Reports of the Academy of Sciences of the USSR

1965, Volume 160, No. 6

MATHEMATICS

R. A. KORDZADZE

ON SINGULAR INTEGRAL EQUATIONS WITH SHIFT

(Presented by Academician I. N. Vekua, 7 VII 1964)

Let \(\Gamma\) be a collection of a finite number of simple closed Lyapunov contours having no common points; let \(\alpha(t)\) be a function given on \(\Gamma\) which maps \(\Gamma\) homeomorphically onto itself, preserving the direction of traversal, and has a derivative \(\alpha'(t)\in H\), \(\alpha'(t)\ne 0\) \((t\in\Gamma)\). In addition, it is assumed that \(\alpha(t)\) generates a cyclic group of order \(n\), whose elements we shall denote by

\[ \alpha_0(t)\equiv t,\quad \alpha_1(t)\equiv \alpha(t),\ldots,\alpha_{n-1}(t). \]

In the paper \((^2)\) the singular equation with shift

\[ T\varphi \equiv \sum_{l=0}^{n-1}\left\{A_l(t_0)\varphi(\alpha_l(t_0))+\frac{1}{\pi i}\int_\Gamma \frac{K_l(t_0,t)\varphi(t)\,dt}{t-\alpha_l(t_0)}\right\}=f(t_0) \]

was considered (we use the notation and numbering adopted in \((^2)\)), as well as the adjoint equation \(T'\psi=0\).

In the proof of Theorem 6 an inaccuracy was made, to which N. E. Tövmassian drew our attention. As a consequence, the index formula given in \((^2)\) requires clarification. The remaining results of the paper remain valid. Below we give the proof of Theorem 4 and the index formula for the operator \(T\).

We shall assume that the operator \(T\) is of normal type; hence, as is easy to see, the operator \(T'\) is normal.

Let us prove Theorem 4. As shown in \((^2)\), with the equation \(T\varphi=f\) \((T'\psi=0)\) there is associated a system of singular integral equations

\[ \vec D\vec\Phi=\mathbf F \quad (D^*\mathbf X=0), \]

where

\[ \mathbf F(t_0)=\{f(t_0),\,f(\alpha(t_0)),\ldots,f(\alpha_{n-1}(t_0))\}. \tag{1} \]

To prove Theorem 4, by virtue of Theorem 5, we need to show that from the conditions (3.2) there follows the solvability of the system \(\vec D\vec\Phi=\mathbf F\), and conversely (see \(^1,{}^2\)). Let \(\psi^\delta(t)\) \((\delta=1,2,\ldots,k')\) be a complete system of linearly independent solutions of the equation \(T'\psi=0\). Then the vectors

\[ \{\psi^\delta(t),\psi^\delta(\alpha(t)),\ldots,\psi^\delta(\alpha_{n-1}(t))\} \]

will be solutions of the associated system of equations \(D^*\mathbf X=0\), and the vectors

\[ \mathbf X^\delta(t)=\{\psi^\delta(t),\alpha'(t)\psi^\delta(\alpha(t)),\ldots, \alpha'_{n-1}(t)\psi^\delta(\alpha_{n-1}(t))\} \quad (\delta=1,2,\ldots,k') \]

will be solutions of the system \(D'\psi=0\), adjoint to the system \(\vec D\vec\Phi=0\). Taking into account the equalities

\[ \int_\Gamma \mathbf F(t)\mathbf X^\delta(t)\,dt = n\int_\Gamma f(t)\psi^\delta(t)\,dt \quad (\delta=1,2,\ldots,k'), \]

we immediately obtain the conditions (3.2) as necessary.

Let us prove the sufficiency of the conditions (3.2). Let

\[ \mathbf y^\delta(t)=\{y_0^\delta(t),\ldots,\ldots,y_{n-1}(t)\} \quad (\delta=1,2,\ldots,\tilde k') \]

be a complete system of linearly independent

solutions of \(D'\vec\psi=0\). Then, as is not difficult to verify, the vectors

\[ \left\{y_0^\delta(t),\ \frac{1}{\alpha'(t)}y_1^\delta(t),\ldots,\frac{1}{\alpha'_{n-1}(t)}y_{n-1}^\delta(t)\right\} \]

will be solutions of the system \(D^*X=0\), and the functions

\[ \omega^\delta(t)=y_0^\delta(t)+\frac{1}{\alpha'_{n-1}(\alpha(t))}y_{n-1}^\delta(\alpha(t))+\cdots+ \frac{1}{\alpha'(\alpha_{n-1}(t))}y_1^\delta(\alpha_{n-1}(t)) \]

\[ (\delta=1,2,\ldots,\tilde k') \]

will be solutions of \(T'\psi=0\). Taking into account the equalities

\[ \int_\Gamma F(t)y^\delta(t)\,dt=\int_\Gamma f(t)\omega^\delta(t)\,dt \qquad (\delta=1,2,\ldots,\tilde k'), \]

by virtue of (3.2), we obtain that the system \(D\Phi=F\) is solvable. Theorem 4 is proved.

Let \(H\) be the space of all vectors with \(n\) components satisfying the Hölder condition, and let \(\tilde H\) be the subspace of the space \(H\) whose elements are all vectors of the form (1).

Denote by \(k,k',\tilde k,\tilde k'\) the numbers of zeros of the operators \(T,T',D\), and \(D'\), respectively. It is clear that the numbers of zeros of the operators \(D\) and \((D^*)'\)* coincide (see (2)).

Let

\[ \vec\Phi^\delta(t)=\{\Phi_0^\delta(t),\ldots,\Phi_{n-1}^\delta(t)\} \quad(\delta=1,2,\ldots,\tilde k) \]

and

\[ X^l(t)=\{X_0^l(t),\ldots,X_{n-1}^l(t)\}\quad(l=1,2,3,\ldots,\tilde k') \]

be complete systems of linearly independent solutions of the systems of equations \((D^*)'\vec\Phi=0\) and \(D'X=0\), respectively. Denote by \(d\) (\(d'\)) the maximal number of vectors (if such exist) among \(\{\vec\Phi^\delta(t)\}\) (\(\{X^l(t)\}\)) orthogonal to the subspace \(\tilde H\). Without loss of generality one may assume that these are \(\vec\Phi^i(t)\) \((i=1,2,\ldots,d)\) (\(X^j(t)\) \((j=1,2,\ldots,d')\)). The components of these vectors satisfy the relations

\[ \Phi_0^i(t)+\alpha'_{n-1}(t)\Phi_1^i(\alpha_{n-1}(t))+\cdots+ \alpha'(t)\Phi_{n-1}^i(\alpha(t))\equiv0 \quad(i=1,2,\ldots,d), \]

\[ X_0^j(t)+\alpha'_{n-1}(t)X_1^j(\alpha_{n-1}(t))+\cdots+ \alpha'(t)X_{n-1}^j(\alpha(t))\equiv0 \quad(j=1,2,\ldots,d'). \]

From Theorem 4, as is easily seen, (2) implies that \(\tilde k'=k'+d'\), \(\tilde k=k+d\), and, consequently, for the index of the operator \(T\) we obtain the formula

\[ \operatorname{ind}T=\frac{1}{2\pi} \left\{\arg\frac{\det[G(t)-K(t,t)]}{\det[G(t)+K(t,t)]}\right\}_{\Gamma} -\chi(T), \tag{2} \]

where \(\chi(T)=d-d'\).

Applying the results of F. V. Atkinson \({}^{4}\), from (2) we obtain:

Theorem. If \(V\) is an arbitrary completely continuous operator mapping the Hölder space into itself, then \(\operatorname{ind}(T+V)=\operatorname{ind}T\).

In accordance with formula (2), the results of \({}^{3}\) must be modified: \(x^*\) should everywhere be replaced by \(x^*-\frac12\chi(T)\), where \(\chi(T)\) corresponds to equation (1.7). In addition, in the conditions of Theorems 1–3 the necessity requirements should be omitted.

Novosibirsk State University

Received
2 VII 1964

CITED LITERATURE

\({}^{1}\) N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
\({}^{2}\) R. A. Kordzadze, DAN, 154, No. 6 (1964).
\({}^{3}\) R. A. Kordzadze, DAN, 155, No. 4 (1964).
\({}^{4}\) F. V. Atkinson, Mat. sbornik, 28, No. 1 (1951).

\[ \text{* }(D)^*\text{ is the operator adjoint to the operator }D^*. \]