Boundary problem with discontinuous translation for two functions which are analytic in domains of different connectivity

In the space L_p(), 1 < p < we consider the problem of finding functions and which are analytic, respectively, in a doubly connected domain with boundary and a simply connected domain with boundary ( and are closed Lyapunov contours) with respect to the boundary condition , where the orientation-preserving translation and the coefficients a, b, c, d admit discontinuities at certain points of . Under specific conditions on the limit values of the derivative of the translation at the points of discontinuity we get necessary and sufficient conditions for the problem to be Noetherian and a formula for calculating its index.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 262–266, February, 1990.