Semi-Fredholmness of weighted singular integral operators with shifts and slowly oscillating data

Let $\alpha ,\beta$ be orientation-preserving homeomorphisms of $[0,\mathrm{\infty }]$ onto itself, which have only two fixed points at $0$ and $\mathrm{\infty }$, and whose restrictions to ${\mathbb{R}}_{+}=(0,\mathrm{\infty })$ are diffeomorphisms, and let $U_{\alpha },U_{\beta }$ be the corresponding isometric shift operators on the space $L^p({\mathbb{R}}_{+})$ given by $U_{\mu }f=({\mu }^{\mathrm{\prime }}{)}^{1\mathrm{/}p}(f\circ \mu )$ for $\mu \in \{\alpha ,\beta \}$. We prove sufficient conditions for the right and left Fredholmness on $L^p({\mathbb{R}}_{+})$ of singular integral operators of the form $A_{+}{P}_{\gamma }^{+}+A_{-}{P}_{\gamma }^{-}$, where $P_{\gamma }^{\pm }=(I\pm S_{\gamma })\mathrm{/}2$, $S_{\gamma }$ is a weighted Cauchy singular integral operator, $A_{+}={\sum }_{k\in \mathbb{Z}}{a}_k{U}_{\alpha }^k$ and $A_{-}={\sum }_{k\in \mathbb{Z}}{b}_k{U}_{\beta }^k$ are operators in the Wiener algebras of functional operators with shifts. We assume that the coefficients $a_k,b_k$ for $k\in \mathbb{Z}$ and the derivatives of the shifts ${\alpha }^{\mathrm{\prime }},{\beta }^{\mathrm{\prime }}$ are bounded continuous functions on ${\mathbb{R}}_{+}$ which may have slowly oscillating discontinuities at $0$ and $\mathrm{\infty }$.