Systems of singular integral equations with a~shift

Yu. I. Karlovich ; V. G. Kravchenko

Sbornik. Mathematics, Tome 44 (1983) no. 1, pp. 75-95

Résumé

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 $L^n_p(\Gamma)$, $1$, 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.