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6 papers
Systems of singular integral equations with a shift
Yu. I. Karlovich,
V. G. Kravchenko
Abstract:
Let
$\Gamma$ be a simple closed oriented Lyapunov curve and let
$\alpha(t)$ be an
$H$-smooth diffeomorphism of
$\Gamma$ onto itself whose set of fixed points is nonempty and finite. The system of equations
$$
T\varphi\equiv A_1P\varphi+A_2Q\varphi=g
$$
is considered in the space
$L^n_p(\Gamma)$,
$1<p<\infty$, where
$P+Q$ is the identity operator,
$P-Q=S$ is a singular integral operator with Cauchy kernel,
$A_k$ (
$k=1,2$) are polynomials of positive and negative degree in the shift operator
$U$ defined by $(U\varphi)(t)=|\alpha'(t)|^{1/p}\varphi[\alpha(t)]$, and the coefficients in the
$A_k$ are matrix-valued functions that are continuous on
$\Gamma$.
The authors obtain conditions for the operator
$T$ to be Fredholm, and the same for generalizations of
$T$ to a shift preserving or changing the orientation and having a finite set of periodic points whose multiplicity is not necessarily equal to one.
Bibliography: 21 titles.
UDC:
517.948.32
MSC: Primary
45E05,
45F15,
47A53; Secondary
30E25 Received: 19.07.1980