Abstract This paper establishes cut-elimination for $\mathsf {\mu LL^\infty }$ , $\mathsf {\mu LK^\infty }$ and $\mathsf {\mu LJ^\infty }$ , that are non-wellfounded sequent calculi with least and greatest fixed-points, by expanding on prior works by Santocanale and Fortier [20] as well as Baelde et al. [3, 4]. The paper studies a fixed-point encoding of $\textsf{LL}$ exponentials in order to deduce those cut-elimination results from that of $\mathsf {\mu MALL^\infty }$ . Cut-elimination for $\mathsf {\mu LK^\infty }$ and $\mathsf {\mu LJ^\infty }$ is obtained by developing appropriate linear decorations for those logics.