Construction of graph-directed invariant sets of weak contractions on semi-metric spaces

We present a construction of graph-directed invariant sets of weak contractions in the sense of Matkowski-Rus on semi-metric spaces. We follow the approach by Bessenyei and Pénzes, which applies the Kuratowski noncompactness measure without relying on Blascke’s completeness theorem. We also establish a relationship between this approach and a generalized de Rham’s functional equation indexed by a finite directed graph.