ON A TOPOLOGICAL METHOD FOR CALCULATING THE INDEX OF QUASI-SINGULAR INTEGRAL EQUATION

Abstract

A reduced spectral problem is considered for an elliptic type operator, in particular, for the Cauchy - Riemann operator with regular boundary value conditions to a quasi-singular integral equation with continuous kernel. Structure of the kernel of the quasi-singular integral equation is studied in explicit form. Index is calculated and condition for Noetherity of the investigated quasi-singular integral equation is established by the topological method on the complex plane. In this case, the spectral parameters are described under which the nonhomogeneous boundary value problem with shift for the Cauchy-Riemann equations is solvable everywhere in the class of continuous functions on the unit circle. The given problem is a nonlocal nature, and similar problems for the Cauchy-Riemann operation are described by M. Otelbaev in 1982. The solution of some singular integral equations with kernels depending on the difference and the sum of the arguments were studied in the works of I.I. Kalmushevsky. This interest is explained both by the theoretical significance of the results obtained and by the possibilities of important applications. Similar boundary value problems arise in the mathematical modeling of gas dynamics problems, soil moisture prediction, radiation transfer, population genetics, etc. The simplest examples of these boundary conditions were formulated by V.A. Steklov 1922. In this paper, the fundamental difference from the above work is the topological approach to calculating the index and establishing the Noether condition on the complexs plane.