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MATHEMATICS
G. S. LITVINCHUK
ON THE INDEX AND NORMAL SOLVABILITY OF ONE CLASS OF FUNCTIONAL EQUATIONS
(Presented by Academician V. I. Smirnov on 1 XI 1962)
On a simple closed Lyapunov contour \(L\) there is given a function \(\alpha(t)\), having a derivative \(\alpha'(t)\), satisfying on \(L\) a Hölder condition \((\alpha'(t)\subset H)\), and moreover \(\alpha'(t)\ne 0\). Let the function \(\alpha(t)\) effect a homeomorphic mapping of the contour \(L\) onto itself (with preservation or change of orientation on \(L\)) and satisfy T. Carleman’s condition \((^{1})\)
\[ \alpha[\alpha(t)]\equiv t. \tag{1} \]
Consider the functional equation
\[ \begin{aligned} K_\mu \varphi \equiv{}& a(t)\varphi(t)+\mu b(t)\varphi[\alpha(t)] +\frac{c(t)}{\pi i}\int_L \frac{\varphi(\tau)}{\tau-t}\,d\tau +\mu\frac{d(t)}{\pi i}\int_L \frac{\varphi(\tau)}{\tau-\alpha(t)}\,d\tau \\ &+\int_L m_1(t,\tau)\varphi(\tau)\,d\tau +\mu\int_L m_2[\alpha(t),\tau]\varphi(\tau)\,d\tau = h(t),\qquad \mu=\pm1. \end{aligned} \tag{2} \]
Here \(a(t), b(t), c(t), d(t), h(t)\subset H\) on \(L\); \(m_1(t,\tau), m_2(t,\tau)\) are Fredholm kernels.
The functional equation (2), under certain additional conditions imposed on its coefficients, admits a complete investigation, carried out earlier by the author \((^{2,3})\). In the present article a new approach is proposed to the study of equation (2), based on reducing this equation to a system of singular integral equations with Cauchy kernels. This method makes it possible to find exact limits for the applicability of Noether theory to equation (2).
§ 1. Consider the system of singular equations
\[ \begin{aligned} &a(t)\rho_1(t)+\nu b(t)\rho_2(t) +\frac{c(t)}{\pi i}\int_L \frac{\rho_1(\tau)}{\tau-t}\,d\tau +\frac{\nu\lambda d(t)}{\pi i}\int_L \frac{\alpha'(\tau)\rho_2(\tau)}{\alpha(\tau)-\alpha(t)}\,d\tau \\ &\qquad +\int_L m_1(t,\tau)\rho_1(\tau)\,d\tau +\nu\lambda\int_L m_2[\alpha(t),\alpha(\tau)]\alpha'(\tau)\rho_2(\tau)\,d\tau = h(t), \end{aligned} \tag{3} \]
\[ \begin{aligned} &b[\alpha(t)]\rho_1(t)+\nu a[\alpha(t)]\rho_2(t) +\frac{d[\alpha(t)]}{\pi i}\int_L \frac{\rho_1(\tau)}{\tau-t}\,d\varepsilon \\ &\qquad +\frac{\nu\lambda c[\alpha(t)]}{\pi i}\int_L \frac{\alpha'(\tau)\rho_2(\varepsilon)}{\alpha(\tau)-\alpha(t)}\,d\tau +\int_L m_2(t,\tau)\rho_1(\tau)\,d\tau \\ &\qquad +\nu\lambda\int_L m'_1[\alpha(t),\alpha(\tau)]\alpha'(\tau)\rho_2(\tau)\,d\varepsilon = \nu h[\alpha(t)],\qquad \nu=\pm1 . \end{aligned} \]
Here \(\lambda=1\) if \(\alpha(t)\) preserves the orientation on \(L\), and \(\lambda=-1\) if the orientation on \(L\) changes.
Let \(l\) be the number of linearly independent solutions of the homogeneous system of equations (3), and let \(k_+\) and \(k_-\) be the numbers of linearly independent solutions of the homogeneous equation (2), respectively for \(\mu=1\) and \(\mu=-1\). The following assertions are valid: a) \(l=k_+ + k_-\); b) there exist \(l\) linearly indepen-
independent solutions of the homogeneous system (3) such that \(k_+\) solutions satisfy the condition \(\rho_2[\alpha(t)]-\nu\rho_1(t)=0\), and the remaining \(k_-\) solutions satisfy the condition \(\rho_2[\alpha(t)]+\nu\rho_1(t)=0\); c) the nonhomogeneous equation (2) is solvable if and only if the nonhomogeneous system (3) is solvable: if equation (2) is solvable and \(\varphi(t)\) is its solution, then the functions \(\rho_1(t)=\varphi(t)\), \(\rho_2(t)=\mu\nu\varphi[\alpha(t)]\) are a solution of system (3); conversely, if system (3) is solvable and \(\{\rho_1(t),\rho_2(t)\}\) is a solution of this system, then the function \(\varphi(t)=\frac12\{\rho_1(t)+\mu\nu\rho_2[\alpha(t)]\}\) is a solution of equation (2).
Let us also consider the functional equation
\[ \begin{aligned} K'_\mu\psi \equiv{}& a(t)\psi(t)+\mu\lambda\alpha'(t)b[\alpha(t)]\psi[\alpha(t)] -\frac{1}{\pi i}\int_L \frac{c(\tau)\psi(\tau)}{\tau-t}\,d\tau \\ &-\frac{\mu\lambda}{\pi i}\int_L \frac{d[\alpha(\tau)]\alpha'(\tau)\psi[\alpha(\tau)]}{\tau-t}\,d\tau +\int_L m_1(\tau,t)\psi(\tau)\,d\tau \\ &+\mu\lambda\int_L m_2(\tau,t)\alpha'(\tau)\psi[\alpha(\tau)]\,d\tau=0, \end{aligned} \tag{4} \]
adjoint to equation (2). Let \(k'_+\) and \(k'_-\) be the numbers of linearly independent solutions of equation (4), respectively for \(\mu=1\) and \(\mu=-1\), and let \(l'\) be the number of linearly independent solutions of the system
\[ a(t)\omega_1(t)+\nu b[\alpha(t)]\omega_2(t) -\frac{1}{\pi i}\int_L \frac{c(\tau)\omega_1(\tau)}{\tau-t}\,d\tau -\frac{\nu}{\pi i}\int_L \frac{d[\alpha(\tau)]\omega_2(\tau)}{\tau-t}\,d\tau +\int_L m_1(\tau,t)\omega_1(\tau)\,d\tau +\nu\int_L m_2(\tau,t)\omega_2(\tau)\,d\tau=0, \tag{5} \]
\[ \begin{aligned} b(t)\omega_1(t)+\nu a[\alpha(t)]\omega_2(t) &-\frac{\lambda\alpha'(t)}{\pi i}\int_L \frac{d(\tau)\omega_1(\tau)}{\alpha(\tau)-\alpha(t)}\,d\tau \\ &-\frac{\nu\lambda\alpha'(t)}{\pi i}\int_L \frac{c[\alpha(\tau)]\omega_2(\tau)}{\alpha(\tau)-\alpha(t)}\,d\tau +\lambda\alpha'(t)\int_L m_2[\alpha(\tau),\alpha(t)]\omega_1(\tau)\,d\tau \\ &+\nu\lambda\alpha'(t)\int_L m_1[\alpha(\tau),\alpha(t)]\omega_2(\tau)\,d\tau=0, \end{aligned} \]
adjoint to system (3). We have \(l'=k'_+ + k'_-\), where \(k'_+\) and \(k'_-\) solutions satisfy, respectively, the conditions
\[ \omega_2(t)-\nu\lambda\alpha'(t)\omega_1[\alpha(t)]=0,\qquad \omega_2(t)+\nu\lambda\alpha'(t)\omega_1[\alpha(t)]=0. \]
Denote \(A(t)=a(t)+c(t)\), \(C(t)=c(t)-a(t)\), \(B(t)=b(t)+d(t)\), \(D(t)=d(t)-b(t)\).
Using the known condition for a system of singular equations with Cauchy kernels to belong to the normal Noetherian type \((^4)\), we obtain the following principal theorem.
Theorem 1. In order that the equation \(K_\mu\varphi=h(t)\) be a Noether equation, it is necessary and sufficient that the following conditions hold on \(L\):
a) \(\theta(t)=B(t)D[\alpha(t)]-A[\alpha(t)]C(t)\ne0\), if \(\lambda=-1\);
b) \(\theta_1(t)=C(t)C[\alpha(t)]-D(t)D[\alpha(t)]\ne0\), \(\theta_2(t)=A(t)A[\alpha(t)]-B(t)B[\alpha(t)]\ne0\), if \(\lambda=1\).
It is easy to formulate Noether theorems for the functional equation (2).
Theorem 2. The index of the functional equation \(K_\mu \varphi = 0\) over the field of real numbers is expressed by the formulas:
\[ I=2\operatorname{Ind}\theta(t), \qquad \text{if } \lambda=-1; \tag{6} \]
\[ I=\operatorname{Ind}\theta_1(t)-\operatorname{Ind}\theta_2(t), \qquad \text{if } \lambda=1. \tag{7} \]
§ 2. The numbers of linearly independent solutions of the characteristic \((m_1(t,\tau)=m_2(t,\tau)\equiv0)\) equations \(K_\mu\varphi=0\) and \(K'_\mu\psi=0\) (and these solutions themselves) are completely determined in the following cases:
1) \(\lambda=-1,\ \theta(t)\ne0,\ \theta_1(t)=\theta_2(t)\equiv0;\)
2) \(\lambda=1,\ \theta_1(t)\ne0,\ \theta_2(t)\ne0,\ \theta(t)\equiv0;\)
3) \(\Lambda(t)=B(t)C[\alpha(t)]-D(t)A[\alpha(t)]\equiv0\) on \(L\)
and one of the conditions a) or b) of Theorem 1 must be satisfied. Indeed, the characteristic equation (2) is equivalent to a system of two Carleman problems \(\left({}^3\right)\) in case 1), to a Haseman problem \(\left({}^5\right)\) in case 2), and to a Riemann problem in case 3). Let us note* that case 1) was investigated in the author’s paper \(\left({}^3\right)\). There the formula (6) was also established for this case.
§ 3. Let us consider several special cases of the functional equation (2).
- Let \(b(t)=c(t)\equiv0\) on \(L\). If the conditions
\[ \theta(t)=a(t)a[\alpha(t)]+d(t)d[\alpha(t)]\ne0 \qquad \text{for } \lambda=-1, \]
\[ \theta_1(t)\equiv\theta_2(t)=a(t)a[\alpha(t)]-d(t)d[\alpha(t)]\ne0 \qquad \text{for } \lambda=1, \]
are satisfied, then for equation (2) the Fredholm alternative is valid \((I=0)\) (see also \(\left({}^2\right)\)).
- Let \(c(t)=b(t),\ d(t)=-a(t)\). Suppose that \(\lambda=-1\). Then condition a) of Theorem 1 takes the form \(a^2(t)-b^2(t)\ne0\);
\[ I=2\operatorname{Ind}\frac{a(t)-b(t)}{a(t)+b(t)}. \]
If \(\lambda=1\), then
\[ \theta_1(t)=-\theta_2(t)=-2\{a(t)b[\alpha(t)]+a[\alpha(t)]b(t)\}\ne0; \]
\[ I=0. \]
Results of an analogous character are obtained if \(c(t)=-b(t),\ d(t)=a(t)\). Let also: a) \(c(t)=b(t),\ d(t)=a(t)\) or b) \(c(t)=-b(t),\ d(t)=-a(t)\). Then for \(\lambda=-1\) we have
\[ \theta(t)=2\{a(t)a[\alpha(t)]-b(t)b[\alpha(t)]\}\ne0 \]
and \(I=0\), while for \(\lambda=1\) equation (2) is not a Noether equation, since \(\theta_1(t)=\theta_2(t)\equiv0\).
- Let \(c(t)=-a(t),\ d(t)=b(t)\). If \(\lambda=-1\), then \(\theta(t)\equiv0\), and equation (2) does not belong to the Noether type. If \(\lambda=1\), then condition b) of Theorem 1 gives \(a(t)\ne0,\ b(t)\ne0\), and from Theorem 2 we have that
\[ I=2\operatorname{Ind}\frac{a(t)}{b(t)} \]
(see (4), p. 227). We have analogous results if \(c(t)=a(t),\ d(t)=-b(t)\). Finally, if: a) \(c(t)=a(t), d(t)=-b(t)\) or b) \(c(t)=-a(t), d(t)=b(t)\), then the corresponding equation (2) is not a Noether equation either for \(\lambda=1\) or for \(\lambda=-1\).
- Let: 1) \(c(t)=d(t)\equiv0\) or 2) \(a(t)=b(t)\equiv0\). Both functional equations belong to the Fredholm type.
§ 4. The characteristic equation (2) is reduced to the problem of finding a piecewise-analytic function \(\{\Phi^+(z),\Phi^-(z)\}\), vanishing at infinity, from the boundary condition on \(L\)
\[ A(t)\Phi^+(t)+\mu B(t)\Phi^+[\alpha(t)]+C(t)\Phi^-(t)+\mu D(t)\Phi^-[\alpha(t)]=h(t). \tag{8} \]
If in (8) we put \(C(t)=D(t)\equiv0\) or \(A(t)=B(t)\equiv0\), then we arrive at the boundary-value problems on \(L\)
\[ A(t)\Phi^+(t)+\mu B(t)\Phi^+[\alpha(t)]=h(t),\quad C(t)\Phi^-(t)+\mu D(t)\Phi^-[\alpha(t)]=ht. \tag{9} \]
* In the paper \(\left({}^3\right)\), to the condition \(a(t), b(t), c(t), d(t)\ne0\) on \(L\), one should add the condition: the inequality \(\theta(t)\ne0\) is satisfied on \(L\) everywhere except for a finite number of points of \(L\), \(g_+(t), g_-(t)\in H\) on \(L\). Of course, all the theorems of paper \(\left({}^3\right)\) are valid if the condition \(a(t), b(t), c(t), d(t)\ne0\) on \(L\) is replaced by condition a) of the present paper.
The conditions of Theorem 1 for the boundary value problems (9) are not satisfied and, consequently, the problems (9) do not belong to the normal Noetherian type. This result is easy to explain. If \(\lambda=1\), then the problems (9) belong to the type of so-called one-sided problems considered by E. I. Zverovich and the author \({}^{6}\). As shown in \({}^{6}\), such problems may have an infinite set of linearly independent solutions. If \(\lambda=-1\), then we have the well-known problems of T. Carleman. The violation of Noether’s theorems here consists in the fact that fulfillment of the conditions of normal solvability (the second Noether theorem) turns out to be insufficient for solvability of these problems; in addition (for example, for the first problem (9)) \(L\) conditions must be satisfied:
\[ A(t)A[\alpha(t)] = B(t)B[\alpha(t)], \qquad \mu B[\alpha(t)]h(t) + A(t)h[\alpha(t)] = 0. \]
Received
26 X 1962
CITED LITERATURE
\({}^{1}\) T. Carleman, Verhandl. des Internat. Mathem. Kongr., Zürich, 1, 1932.
\({}^{2}\) G. S. Litvinchuk, Izv. AN SSSR, ser. matem., 25, No. 6 (1961).
\({}^{3}\) G. S. Litvinchuk, DAN, 139, No. 2 (1961).
\({}^{4}\) N. P. Vekua, Systems of Singular Integral Equations, Moscow–Leningrad, 1950.
\({}^{5}\) D. A. Kveselava, DAN, 53, No. 8 (1946).
\({}^{6}\) E. I. Zverovich, G. S. Litvinchuk, DAN, 145, No. 2 (1962).