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In this paper we correct an inaccuracy that appears in the proof of Theorem 1. in Czerwik's article "Contraction mappings in $b$-metric spaces.", Acta Math. Inform. Univ. Ostraviensis, 1:5--11, 1993., 3 pages

We prove a fixed-point theorem that generalizes and simplifies a number of results in the theory of F-contractions. We show that all of the previously imposed conditions on the operator can be either omitted or relaxed. Furthermore, our result is formulated in the more general context of b-metric spaces and $$\varphi $$ -contractions. We also point out that the framework of F-contractions can be reformulated in an equivalent way that is both closer in spirit to the classical syntax of Banach-type fixed point theorems, and also more natural and easier to deal with in the proofs.

We prove a Dold–Kan type correspondence between the category of planar dendroidal abelian groups and a suitably constructed category of planar dendroidal chain complexes. Our result naturally extends the classical Dold–Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes.