Let be an orientation-preserving homeomorphism of onto itself with only two fixed points at and , whose restriction to is a diffeomorphism, and let be the isometric shift operator acting on the Lebesgue space with by the rule . We establish criteria of the two-sided and one-sided invertibility of functional operators of the form on the spaces under the assumptions that the functions and for all are bounded and continuous on and may have slowly oscillating discontinuities at and . The unital Banach algebra of such operators is inverse-closed: if is invertible on for , then . Obtained criteria are of two types: in terms of the two-sided or one-sided invertibility of so-called discrete operators on the spaces and in terms of conditions related to the fixed points of and the orbits of points .
"Two-sided and one-sided invertibility of Wiener-type functional operators with a shift and slowly oscillating data." Banach J. Math. Anal. 11 (3) 554 - 590, July 2017. https://doi.org/10.1215/17358787-2017-0006