On solvability of a boundary value problem for singular integral equations
Let \(\Gamma\) be a simple regular closed arc on the complex plane and let \(X= H^\mu(\Gamma)\), \(0<\mu< 1\). Consider the following boundary value problem: \[ [(K^n+ K^{n-1} K_c)\varphi](t)= f(t), \] \((F_j K^j\varphi)(t)= \varphi_j(t)\), \(j= 0,1,\dots, n-1\), \(\varphi_j(t)\in\ker K\) if \(\kappa> 0\), \((G_0 f)(t)=0\), \((G_j R_{j-1}\cdots R_0)f(t)= 0\), \(j= 1,\dots, n-1\), if \(\kappa< 0\), \(K_c\) is the operator of multiplication by a function \(c(t)\), \(c,f\in X\), \[ (K\varphi)(t)= a(t)\varphi(t)+ {b(t)\over\pi i} \int_\Gamma {\varphi(\tau)\over \tau-t}\,d\tau,\quad \kappa= \text{ind\,}K, \] \(F_j\), \(G_j\) are initial and \(\omega\)-initial operators for \(K\) induced by right (left) inverses \(R_j\) of \(K\), respectively. The problem under question is reduced to a system of algebraic linear equations and then all its solutions are given in a closed form.
scientific article; zbMATH DE number 2004107