LAUSR

Abstract Let α and β be orientation-preserving diffeomorphism (shifts) of R + = ( 0 , ∞ ) onto itself with the only fixed points 0 and ∞. We establish a Fredholm criterion and calculate the index of the weighted singular integral operator with shifts ( a I − b U α ) P γ + + ( c I − d U β ) P γ − , acting on the space L p ( R + ) , where P γ ± = ( I ± S γ ) / 2 are the operators associated to the weighted Cauchy singular integral operator S γ given by ( S γ f ) ( t ) = 1 π i ∫ R + ( t τ ) γ f ( τ ) τ − t d τ with γ ∈ C satisfying 0 1 / p + ℜ γ 1 , and U α , U β are the isometric shift operators given by U α f = ( α ′ ) 1 / p ( f ∘ α ) , U β f = ( β ′ ) 1 / p ( f ∘ β ) , under the assumptions that the coefficients a , b , c , d and the derivatives α ′ , β ′ of the shifts are bounded and continuous on R + and admit discontinuities of slowly oscillating type at 0 and ∞.