1 INTRODUCTION

Many mathematical models of natural phenomena, related with a liquid medium, in supersonic and transonic flows of gases, arising in the theory of infinitesimal bending of surfaces, as well as, in the membrane theory of shells with curvature of a variable sign, are directly connected with the mixed type partial differential equations (PDEs) (see, [1–3]). Along with studying the basic boundary-value problems for the mixed type PDEs, interest in investigations of nonlocal problems that arise in the study of various problems of mathematical biology, forecasting soil moisture, and problems of physics and plasma has grown. General defnitions and classifications of such problems can be found in [4]. More detailed information about nonlocal problems for mixed type PDEs, concerning the work of recent years, can be found in [5–9].

Note that at present, from the point of view of physical applications, nonlocal problems are of a great interest, where the boundary conditions are specified in the form of an integral operator. When the boundary of the physical process is not available for measurement, as an additional information for the unique solvability of the problem, nonlocal conditions in the integral form can be used (see, for example, [10–12]).

A new nonlocal boundary-value problem for elliptic equations arising in the plasma theory was formulated in [1], which is now called the Bitsadze–Samarskii problem. Such types of problems and various generalizations of these problems were considered by many authors (for this topic, we can mention works [13–16]). Even though the list of works with Bitsadze–Samarskii problems is quite large, similar problems for mixed equations with conditions in the hyperbolic part of the domain are poorly studied. This is due to the fact that the study of such problems usually leads to the solution of the Fredholm or Voltaire integral equations, or to singular integral equations with one unknown, the solvability problems for which have been studied sufficiently (see, for example, [9, 17, 18]).

In this work the Bitsadze–Samarskii (BS) type problem in the domains with the deviation from the characteristic, when the conditions on deviation and on characteristics are given in the form of a first-order differential operator, is investigated.

2 FORMULATION OF THE PROBLEM

The Bitsadze–Samarsky (BS) problem consists in finding a solution \(u=u(x,y)\) that satisfies the equation

$$-\textrm{sgn}\!\left(y\right)u_{xx}\left(x,y\right)-\textrm{sgn}\!\left(x\right)u_{yy}\left(x,y\right)=f(x,y)$$
(1)

in a mixed domain \(\Omega\subset R^{2}\) which is simply connected, bounded by a Lyapunov’s curve \(\sigma\) (for \(x>0\), \(y>0\)) with end points \(A(1,0)\) and \(B(0,1)\), by the real characteristics \(OD:x+y=0\) and \(DA:x-y=1\) for \(x>0\), \(y<0\); \(OC:x+y=0\) and \(CB:x-y=-1\) for \(x<0\), \(y>0\) of the homogeneous equation (1) (\(f(x,y)=0\)) and that these characteristics meet at a points \(D\) (for \(y<0\)) and \(C\) (for \(y>0\)), and fulfills the following boundary conditions

$$u(x,y)=0,\quad(x,y)\in\sigma,$$
(2)
$$a_{1}(t)[u_{y}+u_{x}](\theta_{1}(t))=b_{1}(t)[u_{y}+u_{x}](\theta_{1}^{*}(t)),$$
(3)
$$a_{2}(\tau)[u_{y}-u_{x}](\theta_{2}(\tau))=b_{2}(\tau)[u_{y}-u_{x}](\theta_{2}^{*}(\tau)),$$
(4)

where \(a_{i}(t)\), \(b_{i}(t)\), \(i=1,2\), are given functions.

The smooth curves \(y=\gamma_{1}(x)\), \(0\leq x\leq 1\) and \(x=\gamma_{2}(y)\), \(0\leq y\leq 1\), \(\gamma_{i}(0)=\gamma_{i}(1)=0\), \(i=1,2\), lying strictly inside the characteristic triangles \(\Delta OAD:0\leq x+y\leq x-y\leq 1\) and \(\Delta OBC:0\leq x+y\leq y-x\leq 1\) respectively . Also, we suppose that \(\gamma_{i}(t)\in C^{2}[0,1]\) for \(i=1,2\) and that the functions \(t+\gamma_{1}(t)\) and \(t-\gamma_{2}(t)\) monotonically increases.

As an example for a such function one can consider \(\gamma_{1}(x)=a\sin(\pi x)\), where \(x\in[0,1]\), \(a<0\). Here, \(a\) can be chosen so that \(\gamma_{1}\) strongly lie inside the characteristic triangle \(0\leq x+y\leq x-y\leq 1\).

Denote the elliptic region of the mixed domain \(\Omega\) by \(\Omega_{3}=\Omega\cap\{x>0,y>0\}\) and the hyperbolic regions by \(\Omega_{2}=\Omega\cap\{x<0,y>0\}\) and \(\Omega_{1}=\Omega\cap\{x<0,y<0\}\) respectively. \(\theta_{1}(t)\), \(\theta_{1}^{*}(t)\), \(\theta_{2}(t)\), and \(\theta_{2}^{*}(t)\) are the coordinate points obtained from the intersection of the characteristics \(x-y=1\) and \(x+y=t\), the curve \(y=\gamma_{1}(t)\) with the characteristic \(x+y=t\) outgoing from the points \((t,0)\), \(0<t<1\), the characteristics \(x+y=0\) and \(y-x=\tau\), and the curve \(x=\gamma_{2}(\tau)\) with the characteristic \(y-x=\tau\) outgoing from the points \((0,\tau)\), \(0<\tau<1\) respectively.

Note that in the cases of \(b_{i}(t)\equiv 0\), \(i=1,2\) (BS) problem coincides with the known Tricomi problem for the equation (1) (see, [1]), while for the case of \(a_{i}(t)\equiv 0\), \(i=1,2\), regular solvability of this problem was investigated in [9]. Prior to our main result, we need to define regular solution for the (BS) Problem.

Definition 1. A function \(u=u(x,y)\) is a regular solution of the homogeneous problem (BS) if:

(i) \(u\in W\equiv C(\bar{\Omega})\cap C^{1}(\overline{\Omega}\backslash(BC\cup OD)\cap C^{2}(\Omega_{1}\cup\Omega_{2}\cup\Omega_{3});\)

(ii) \(u\)satisfies equation (1) in\(\Omega_{i}\),\(i=1,2,3\)and conditions (2)–(4);

(iii) \(u_{x}(0,y)\)and\(u_{y}(x,0)\) may go to infinity of order \({1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}\) respectively at the points B and A.

where\(\bar{\Omega}=\Omega\cup\partial\Omega\) is the closure of\(\Omega\).

Let \(L\) be a smooth contour and \(\phi(t)\) a function of position on \(L\). Then

Definition 2. The function \(\phi(t)\) is said to satisfy on the curve the Holder condition ( \(H(\lambda)\) condition), if for two arbitrary points of this curve \(|\phi(t_{1})-\phi(t_{2})|\leq k|t_{1}-t_{2}|^{\lambda},\) where the Holder constant \(k\) and the Holder index \(\lambda\) are positive numbers.

3 MAIN RESULT

The existence and uniqueness of regular solution to the problem (BS) is given by the following

Theorem 1. Suppose that the following conditions holds

$$a_{i}(t),b_{i}(t)\in C^{2}[0,1],\quad a_{i}(t)\neq b_{i}(t),\quad i=1,2.$$

Then for any function \(f(x,y)\in C^{1+\delta}(\bar{\Omega})\) , \(0<\delta<1\) there is a unique regular solution of the (BS) problem.

Proof. A solution of the problem (BS) in \(\Omega_{1}\) can be represented by the D’ Alembert’s formula

$$u(\xi,\eta)=\frac{1}{2}\left[\tau_{1}(\xi)+\tau_{1}(\eta)-\int\limits_{\xi}^{\eta}v_{1}(t)dt\right]-\int\limits_{\xi}^{\eta}\int\limits_{t}^{\eta}f_{1}(t,\tau)dtd\tau,$$

where

$$f_{1}(\xi,\eta)=\frac{1}{4}f\left(\frac{\xi+\eta}{2},\quad\frac{\xi-\eta}{2}\right),\quad\xi=x+y,\quad\eta=x-y,$$

and \(\tau_{1}(x)=u(x,0),\nu_{1}(x)=u_{y}(x,0).\) Thus, we are now solving the Cauchy problem for the functions \(\tau_{1}(x)\) and \(\nu_{1}(x)\) in the domain \(\Omega_{1}\).

The equation of the curve \(\gamma_{1}\) in characteristic variables can be written as \(\eta=\lambda_{1}(\xi)\), \(0\leq\xi\leq 1\). Then using condition (3) from (4) we get the following relation

$$\tau_{1}^{{}^{\prime}}(t)+v_{1}(t)=F_{1}(t),\quad 0<t<1,$$
(5)

where

$$F_{1}(t)=\frac{2b_{1}(t)}{a_{1}(t)-b_{1}(t)}\int\limits_{t}^{\lambda_{1}(t)}f_{1}(t,\tau)d\tau-\frac{2a_{1}(t)}{a_{1}(t)-b_{1}(t)}\int\limits_{t}^{1}f_{1}(t,\tau)d\tau.$$
(6)

Similarly, from the D’ Alembert’s formula

$$u(\xi,\eta)=\frac{1}{2}\left[\tau_{2}(\xi)+\tau_{2}(\eta)-\int\limits_{\xi}^{\eta}v_{2}(t)dt\right]-\int\limits_{\xi}^{\eta}dt\int\limits_{t}^{\eta}f_{2}(t,\tau)d\tau,$$

considering (4), from the domain \(\Omega_{2}\) one can deduce the following relation between functions \({\tau_{2}(y)=u(0,y)}\) and \(\nu_{2}(y)=u_{x}(0,y)\)

$$\tau_{2}^{{}^{\prime}}(t)-v_{2}(t)=F_{2}(t),\quad 0<t<1,$$
(7)

where

$$F_{2}(t)=\frac{2a_{2}(t)}{a_{2}(t)-b_{2}(t)}\int\limits_{0}^{t}f_{2}(\tau,t)d\tau-\frac{2b_{2}(t)}{a_{2}(t)-b_{2}(t)}\int\limits_{\lambda_{2}(t)}^{t}f_{2}(\tau,t)d\tau,$$
$$\xi=x+y,\quad\eta=y-x,\quad f_{2}(\xi,\eta)=\frac{1}{4}f\left(\frac{\xi-\eta}{2},\quad\frac{\xi+\eta}{2}\right),$$

\(\xi=\lambda_{2}(\eta)\), \(0\leq\eta\leq 1\) is an equation of the curve \(\gamma_{2}\) in characteristic variables \(\xi\), \(\eta\), existence of which provided by conditions imposed on \(\gamma_{2}\).

3.1 The Uniqueness of the Solution

For the uniqueness of the regular solution of the problem (BS), we suppose that the problem has two different regular solutions \(u_{1}(x,y)\) and \(u_{2}(x,y)\). Then \(u(x,y)=u_{1}(x,y)-u_{2}(x,y)\) is the regular solution of the homogeneous problem (BS) (\(f(x,y)=0\)). We prove that \(u(x,y)\equiv 0\) in \(\bar{\Omega}\), which provides the uniqueness of the solution of the problem (BS).

Let \(u(x,y)\) be a regular solution of the homogeneous problem (BS) in the domain \(\Omega_{3}\). Multiplying the equation \((u_{xx}+u_{yy})=0\) by \(u\), integrating this identity along the domain \(\Omega_{3}\), applying Green’s formula, and using the condition (2), we obtain

$$\iint\limits_{\Omega_{3}}\left[u_{x}^{2}+u_{y}^{2}\right]dxdy+\int\limits_{0}^{1}\tau_{1}(x)v_{1}(x)dx+\int\limits_{0}^{1}\tau_{2}(y)v_{2}(y)dy=0.$$
(8)

On the other hand, from the fundamental relations (5) and (7) it follows that

$$\int\limits_{0}^{1}\tau_{1}(t)\cdot v_{1}(t)dt=\frac{1}{2}\tau_{1}^{2}(0),\quad\int\limits_{0}^{1}\tau_{2}(t)\cdot v_{2}(t)dt=-\frac{1}{2}\tau_{2}^{2}(0).$$

Substituting the last equalities in the equation (8) we obtain

$$\iint\limits_{\Omega_{3}}\left[u_{x}^{2}+u_{y}^{2}\right]dxdy+\frac{1}{2}\tau_{1}^{2}(0)-\frac{1}{2}\tau_{2}^{2}(0)=0.$$

From the continuity of the solution in \(\bar{\Omega}_{3}\) it follows that \(\tau_{1}(0)=\tau_{2}(0)\) and considering (2) we have \(u(x,y)\equiv 0\) on \(\bar{\Omega}_{3}\). Further, since \(\tau_{i}(t)=0\), \(v_{i}(t)=0\), \(i=1,2\) which follows from \(u(x,y)\equiv 0\) and (5), (7), we obtain \(u(x,y)\equiv 0\) on the the domains \(\bar{\Omega}_{1}\) and \(\bar{\Omega}_{2}\). Therefore, \(u(x,y)\equiv 0\) in \(\bar{\Omega}\). This completes the proof of the uniqueness of the solution for the problem (BS). \(\Box\)

3.2 The Existence of the Solution

The proof of the existence of the solution is mainly based on the method of singular integral equations.

It is known that the solution to the problem (BS) in \(\Omega_{3}\) can be represented by the formula (see, [9])

$$u(x,y)=-\int\limits_{0}^{1}G(x,y,t,0)v_{1}(t)dt-\int\limits_{0}^{1}G(x,y,0,t)v_{2}(t)dt+\iint\limits_{\Omega_{3}}G(x,y,\xi,\eta)f(\xi,\eta)d\xi d\eta,$$
(9)

where \(G(x,y,\xi,\eta)\) is the known Green’s function of a problem (N):

Find a regular solution of the equation (1) in \(\Omega_{3}\), satisfying the boundary conditions

$$(\mathbf{N})\begin{cases}u|_{\sigma}=0,\\ u_{y}(x,0)=\nu_{1}(x),\quad 0<x<1,\\ u_{x}(0,y)=\nu_{2}(y),\quad 0<y<1.\end{cases}$$

Differentiating the equation (9) with respect to \(x\) and \(y\) and taking the limits as \(y\to+0\) and \(x\to+0\), we get the functional relations between the functions \(\tau_{i}(t)\) and \(\nu_{i}(t)\), \(i=1,2\)

$${\tau_{1}^{{}^{\prime}}(x)=-\frac{d}{dx}\int\limits_{0}^{1}G(x,0,t,0)v_{1}(t)dt-\frac{d}{dx}\int\limits_{0}^{1}G(x,0,0,t)v_{2}(t)dt}$$
$${}+\frac{d}{dx}\left(\iint\limits_{\Omega_{3}}G(x,0,\xi,\eta)f(\xi,\eta)d\xi d\eta\right),\quad 0<x<1,$$
(10)

and

$$\tau_{2}^{{}^{\prime}}(y)=-\frac{d}{dy}\int\limits_{0}^{1}G(0,y,t,0)v_{1}(t)dt-\frac{d}{dy}\int\limits_{0}^{1}G(0,y,0,t)v_{2}(t)dt$$
$${}+\frac{d}{dy}\left(\iint\limits_{\Omega_{3}}G(0,y,\xi,\eta)f(\xi,\eta)d\xi d\eta\right),\quad 0<y<1$$
(11)

on \(OA\) and \(OB\) respectively.

We prove the existence of solution for a special case, where the curve \(\sigma\) coincides with an arc of the unit circle \(x^{2}+y^{2}=1\). It is known that in this case the Green-function of the Neumann problem (N) has a form [9]

$$G(x,y,\xi,\eta)=\frac{1}{2\pi}\ln\left|\frac{1-z^{2}\bar{\zeta}^{2}}{z^{2}-\bar{\zeta}^{2}}\cdot\frac{1-z^{2}\zeta^{2}}{z^{2}-\zeta^{2}}\right|,\quad z=x+iy,\quad\zeta=\xi+i\eta,$$

and, as we will show below, that the problem (BS) is equivalently reduced to the system of singular integral equations, whose solution can be written in a quadrature through the known functions.

Note that in the case of the arbitrary contour \(\sigma\) satisfying Lyapunov’s condition and ending of as much as small length in small arcs of a normal contour, the Green function can be constructed by the method of conformal mappings (see, [1]). In this case, the considered problem is equivalently reduced to a system of singular integral equations, which contains both singular and regular kernels. Then using regularization, we reduce this system to the system of Fredholm equations of the second kind and unique solvability follows from the uniqueness of the problem (BS).

Having eliminated \(\tau_{1}^{{}^{\prime}}(x)\) and \(\tau_{2}^{{}^{\prime}}(y)\) from (5), (7), (10), (11) and considering the identities

$$G(x,0,t,0)=G(0,x,0,t),\quad G(0,x,t,0)=G(x,0,0,t),$$
$$\frac{\partial}{\partial x}G(x,0,t,0)=\frac{2x}{\pi}\left(\frac{1}{t^{2}-x^{2}}-\frac{t^{2}}{1-t^{2}x^{2}}\right),$$
$$\frac{\partial}{\partial x}G(0,x,t,0)=-\frac{2x}{\pi}\left(\frac{1}{t^{2}+x^{2}}-\frac{t^{2}}{1+t^{2}x^{2}}\right)$$

we obtain the following system in \(v_{1}(x)\) and \(v_{2}(x)\) :

$$\begin{cases}{{\displaystyle v_{1}(x)-\frac{2x}{\pi}\int\limits_{0}^{1}\frac{(1-t^{4})v_{1}\left(t\right)dt}{(t^{2}-x^{2})(1-x^{2}t^{2})}+\frac{2x}{\pi}\int\limits_{0}^{1}\frac{(1-t^{4})v_{2}\left(t\right)dt}{(t^{2}+x^{2})(1+x^{2}t^{2})}}=L_{1}\left(x\right),}\\ \\ {{\displaystyle v_{2}(x)-\frac{2x}{\pi}\int\limits_{0}^{1}\frac{(1-t^{4})v_{1}\left(t\right)dt}{(t^{2}+x^{2})(1+x^{2}t^{2})}+\frac{2x}{\pi}\int\limits_{0}^{1}\frac{(1-t^{4})v_{2}\left(t\right)dt}{(t^{2}-x^{2})(1-x^{2}t^{2})}}=L_{2}\left(x\right),}\end{cases}$$
(12)

where

$$L_{1}\left(x\right)=F_{1}\left(x\right)-\frac{d}{dx}\iint\limits_{\Omega_{3}}G\left(x,0,\xi,\eta\right)f\left(\xi,\eta\right)d\xi d\eta,$$
$$L_{2}\left(x\right)=-F_{2}\left(x\right)+\frac{d}{dx}\iint\limits_{\Omega_{3}}G\left(0,x,\xi,\eta\right)f\left(\xi,\eta\right)d\xi d\eta.$$
(13)

Thus, in the case of a normal contour the problem (BS) is reduced to a system of singular integral equations (12). Using the change of variables

$$x=\frac{y^{\frac{1}{2}}}{\left(1+\sqrt{1-y^{2}}\right)^{\frac{1}{2}}},\quad t=\frac{\tau^{\frac{1}{2}}}{\left(1+\sqrt{1-\tau^{2}}\right)^{\frac{1}{2}}}$$
(14)

the system (12) can be rewritten in the form

$$\begin{cases}{{\displaystyle\rho_{1}\left(y\right)-\frac{1}{\pi}\int\limits_{0}^{1}\frac{\rho_{1}\left(\tau\right)d\tau}{\tau-y}+\frac{1}{\pi}\int\limits_{0}^{1}\frac{\rho_{2}\left(\tau\right)d\tau}{\tau+y}}=R_{1}\left(y\right)}\\ \\ {{\displaystyle\rho_{2}\left(y\right)-\frac{1}{\pi}\int\limits_{0}^{1}\frac{\rho_{2}\left(\tau\right)d\tau}{\tau-y}+\frac{1}{\pi}\int\limits_{0}^{1}\frac{\rho_{1}\left(\tau\right)d\tau}{\tau+y}}=R_{2}\left(y\right),}\end{cases}$$
(15)

where

$$\rho_{i}\left(y\right)=\frac{1+x^{4}}{x}v_{i}\left(x\right),\quad R_{i}\left(y\right)=\frac{1+x^{4}}{x}L_{i}\left(x\right),\quad i=1,2.$$
(16)

The behaviors of terms in the right-hand side of (15) are given in the following lemma.

Lemma 1. Let \(f(x,y)\in C^{1+\delta}\left(\bar{\Omega}\right)\), \(0<\delta<1\). Then \(R_{i}(y)\in C^{1+\delta}(0,1)\), \(i=1,2\) and have singularity of order \(\frac{1}{2}\) as \(y\to 0\).

Proof. Consider the function \(L_{1}(x)\) determined by (13). From (6) considering smoothness of \(\gamma_{i}(x)\), \(i=1,2\) and imposing smoothness condition to the given functions \(a_{1}(t),b_{1}(t)\in C^{2}[0,1]\), it is easy to get \(F_{1}(x)\in C^{1+\delta}\left[0,1\right]\). According to [9] the second term in (13) belongs to the class of functions \(C[0,1]\cap C^{1+\delta}(0,1)\). Hence, \(L_{1}(x)\in C[0,1]\cap C^{1+\delta}(0,1)\). Due to (14), (16) we will get the proof of the Lemma 2 in the case \(i=1\). The case \(i=2\) can be proved in a similar way. The lemma was proved. \(\Box\)

The system (15), as well as, in case of the Tricomi or Gellerstedt equations [1], except a singular kernel, contains also a nonsingular part \(\frac{1}{\tau+y}\), which is continuous if one of variables \(\tau\) and \(y\) changes strictly in an interval \((0,1]\). But if \(\tau=y=0\), it goes to infinity and hence it is not summarized in a square \(0\leq\tau\), \(y\leq 1\). Nevertheless, we will use special approach for the solvability of this system, which allows obtaining a solution in an explicit form.

We introduce the following notation

$$\rho(y)=\begin{cases}{\rho_{1}(y),\quad 0<y<1,}\\ {-\rho_{2}(-y),\quad-1<y<0,}\end{cases}$$
(17)
$$c(y)=\begin{cases}{-i,\quad 0<y<1,}\\ {i,\quad-1<y<0,}\end{cases}\quad R(y)=\begin{cases}{R_{1}(y),\quad 0<y<1,}\\ {-R_{2}(-y),\quad-1<y<0.}\end{cases}$$
(18)

Then the system of equations (15) can be rewritten in a form

$$A(y)\varphi(y)+\frac{B}{\pi i}\int\limits_{L}\frac{\varphi(\tau)d\tau}{\tau-y}=H(y),$$
(19)

where \(L=(-1,0)\cup(0,1),\)

$$\varphi(y)=\left[\begin{matrix}{\rho(y)}\\ {signy\rho(y)}\end{matrix}\right],\quad H(y)=\left[\begin{matrix}{R(y)}\\ {0}\end{matrix}\right],$$
(20)
$$A(y)=\left[\begin{matrix}{10}\\ {1-signy}\end{matrix}\right],\quad B(y)=\left[\begin{matrix}{0c(y)}\\ {00}\end{matrix}\right].$$
(21)

Thus, in the case when \(\sigma\) coincides with an arc of a unit circle, the solvability of a problem BS for the equation (1) is reduced to the system of singular integral equations (19).

From (18) and (21) it follows that determinants of matrixes \(A(y)\pm B(y)\) are not zero everywhere on \(L\), that is the system (19) is normal type and its solution can be constructed according to the general theory of singular integral equations [19].

Let \(z\) the arbitrary point of a complex plane. Let us introduce the piecewise holomorphic vector function represented by an integral of Cauchy-type

$$\Phi(z)={\displaystyle\frac{1}{2\pi i}\int\limits_{L}\frac{\varphi(\tau)d\tau}{\tau-z}},$$

where

$$\Phi(z)=\left[\begin{matrix}{\Phi_{1}(z)}\\ {\Phi_{2}(z)}\end{matrix}\right],\quad\varphi(z)=\left[\begin{matrix}{\varphi_{1}(z)}\\ {\varphi_{2}(z)}\end{matrix}\right].$$
(22)

It is apparent that \(\Phi(z)\) is the holomorphic vector function both in upper and in the lower plane. In accordance to Sokhotski-Plemelj formulae ([19], p. 512), the limiting values \(\Phi^{+}(y)\), \(\Phi^{-}(y)\) on approaching the curve from the left and from the right are expressed by

$$\begin{cases}{\Phi^{+}(y)={\displaystyle\frac{1}{2}\varphi(y)+\frac{1}{2\pi i}\int\limits_{L}\frac{\varphi(\tau)}{\tau-y}d\tau,}}\\ \\ {\Phi^{-}(y)=-{\displaystyle\frac{1}{2}\varphi(y)+\frac{1}{2\pi i}\int\limits_{L}\frac{\varphi(\tau)}{\tau-y}d\tau.}}\end{cases}$$

The system (19) is reduced to the Hilbert problem consisting to find a piecewise holomorphic function \(\Phi(z)\) under the boundary conditions

$$\Phi_{1}^{+}(y)=\Phi_{1}^{-}(y)+a_{1}(y)\Phi_{2}^{-}(y)+b_{1}(y),$$
(23)
$$\Phi_{2}^{+}(y)=a_{2}(y)\Phi_{2}^{-}(y)+b_{2}(y),$$
(24)

where

$$a_{1}(y)=\begin{cases}{1-i,\quad-1<y<0}\\ {i-1,\quad 0<y<1,}\end{cases}\quad a_{2}(y)=i,$$
(25)
$$b_{1}(y)=\begin{cases}{\displaystyle{-\frac{R_{2}(-y)}{1-i},\quad-1<y<0}}\\ {{\displaystyle\frac{R_{1}(y)}{1-\alpha i}},\quad 0<y<1,}\end{cases}\quad b_{2}(y)=\begin{cases}{\displaystyle{\frac{R_{2}(-y)}{1-i}},\quad-1<y<0}\\ {\displaystyle{\frac{R_{1}(y)}{1-i},\quad 0<y<1.}}\end{cases}$$
(26)

The equation (24) contains only one unknown function \(\Phi_{2}(z)\). Therefore it is possible to find unknown function \(\Phi_{2}(z)\). For this purpose, we will define real numbers \(\sigma_{k}\) and \(\mu_{k}\) from a condition ([19], p. 313)

$$\sigma_{k}+i\mu_{k}=\sum_{j}\frac{\mp\ln a_{2}\left(c_{k}\right)}{2\pi i},$$

where the sum extends to every number j of arcs L\({}_{j}\) , meeting in \({c}_{k}\) , and the upper signs correspond to the outgoing arcs, and lower for ingoing arcs, \(j=1,2\), \(k=1,2,3\), \(L_{1}=(-1,0)\), \(L_{2}=(0,1)\), \(c_{1}=-1\), \(c_{2}=0\), \(c_{3}=1\).

From this formula, considering (26), we have

$$\sigma_{1}=-\frac{1}{4},\quad\mu_{1}=0,\quad\sigma_{2}=0,\quad\mu_{2}=0,\quad\sigma_{3}=\frac{1}{4},\quad\mu_{3}=0.$$
(27)

Now, we introduce the index \(\chi\) of a class \(h_{1}\) of the problem (24) which is defined by ([19], p. 315)

$$\chi=-\sum_{k=1}^{3}\lambda_{k},$$
(28)

where \(\lambda_{k}\) are integer numbers, satisfying \(-1<\lambda_{k}+\sigma_{k}<1\), \(k=\overline{1,3}\).

We select \(\lambda_{k}\) satisfying

$$-1<\lambda_{k}+\sigma_{k}<0$$
(29)

in the case of unbounded solution to (24),

$$0<\lambda_{k}+\sigma_{k}<1$$
(30)

in the case of bounded solution to (24) at nonsingular points, and \(\lambda_{k}=0\) at singular points, \(k=\overline{1,3}\).

Let us remind that those ends, on which \(\sigma_{k}\) are integers, are called singular, and other ends nonsingular.

From (27) follows that the point \({c}_{2}=0\) is singular, and the points \(c_{1}=-1\) and \(c_{3}=1\) are nonsingular. Then from (28) considering (29) and (30) we get \(\chi=0\), that is the index of the class \(h_{1}\) is equal to zero. Therefore, the Hilbert problem corresponding to the boundary condition (24) has a unique solution of class \(h_{1}\) of the form

$$\Phi_{2}(z)=\frac{X(z)}{2\pi i}\int\limits_{L}\frac{b_{2}(\tau)d\tau}{X^{+}(\tau)(\tau-y)},$$
(31)

where \(X(z)={\displaystyle\left(\frac{1-z}{1+z}\right)^{\frac{1}{4}}}\) is a canonical function of the class \(h_{1}\) ([19], p. 314,394). In accordance with the Sokhotski–Plemelj formulae and in view of the expression (31),we obtain the following explicit expression of the nonhomogeneous boundary condition (24)

$$\varphi_{2}(y)=\frac{1}{2}\frac{b_{2}(y)}{X^{+}(y)}\left(X^{+}(y)+X^{-}(y)\right)+\frac{\left(X^{+}(y)-X^{-}(y)\right)}{2\pi i}\int\limits_{L}\frac{b_{2}(\tau)d\tau}{X^{+}(y)(\tau-y)}.$$

In the same way, considering (22), (23), (26) and (31), we find \(\varphi_{1}(y)\).

These solutions, according to formulae (17), (18), (20)–(22), (25) and (26) relatively functions \(\rho_{i}(y)\), \({i=1,2}\), have the following expressions

$$\rho_{1}(y)=\frac{1}{2}R_{1}(y)+\frac{1}{2\pi}\int_{0}^{1}\left[\frac{(1-y)(1+\tau)}{(1+y)(1-\tau)}\right]^{\frac{1}{4}}\frac{R_{1}(\tau)d\tau}{\tau-y}-\frac{1}{2\pi}\int\limits_{0}^{1}\left[\frac{(1-y)(1-\tau)}{(1+y)(1+\tau)}\right]^{\frac{1}{4}}\frac{R_{2}(\tau)d\tau}{\tau+y},$$
(32)
$$\rho_{2}(y)=\frac{1}{2}R_{2}(y)-\frac{1}{2\pi}\int_{0}^{1}\left[\frac{(1+y)(1-\tau)}{(1-y)(1+\tau)}\right]^{\frac{1}{4}}\frac{R_{2}(\tau)d\tau}{\tau-y}+\frac{1}{2\pi}\int\limits_{0}^{1}\left[\frac{(1+y)(1+\tau)}{(1-y)(1-\tau)}\right]^{\frac{1}{4}}\frac{R_{1}(\tau)d\tau}{\tau+y},$$
(33)

giving the solution of the system (15) in an explicit form.

Now we will study behavior of these solutions on the segment [0]. From (32) and (33) based on Lemma 1 and properties of the Cauchy type integrals it follows that \(\rho_{i}(y)\in C^{1}(0,1)\).

Now let us investigate the behavior of the solution \(\rho_{1}(y)\) near the ends at \(y=0\) and at \(y=1\). Using the following notations

$$T_{1}(y)=\left[\frac{1-y}{1+y}\right]^{\frac{1}{4}}\int_{0}^{1}\left[\frac{1+\tau}{1-\tau}\right]^{\frac{1}{4}}\frac{R_{1}(\tau)d\tau}{\tau-y},\quad T_{2}(y)=\left[\frac{1-y}{1+y}\right]^{\frac{1}{4}}\int_{0}^{1}\left[\frac{1-\tau}{1+\tau}\right]^{\frac{1}{4}}\frac{R_{2}(\tau)d\tau}{\tau+y},$$

we rewrite (32) in the form

$$\rho_{1}(y)=\frac{1}{2}R_{1}(y)+\frac{1}{2\pi}T_{1}(y)-\frac{1}{2\pi}T_{2}(y),$$
(34)

which is convenient in investigating each item in a right side of this ratio.

As the function \(R_{1}(y)\) is bounded at \(y=1\) and can have singularity of order \(\frac{1}{2}\) at \(y=0\), the function \({T}_{1}(y)\) will be presented in the neighborhood of a point \(y=0\) as (see [19], p. 90)

$$T_{1}(y)=\left[\frac{1-y}{1+y}\right]^{\frac{1}{4}}\int\limits_{0}^{1}\frac{T_{1}^{*}(\tau)d\tau}{\tau^{\frac{1}{2}}(\tau-y)},$$

where \(T_{1}^{*}(y)\) is bounded function in the neighborhood of \(y=0\).

In this case, as it appears from properties of Cauchy integral ([19], p. 92), that the function \({T}_{1}(y)\) can go to infinity of an order less then \(\frac{1}{2}\) as \(y\rightarrow 0\), and is bounded in the neighborhood of the point \(y=1\). Similarly, the function \({T}_{2}(y)\) go to infinity of order \(\frac{1}{2}\) as \(y\rightarrow 0\), and is bounded in the neighborhood of the point \(y=1\).

Considering properties of the functions \(R_{1}(y)\), \({T}_{1}(y)\) and \({T}_{2}(y)\), it follows from (34) that the function \(\rho_{1}(y)\) can go to infinity of order \(\frac{1}{2}\) as \(\rightarrow\) 0 and it is bounded at \(y=1\). Similarly, we can prove that the function \(\rho_{2}(y)\) can have singularity of an order \(\frac{1}{2}\) as \(y\rightarrow 0\) and an order 1/4 as \(y\rightarrow 1\).

Using the expressions of \(\rho_{1}(y)\), \(\rho_{2}(y)\), and the change of variables given respectively by the equations (14), (32) and (33), we will obtain from (16), the expressions of the functions \(v_{i}(x),i=1,2\)

$$v_{1}(x)=\frac{1}{2}L_{1}(x)+\frac{1}{\pi}\int\limits_{0}^{1}\left[\frac{\left(1-x^{2}\right)\left(1+t^{2}\right)}{\left(1+x^{2}\right)\left(1-t^{2}\right)}\right]^{\frac{1}{2}}N_{1}(x,t)L_{1}(t)dt$$
$${}-\frac{1}{\pi}\int\limits_{0}^{1}\left[\frac{\left(1-x^{2}\right)\left(1-t^{2}\right)}{\left(1+x^{2}\right)\left(1+t^{2}\right)}\right]^{\frac{1}{2}}N_{2}(x,t)L_{2}(t)dt,$$
$${v_{2}(x)=\frac{1}{1+\alpha^{2}}L_{2}(x)-\frac{1}{\pi}\int\limits_{0}^{1}\left[\frac{\left(1+x^{2}\right)\left(1-t^{2}\right)}{\left(1-x^{2}\right)\left(1+t^{2}\right)}\right]^{\frac{1}{2}}N_{1}(x,t)L_{2}(t)dt}$$
$${}+\frac{1}{\pi}\int\limits_{0}^{1}\left[\frac{\left(1+x^{2}\right)\left(1+t^{2}\right)}{\left(1-x^{2}\right)\left(1-t^{2}\right)}\right]^{\frac{1}{2}}N_{2}(x,t)L_{1}(t)dt,$$

where

$$N_{i}(x,t)=\frac{2\alpha x}{1+\alpha^{2}}\left[\frac{1}{t^{2}\mp x^{2}}-\frac{t^{2}}{1\mp t^{2}x^{2}}\right],\quad i=1,2.$$

Here the upper sign is taken at \(i=1\) and the lower sign at \(i=2\).

From properties of the functions \(\rho_{1}(x)\), \(\rho_{2}(x)\), and the expressions given by (16) it follows that \({v}_{i}(x)\in C^{1}(0,1)\), \(i=1,2\). Also, \(v_{1}(x)\) is bounded at the points \(x=0\) and \(x=1\), while \(v_{2}(x)\) is bounded at \(x=0\) and can go to infinity of order 1/2.

As \(v_{1}(x)\) and \(v_{2}(x)\) are now known, we find from (5), (7), the functions \(\tau_{1}(x)\) and \(\tau_{2}(x)\). Considering properties of the functions \(v_{i}(x)\), \(i=1,2\), we will also get \(\tau_{i}(x)\in C[0,1]\cap C^{1}(0,1)\), \(i=1,2\). Thus, it follows that the regular solution of the problem (BS) exists and this solution can be found in the domains \(\Omega_{1}\) and \(\Omega_{2}\) as the solution of the Cauchy problem, and in the domain \(\Omega_{3}\) as the solution of the problem (N) corresponding to the equation (1) by the expression (9). The theorem was proved. \(\Box\)