Solvability theory of general characteristic singular integral equations with a Carleman fractional linear shift on the unit circle

Abstract

In the preceding Chapter we obtained some results on the solvability theory of characteristic singular integral equations (polynomial boundary value problems) with a Carlcman shift, which can be reduced to binomial boundary value problems by imposing very hard restrictions on the given coefficients, that is we considered the so-called degenerated cases. The present Chapter is also devoted to the solvability theory of characteristic singular integral equations with a Carleman shift. However, as distinct from the previous chapter, now we use another way which consists in, without restricting the coefficients of the equation, using some identities of a type of the mentioned degenerated cases; we put the restrictions on the shift function α, on the shift operator U and on the contour, being Γ such that US = ±SU (S is the operator of singular integration) for α = α₊(t) and α = α₋(t), respectively. In this way we get rid of the compact operator in the equality (6.14) and we arrive at the corresponding (2 × 2)-matrix of singular integral operators without shift in the characteristic form M = AP₊+BP₋. More precisely, here we consider a characteristic singular integral operator with a Carleman fractional linear shift on the unit circle.