UDC 517.948.33

MATHEMATICS

V. ZHAKOVSKII, G. A. MAGOMEDOV

INVESTIGATION OF A CERTAIN GENERALIZED SYSTEM OF SINGULAR INTEGRAL EQUATIONS IN POGORZELSKI’S CLASS OF FUNCTIONS $\mathfrak{H}$

(Presented by Academician I. N. Vekua on 25 IX 1965)

V. Pogorzelski (¹, ²) introduced into consideration a class of functions $\mathfrak{H}$ and proved a theorem on its transformation into itself by an integral of Cauchy type. These results were a generalization to the complex case of the corresponding results of A. I. Guseinov (³). In the present paper one generalization of the indicated theorem of V. Pogorzelski is made, and the theorem proved is applied to the investigation of a generalized system of singular integral equations with shift.

Let a curve
\[ L=\sum \widehat{c_\sigma c_{\sigma'}}, \]
be given, consisting of a system of a finite number of directed closed and open arcs $\widehat{c_\sigma c_{\sigma'}}$, having no common points except, possibly, common endpoints (for a closed arc the endpoints $c_\sigma$ and $c_{\sigma'}$ coincide). Suppose that the arcs $\widehat{c_\sigma c_{\sigma'}}$ have continuous tangents at every interior point, and one-sided tangents at the endpoints. Number in an arbitrary way the endpoints of the arcs $c_1,c_2,\ldots,c_p$, which may be ordinary endpoints, corner points, or nodal points; the cusp case (return points) is excluded.

We give the definition of Pogorzelski’s class of functions $\mathfrak{H}_\alpha^\mu$ (¹, ²) with respect to the system of arcs specified above.

Definition. By the class $\mathfrak{H}_\alpha^\mu$ we shall mean the set of all functions of a complex variable $\varphi(t)$, defined and continuous at every point $t\in L$ except, possibly, the points of discontinuity $c_1,c_2,\ldots,c_p$, which satisfy the inequalities
\[ |\varphi(t)|\prod_{\sigma=1}^{p}|t-c_\sigma|^\alpha \le \rho,\qquad W(t,t_1)|\varphi(t_1)-\varphi(t)|\le \chi |t_1-t|^\mu, \tag{1} \]
where $\rho$ and $\chi$ are arbitrary positive constants, and the real parameters $\alpha$ and $\mu$ satisfy the conditions $0\le \alpha<1$, $0<\mu<1$, $\alpha+\mu<1$; the second of inequalities (1) holds for each pair of interior points $t$ and $t_1$ of an arbitrary arc $\widehat{c_\sigma c_{\sigma'}}$, and the auxiliary function $W(t,t_1)$ is defined as follows:
\[ W(t,t_1)= \begin{cases} |t-c_\sigma|^{\alpha+\mu}, & \text{when } t,t_1\in \widehat{c_\sigma c_{\sigma'}} \text{ and } c_\sigma=c_{\sigma'},\\ |t-c_\sigma|^{\alpha+\mu}|t_1-c_{\sigma'}|^{\alpha+\mu}, & \text{when } t,t_1\in \widehat{c_\sigma c_{\sigma'}} \text{ and } c_\sigma\ne c_{\sigma'}. \end{cases} \tag{2} \]
If the points $t$ and $t_1$ lie on an open arc $\widehat{c_\sigma c_{\sigma'}}$, it is assumed that $t_1\in \widehat{t c_{\sigma'}}$; but if $t$ and $t_1$ lie on a closed arc $\widehat{c_\sigma c_{\sigma'}}$, it is assumed that $|t-c_\sigma|\le |t-c_{\sigma'}|$ and, moreover, that the ratio of the length of the arc $\widehat{t t_1}$ on which the common endpoint lies to the length of the entire arc does not exceed $1/2$.

By \(\mathfrak{H}_{\alpha}^{\mu}(\rho,\chi)\) we denote the subset of those functions of the class \(\mathfrak{H}_{\alpha}^{\mu}\) which satisfy condition (1) with prescribed values of \(\rho\) and \(\chi\). Suppose, further, that the complex function \(s(t)\), defined for \(t\in L\), satisfies the following conditions, which we shall call conditions \((S)\): it maps each arc \(\widehat{c_\sigma c_{\sigma'}}\) of the curve \(L\) onto itself one-to-one, preserving orientation; moreover,

\[ 0<m_s\leq \left|\frac{s(t_1)-s(t)}{t_1-t}\right|\leq M_s, \tag{3} \]

for every pair of points \(t,t_1\in L\); \(m_s\leq 1,\ M_s\geq 1\) are positive constants. Then the following holds.

Theorem. If the complex function \(f(t,\tau)\), defined for

\[ t,\tau\in L_0=L-\sum_{\sigma=1}^{p} c_\sigma, \]

is a function of the class \(\mathfrak{H}_{\alpha}^{\mu}(\rho,\chi)\) \((\alpha>0)\) with respect to the variable \(\tau\), and also satisfies the Hölder condition with respect to the variable \(t\), i.e.

\[ |f(t,\tau)|\prod_{\sigma=1}^{p}|\tau-c_\sigma|^\alpha\leq \rho, \tag{4} \]

\[ |f(t,\tau)-f(t_1,\tau_1)|\,W(\tau,\tau_1) \leq \chi\bigl(|\tau-\tau_1|^\mu+|t-t_1|^{\mu_1}\bigr), \tag{5} \]

where \(0<\mu<\mu_1\leq 1,\ \alpha+\mu<1\), and the complex function \(s(t)\), defined for \(t\in L\), satisfies conditions \((S)\), then the function \(F(t)\), defined on the set \(L_0\) by the integral

\[ F(t)=\int_L \frac{f(t,\tau)}{\tau-s(t)}\,d\tau, \]

belongs to the class \(\mathfrak{H}_{\alpha}^{\mu}(k_1\rho+k_2\chi,\ k_3\rho+k_4\chi)\), where \(k_1,k_2,k_3,k_4\) are positive constants independent of the function \(f(t,\tau)\).

Proof. Let \(s(t)=t'\); then \(t=s^{-1}(t')\) (\(s^{-1}(t')\) is the inverse function of \(s(t)\)) and \(f(t,\tau)=f[s^{-1}(t),\tau]=f^*(t',\tau)\); consequently,

\[ F(t)=F^*(t')=\int_L \frac{f^*(t,\tau)}{\tau-t'}\,d\tau. \]

In view of (4) and (5) we have

\[ |f^*(t,\tau)|\prod_{\sigma=1}^{p}|\tau-c_\sigma|^\alpha\leq \rho, \tag{6} \]

\[ |f^*(t',\tau)-f^*(t_1',\tau_1)|\,W(\tau,\tau_1) \leq \frac{\chi}{m_s}\bigl(|t'-t_1'|^{\mu_1}+|\tau-\tau_1|^\mu\bigr). \tag{7} \]

Then, by V. Pogorzelski’s theorem (2), we have:

\[ |F^*(t')|\prod_{\sigma=1}^{p}|t'-c_\sigma|^\alpha \leq c_1\rho+c_2\frac{\chi}{m_s}, \tag{8} \]

\[ |F^*(t')-F^*(t_1')|\,W(t',t_1') \leq \left(c_3\rho+c_4\frac{\chi}{m_s}\right)|t'-t_1'|^\mu, \tag{9} \]

where \(c_1,c_2,c_3,c_4\) are positive constants independent of the function \(f(t,\tau)\).

By inequalities (2) and (3) we obtain

\[ \frac{W(t',t_1')}{W(t,t_1)}\geq m_s^{2(\alpha+\mu)},\qquad \frac{|t'-t_1'|}{|t-t_1|}\leq M_s,\qquad \frac{\displaystyle\prod_{\sigma=1}^{p}|t'-c_\sigma|^\alpha} {\displaystyle\prod_{\sigma=1}^{p}|t-c_\sigma|^\alpha} \geq m_s^{p\alpha}. \tag{10} \]

Finally, taking into account (8), (9), and (10), we obtain

\[ |F(t)|\prod_{\sigma=1}^{p}|t-c_{\sigma}|^{\alpha}\leq k_1\rho+k_2\varkappa, \]

\[ |F(t)-F(t_1)|W(t,t_1)\leq (k_3\rho+k_4\varkappa)|t-t_1|^{\mu}, \]

where the constants independent of the function \(f(t,\tau)\) are

\[ k_1=c_1m_s^{-p\alpha},\qquad k_2=c_2m_s^{-(p\alpha+1)},\qquad k_3=c_3m_s^{\mu}m_s^{-2(\alpha+\mu)},\qquad k_4=c_4m_s^{\mu}m_s^{-(2\alpha+2\mu+1)}. \]

The theorem is proved.

Next we study the nonlinear system of singular integral equations:

\[ \varphi_k(t)=F_k\left[t,\varphi_1(t),\varphi_2(t),\ldots,\varphi_n(t), \int_L \frac{N_{k1}\left[t,\tau,\varphi_1(\tau),\varphi_2(\tau),\ldots,\varphi_n(\tau)\right]}{\tau-s_{k1}(t)}\,d\tau,\ldots\right. \]

\[ \left.\ldots,\int_L \frac{N_{km}\left[t,\tau,\varphi_1(\tau),\varphi_2(\tau),\ldots,\varphi_n(\tau)\right]}{\tau-s_{km}(t)}\,d\tau\right] \equiv \widehat{A}_k[\varphi_1(t),\varphi_2(t),\ldots,\varphi_n(t)], \tag{11} \]

where \(\varphi_1(t),\varphi_2(t),\ldots,\varphi_n(t)\) are unknown functions, and \(L\) is the system of arcs defined above.

We shall study the system under the following assumptions.

I. The complex functions \(F_k(t,u_1,u_2,\ldots,u_n;u_{n+1},\ldots,u_{n+m})\), \(k=1,2,\ldots,n\), are defined in the domain
\(\Omega_1\{t\in L_0;\ u_j\in \Pi,\ j=1,2,\ldots,n+m\}\), where \(\Pi\) is the open complex plane, and satisfy the inequality

\[ |F_k(t,u_1,u_2,\ldots,u_{n+m})| \leq \frac{M_F}{\prod_{\sigma=1}^{p}|t-c_{\sigma}|^{\alpha}} +k_F\sum_{i=1}^{n+m}|u_i|, \tag{12} \]

as well as the generalized Hölder–Lipschitz condition

\[ |F_k(t,u_1,\ldots,u_{n+m})-F_k(t_1,u'_1,\ldots,u'_{n+m})| \leq \frac{k'_F|t-t_1|^{\mu}}{W(t,t_1)} +k_F\sum_{i=1}^{n+m}|u_i-u'_i|, \tag{13} \]

where \(M_F,k'_F\), and \(k_F\) are positive constants; \(W(t,t_1)\) is the function defined by formula (2); the constants \(\alpha\) and \(\mu\) satisfy the conditions \(\alpha>0,\ 0<\mu<1,\ \alpha+\mu<1\).

II. The complex functions \(N_{k_i}(t,\tau,w_1,\ldots,w_n)\), \(k=1,2,\ldots,n;\ i=1,2,\ldots,m\), are defined in the domain
\(\Omega_2\{t,\tau\in L_0,\ w_j\in\Pi,\ j=1,2,\ldots,n\}\), and satisfy the inequality

\[ |N_{k_i}(t,\tau,w_1,\ldots,w_n)| \leq \frac{M_N}{\prod_{\sigma=1}^{p}|\tau-c_{\sigma}|^{\alpha}} +k_N\sum_{i=1}^{n}|w_i| \]

and the generalized Hölder–Lipschitz condition

\[ |N_{k_i}(t,\tau,w_1,\ldots,w_n)-N_{k_i}(t_1,\tau_1,w'_1,\ldots,w'_n)| \leq \frac{k'_N\left[|\tau-\tau_1|^{\mu}+|t-t_1|^{\mu_1}\right]}{W(\tau,\tau_1)} +k_N\sum_{i=1}^{n}|w_i-w'_i|, \]

where \(M_N,k'_N\), and \(k_N\) are positive constants; \(W(\tau,\tau_1)\) is defined by formula (2), \(\mu<\mu_1<1\).

III. The complex functions \(s_{ki}(t)\), \(k=1,2,\ldots,n;\ i=1,2,\ldots,m\), defined on \(L\), satisfy the conditions \((S)\) stated above.

The position of the points \(t,t_1\in L_0\) and \(\tau,\tau_1\in L_0\) is determined in accordance with the qualifications adopted in the definition of the class of functions \(\mathfrak{H}_{\alpha}^{\mu}\).

To prove the existence of a solution of the system (11), we apply Schauder’s fixed-point principle. Consider the functional space \(\Lambda\), consisting of all systems of \(n\) complex functions
\([\varphi_1(t),\varphi_2(t),\ldots,\varphi_n(t)]\), continuous on \(L_0\) and satisfying the condition

\[ \max_{1\le k\le n}\sup_{t\in L_0}\prod_{\sigma=1}^{p}|t-c_\sigma|^{\alpha+\mu}|\varphi_k(t)|<\infty. \]

In the usual way we define the sum of elements of \(\Lambda\) and the product of an element by a number:

\[ [\varphi_1(t),\ldots,\varphi_n(t)]+[\psi_1(t),\ldots,\psi_n(t)] = [\varphi_1(t)+\psi_1(t),\ldots,\varphi_n(t)+\psi_n(t)], \]

\[ \lambda[\varphi_1(t),\ldots,\varphi_n(t)] = [\lambda\varphi_1(t),\ldots,\lambda\varphi_n(t)]. \]

The norm \(\|U\|\) of an element \(U=[\varphi_1(t),\ldots,\varphi_n(t)]\) of the space \(\Lambda\) is defined as follows:

\[ \|U\|=\max_{1\le k\le n}\sup_{t\in L_0}\prod_{\sigma=1}^{p}|t-c_\sigma|^{\alpha+\mu}|\varphi_k(t)|. \]

Then the space \(\Lambda\) will be a Banach space. Next consider, in the space \(\Lambda\), the set \(Z(\rho,\chi)\) of all systems of \(n\) functions belonging to the set \(\mathfrak{H}_{\alpha}^{\mu}(\rho,\chi)\). The set \(Z(\rho,\chi)\) is evidently closed, convex, and compact \((^4)\).

Taking into account the form of the system (11), we transform the set \(Z(\rho,\chi)\) by means of the operator

\[ \psi_k(t)=\hat A_k[\varphi_1(t),\ldots,\varphi_n(t)],\qquad k=1,2,\ldots,n, \tag{14} \]

which assigns to each element \([\varphi_1(t),\varphi_2(t),\ldots,\varphi_n(t)]\) from \(Z(\rho,\chi)\) the element \([\psi_1(t),\ldots,\psi_n(t)]\) of some set \(Z'\).

Lemma 1. If the constant \(K_F\) is sufficiently small, namely

\[ K_F<\min\left[ \frac{1}{n(1+mk_1k_N+mk_3K_N)},\, \frac{1}{n(1+mk_2K_N+mk_4K_N)} \right], \tag{15} \]

then \(\rho=\rho_0\) and \(\chi=\chi_0\) can be chosen so that the set \(Z'\) is a subset of the set \(Z(\rho_0,\chi_0)\).

Lemma 2. The operator (14) is continuous in the space \(\Lambda\).

As a result the following has been proved.

Theorem. If conditions I—III are fulfilled and, in addition, inequality (15) holds, then the system of singular integral equations (11) has at least one solution in the class \(\mathfrak{H}_{\alpha}^{\mu}\).

Imposing on the derivatives of the real and imaginary parts of the function \(F_i\) conditions of the type (12) and (13), according to the scheme of the work \((^5)\), one can prove the existence of a unique solution of equation (11), which is found by the method of successive approximations.

Warsaw Polytechnic Institute
Warsaw, Poland

Dagestan State University
named after V. I. Lenin
Makhachkala

Received
18 IX 1965

CITED LITERATURE

\(^1\) W. Pogorzelski, J. Math. and Mech. Indiana Univ., 9 (1960).
\(^2\) W. Pogorzelski, Rownania calkowe i ich zastosowania, 3, Warszawa, 1960.
\(^3\) A. I. Guseinov, Izv. AN SSSR, ser. matem., 12, 193 (1948).
\(^4\) W. Zakowski, Zeszyty nauk. Politechn. Warsz., Matematyka, 2 (1964).
\(^5\) W. Żakowski, Roczn. Polsk. towarz. mat., Prace Mat., 8 (1963).