis of formulas (19) and (20), these operators have the following properties:

\[ BB^{-1}\psi(t)=B^{-1}B\psi(t)=\psi(t). \tag{23} \]

It follows directly from these equalities that the equations

\[ B_n\psi(t)=0,\qquad B_n^{-1}\psi(t)=0 \tag{24} \]

have only the trivial solution. Consequently, equalities (14) and (16) are equivalent.

§ 4. We rewrite equations (11) and (12) with the aid of the operators (21) and (22):

\[ B_n\psi_n^{(1)}(t)= \int_0^t d\tau\int_C \Phi_n^{(1)}(s_1,t-\tau)\psi(s_1,\tau)\,ds_1+f_n^{(1)}(t), \tag{11_1} \]

\[ B_n\psi_n^{(2)}(t)= \int_0^t d\tau\int_C \Phi_n^{(2)}(s_1,t-\tau)\psi(s_1,\tau)\,ds_1+f_n^{(2)}(t). \tag{12_1} \]

Applying the operator \(B_n^{-1}\) to these equalities, on the basis of (23) we obtain

\[ \psi_n^{(1)}(t) = \int_0^t d\tau \int_C B_n^{-1}\Phi_n^{(1)}(s_1,t-\tau)\psi(s_1,\tau)\,ds_1 + B_n^{-1}f_n^{(1)}(t), \tag{25} \]

\[ \psi_n^{(2)}(t) = \int_0^t d\tau \int_C B_n^{-1}\Phi_n^{(2)}(s_1,t-\tau)\psi(s_1,\tau)\,ds_1 + B_n^{-1}f_n^{(2)}(t). \tag{26} \]

It is clear that these equalities are equivalent to equalities (11) and (12). Substituting (25) and (26) into the series (6) and assuming that the resulting series converges uniformly, we obtain

\[ \psi(s,t) = \int_0^t d\tau \int_C K(s,s_1,t-\tau)\psi(s_1,\tau)\,ds_1 + f_1(s,t), \tag{27} \]

where

\[ K(s,s_1,t-\tau) = \frac{1}{2}\Phi_0(t) + \sum_{n=1}^{\infty} \left[ B_n^{-1}\Phi_n^{(1)}(s_1,t-\tau)\cos\frac{n\pi s}{l} + B_n^{-1}\Phi_n^{(2)}(s_1,t-\tau)\sin\frac{n\pi s}{l} \right], \tag{28} \]

\[ f_1(s,t) = \frac{1}{2}f_0(t) + \sum_{n=1}^{\infty} \left[ B_n^{-1}f_n^{(1)}(t)\cos\frac{n\pi s}{l} + B_n^{-1}f_n^{(2)}(t)\sin\frac{n\pi s}{l} \right]. \tag{29} \]

If \(d_0\leq 0\) and \(\nu<0\), then the operator \(B_n^{-1}\) grows exponentially, or as \(n^2\) as \(n\) increases, and in this case the series (28) and (29) diverge. In order for these series to converge uniformly, it is necessary and sufficient that \(d_0>0\) or \(\nu\geq 0\). Translating these conditions into \(\lambda\), we have:

\[ \lambda < a_1(a_1+a_2)/2\pi^{3/2}=\lambda_0. \tag{30} \]

On the basis of (30) and (13),

\[ |K(s,s_1,t-\tau)|\leq M_1(t-\tau)^{-3/4}, \tag{31} \]

\[ |K(s',s_1,t-\tau)-K(s'',s_1,t-\tau)| \leq M_1(t-\tau)^{-3/4}|s'-s''|^\alpha \quad (0<\alpha<1/2), \tag{32} \]

and on the basis of (8)

\[ |f_1(s,t)|\leq M_2 t^{-\sigma},\quad |f_1(s',t)-f_1(s'',t)| \leq M_2 t^{-\sigma}|s'-s''|^{\alpha_1} \quad (0<\alpha_1<1). \tag{33} \]

The integral equation (27) can now be integrated by the method of successive approximations, and its solution will satisfy condition (5).

Kharkov Polytechnic Institute
named after V. I. Lenin

Received
4 X 1956