Mathematics

G. S. LITVINCHUK

ON A SPECIAL INTEGRAL EQUATION WITH A SHIFT

(Presented by Academician P. Ya. Kochina, 1 VI 1960)

Consider the integral equation

\[ a(t) f(t) + \frac{b(t)}{\pi i} \int_L \frac{f(\tau)\,d\tau}{\tau-\alpha(t)} = h(t), \tag{1} \]

in which the functions \(a(t)\), \(b(t)\), \(h(t)\) satisfy a Hölder condition on the closed Lyapunov contour \(L\), with \(a(t)\ne 0\) and \(b(t)\ne 0\) at the points of \(L\); the function \(\alpha(t)\) maps the contour \(L\) one-to-one onto itself and has a derivative \(\alpha'(t)\) distinct from zero and satisfying a Hölder condition on \(L\); the solution \(f(t)\) is sought in the class of functions satisfying a Hölder condition.

Special integral equations with kernels containing shifts, as is known, arise in the solution of boundary-value problems for differential equations of elliptic-hyperbolic (mixed) type. This above all explains the considerable attention that has been paid in recent years to the study of such equations \((^{1-6})\).

The types of special equations considered in works \((^{1-6})\) are essentially reduced to the Riemann boundary-value problem, and consequently solutions in closed form have been found for them. The special integral equation with shift in the form (1) is reduced to a boundary-value problem for analytic functions that is more general than the Riemann problem.

In general, the solution of equations of the form (1) will not have a closed form.

In the present work, the integral equation (1) is investigated under the following assumptions:

1. The function \(\alpha(t)\) maps the contour \(L\) onto itself with reversal of the direction of traversal on it.

2. The Carleman conditions \((^{7})\) are satisfied:

\[ \alpha[\alpha(t)] = t, \tag{2} \]

\[ \frac{b(t)b[\alpha(t)]}{a(t)a[\alpha(t)]}=1. \tag{3} \]

Using the Sokhotski formulas for the limiting values of the Cauchy-type integral

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

where \(f(t)\) is the unknown function, we reduce the integral equation (1) to the following boundary-value problem for the piecewise-analytic function \(\Phi(z)\):

\[ \Phi^{+}(t)+A(t)\Phi^{+}[\alpha(t)]-\Phi^{-}(t)+A(t)\Phi^{-}[\alpha(t)] =H(t)\quad \text{on } L, \tag{4} \]

where \(A(t)=b(t)/a(t)\), \(H(t)=h(t)/a(t)\), and condition (3) takes the form

\[ A(t)A[\alpha(t)]=1. \tag{3'} \]

To each solution of problem (4) vanishing at infinity there corresponds, by the formula \(\Phi^{+}(t)-\Phi^{-}(t)=f(t)\), a definite solution of equation (1).

Using conditions (2) and (3′), we reduce the boundary-value problem (4) to the equivalent pair of Riemann—Carleman boundary-value problems:

\[ \Phi^{+}[\alpha(t)]=-A[\alpha(t)]\Phi^{+}(t)+\frac12\{A[\alpha(t)]H(t)+H[\alpha(t)]\} \quad \text{on } L; \tag{5} \]

\[ \Phi^{-}[\alpha(t)]=A[\alpha(t)]\Phi^{-}(t)+\frac12\{A[\alpha(t)]H(t)-H[\alpha(t)]\} \quad \text{on } L. \tag{6} \]

The Riemann—Carleman problem for the interior domain was considered by D. A. Kveselava \((^{8})\). The solution of this problem in \((^{8})\) is based on the study of a Fredholm integral equation whose kernel turns out to have no eigenfunctions. The same method can be applied in studying the Riemann—Carleman problem for the exterior (infinite) domain \(D^{-}\). Here we arrive at a homogeneous integral equation having nontrivial solutions. An analogous relation exists between investigations, by the method of integral equations, of the solutions of the interior and exterior Dirichlet problems \((^{9})\).

It is not difficult to verify that condition (3′) is necessary for the solvability of problems (5) and (6) for an arbitrary right-hand side \(H(t)\). Let us also note that the shift \(\alpha(t)\) has on \(L\) two fixed points \(t_{0}^{\prime}\) and \(t_{0}^{\prime\prime}\). Denote \(\operatorname{Ind} A(t)=\operatorname{Ind} a(t)-\operatorname{Ind} b(t)=\varkappa\). We shall call the number \(\varkappa\) the index of the boundary-value problem (4) and of the integral equation (1).

Theorem 1. The homogeneous boundary-value problem (4) is solvable for any index \(\varkappa\) \((\varkappa=2\varkappa'\) or \(\varkappa=2\varkappa'-1,\ \varkappa'=0,\ \pm1,\ \pm2,\ldots,\ \pm p)\).

For \(\varkappa>0\) the homogeneous problem (4) has \(\varkappa'\) linearly independent solutions if \(A(t_{0}^{\prime})=A(t_{0}^{\prime\prime})=-\lambda=1\), and \(\varkappa'+1\) solutions in the remaining cases: \(A(t_{0}^{\prime})=A(t_{0}^{\prime\prime})=-\lambda=-1\) and \(A(t_{0}^{\prime})=-A(t_{0}^{\prime\prime})=\pm1\).

The general solution of the problem is given by the formulas

\[ \Phi^{+}(z)=X^{+}(z)\left\{R_{\varkappa'}(z)+\frac{1}{2\pi i}\int_{L}\frac{\varphi(\tau)}{\tau-z}\,d\tau\right\}, \]

\[ \Phi^{-}(z)=0, \tag{7} \]

where \(R_{\varkappa'}(z)\) is a rational function with arbitrary coefficients, with a pole at the point \(z=0\) of order not exceeding \(\varkappa'\); \(\varphi(t)\) is a solution of the Fredholm integral equation

\[ K_{+}\varphi\equiv \varphi(t)+\frac{1}{2\pi i}\int_{L}\left[\frac{1}{\tau-t}-\frac{\alpha'(\tau)}{\alpha(\tau)-\alpha(t)}\right]\varphi(\tau)\,d\tau =\lambda R_{\varkappa'}[\alpha(t)]-R_{\varkappa'}(t); \]

\(K_{+}\) is an operator without eigenfunctions.

For \(\varkappa<0\) the homogeneous problem (4) has \(-\varkappa'\) linearly independent solutions if \(A(t_{0}^{\prime})=A(t_{0}^{\prime\prime})=\lambda=-1\), and \(-\varkappa'+1\) solutions in the other two cases. The general solution is given by the formulas

\[ \Phi^{+}(z)=0, \]

\[ \Phi^{-}(z)=X^{-}(z)\left\{P_{\varkappa'}(z)+\frac{1}{2\pi i}\int_{L}\frac{\varphi(\tau)}{\tau-z}\,d\tau\right\}, \tag{8} \]

where \(P_{\varkappa'}(z)\) is an arbitrary polynomial of degree not exceeding \(-\varkappa'\); \(\varphi(t)\) is a solution of the Fredholm integral equation

\[ K_{-}\varphi\equiv \varphi(t)-\frac{1}{2\pi i}\int_{L}\left[\frac{1}{\tau-t}-\frac{\alpha'(\tau)}{\alpha(\tau)-\alpha(t)}\right]\varphi(\tau)\,d\tau = P_{\varkappa'}(t)-\lambda P_{\varkappa'}[\alpha(t)]. \]

The operator \(K_-\) has one eigenfunction \(\varphi(t)=1\). \(X^\pm(z)\) are the canonical functions \({}^{(8,10)}\) of problems (5) and (6), determined respectively by the conditions
\[ X^\pm[\alpha(t)]=\mp \lambda A[\alpha(t)]X^\pm(t). \]

For \(\chi=0\) there exists one linearly independent solution of problem (4), represented by formulas (7), where \(R_{\chi'}(z)\equiv C\), if \(A(t_0')=A(t_0'')=-1\), and by formulas (8), where \(P_{\chi'}(z)\equiv C\), if \(A(t_0')=A(t_0'')=1\).

Theorem 2. The nonhomogeneous problem (4) is unconditionally solvable only for \(\chi=0\). The general solution is expressed by the formulas:
\[ \Phi^+(z)=X^+(z)\left\{R_{\chi'}(z)+\frac{1}{2\pi i}\int_L \frac{\varphi(\tau)\,d\tau}{\tau-z}\right\}, \]
\[ \Phi^-(z)=X^-(z)\left\{P_{\chi'}(z)+\frac{1}{2\pi i}\int_L \frac{\varphi^*(\tau)\,d\tau}{\tau-z}\right\}; \]
\(\varphi(t)\) and \(\varphi^*(t)\) are solutions of the Fredholm integral equations
\[ K_+\varphi=\frac{1}{2}\left\{\frac{H(t)}{X^+(t)}-\frac{\lambda H[\alpha(t)]}{X^+[\alpha(t)]}\right\}+\lambda R_{\chi'}[\alpha(t)]-R_{\chi'}(t), \]
\[ K_-\varphi=\frac{1}{2}\left\{\frac{H(t)}{X^-(t)}-\frac{\lambda H[\alpha(t)]}{X^-[\alpha(t)]}\right\}+P_{\chi'}(t)-\lambda P_{\chi'}[\alpha(t)]. \]

For \(\chi>0\), \(P_{\chi'}(z)\equiv 0\), and the solvability conditions are required:
\[ \int_L t^{k-1}\varphi^*(\tau)\,d\tau=0,\quad k=1,2,\ldots,\chi'-1, \]
to which, when \(A(t_0')=A(t_0'')=-1\), the condition is added
\[ \int_L \left\{\frac{H(t)}{X^-(t)}+\frac{H[\alpha(t)]}{X^-[\alpha(t)]}\right\}\psi(t)\,dt=0; \]
\(\psi(t)\) is a nontrivial solution of the equation \(K_-'\psi=0\), adjoint to the equation \(K_-\varphi^*=0\). For \(\chi<0\), \(R_{\chi'}(z)\equiv 0\), and the conditions
\[ \int_L t^{-k}\varphi(t)\,dt=0,\quad k=1,2,\ldots,-\chi' \]
must be satisfied.

Imposing the condition \(\Phi^-(\infty)=0\), we obtain the following conclusions for the integral equation (1):

1. The number of linearly independent solutions of the homogeneous equation (1) is equal (for \(\chi>0\), and also for \(\chi=0\) and \(A(t_0')=A(t_0'')=1\)) or is less by one (for \(\chi<0\), and also for \(\chi=0\) and \(A(t_0')=A(t_0'')=1\)) than the number of linearly independent solutions of the homogeneous problem (4).

Hence the unsolvability of the homogeneous equation follows in the following three cases:

1) \(\chi=0,\quad A(t_0')=A(t_0'')=1;\)

2) \(\chi=-1;\)

3) \(\chi=-2,\quad A(t_0')=A(t_0'')=-1.\)

1. The nonhomogeneous equation (1) is unconditionally solvable and has a unique solution if \(\chi=0\) and \(A(t_0')=A(t_0'')=1\). In the remaining cases, for solvability of this equation it is necessary and sufficient that the conditions
   \[ \int_L h_k(t)h(t)\,dt=0,\quad k=1,2,\ldots,q(\chi'), \]
   be satisfied, where \(h_k(t)\) are completely determined linearly independent functions not depending on \(h(t)\).

Studying the singular equation adjoint to equation (1), we arrive at the following conclusions:

Theorem 3. The indices of the adjoint and the given integral equations are related by the relation \(\chi^*=\chi-\chi_{\alpha'}\), where \(\chi_{\alpha'}=\operatorname{Ind}\alpha'(t)=-2\).

Theorem 4. The given homogeneous integral equation and the homogeneous equation adjoint to it have the same number of solutions.

Thus, the theory of the singular integral equation (1) represents a distinctive interweaving of Noether theory and Fredholm theory.

I express my gratitude to Prof. F. D. Gakhov for his constant interest in, and attention to, this work.

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