One-sided invertibility of binomial functional operators with a shift in   rearrangement-invariant spaces
  Let $\Gamma$ be an oriented Jordan smooth curve and $\alpha$ be a diffeomorphism of $\Gamma$ onto itself which has an arbitrary nonempty set of periodic points. We prove criteria for one-sided invertiblity of the binomial functional operator \[ A=aI-bW \] where $a$ and $b$ are continuous functions, $I$ is the identity operator, $W$ is the shift operator $Wf=f\circ\alpha$, in a reflexive rearrangement-invariant space $X(\Gamma)$ with Boyd indices $\alpha_X,\beta_X$ and Zippin indices $p_X,q_X$ satisfying inequalities \[ 0<\alpha_X=p_X\le q_X=q_X<1. \] 
A. Karlovich, Yu. Karlovich