In complete b-metric spaces, perimeter-contracting maps without 2-cycles have iterates that become graphic contractions when s q^n <1 (yielding 1–2 fixed points); with a 2-cycle, even iterates do likewise under s q^2 <1.
Suzuki,Basic inequality on a b-metric space and its applications, Journal of Inequali- ties and Applications, 2017 (2017), Article 256
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Perimetric Contractions and Their Iterates in Complete $b$-Metric Spaces
In complete b-metric spaces, perimeter-contracting maps without 2-cycles have iterates that become graphic contractions when s q^n <1 (yielding 1–2 fixed points); with a 2-cycle, even iterates do likewise under s q^2 <1.