Effectively constructible fixed points in Sacchetti's modal logics of provability

We give a purely syntactical proof of the fixed point theorem for Sacchetti's modal logics ${\bf K} + \Box(\Box^n p \to p) \to \Box p$ ($n \geq 2$) of provability. From our proof, an effective procedure for constructing fixed points in these logics is obtained. We also show the existence of simple fixed points for particular modal formulas.