A method for determining the roots of some classes of equations with analytic functions and its applications (Q1963404)

Let \(D\) be a simply connected plane domain with piecewise smooth boundary. Given a function \(\varphi(z)\) analytic in \(D\) and continuous in \(\overline{D}\), the problem is to find a fixed point of this function, i.e. for solving the equation \(z-\varphi(z)=0\). It is assumed that \(z-\varphi(z)\neq 0\) for \(z\in \partial D\) and that \(\varphi\) takes \(\overline{D}\) into \(\overline{D}\). It is shown that there exists a unique fixed point \(z_0\) which is representable as \[ z_0=\frac{1}{2\pi i}\int_{\partial D} z \frac{\psi'(z)-1}{\psi(z)-1} dz, \] where \(\psi(z)=\varphi\big(\overline{\varphi(\bar{z})}\big)\). This result is applied to studying the Bitsadze-Samarskij problem for the Laplace equation which can be stated as follows: Find a solution to the equation \(\Delta u=0\) in the domain \(S=\{z:|z|<1\}\) satisfying the boundary condition \[ u(z)=au(\alpha(z)) + b u(\alpha(\overline{z})) + f(z),\quad |z|=1, \] where \(a\) and \(b\) are constants, \(f\) is a Hölder function defined on \(\partial S\), and \(\alpha(z)\) is an analytic function in \(S\) continuous in \(\overline{S}\).