On iterated function systems and algebraic properties of Lipschitz maps in partial metric spaces

The paper states and justifies the Collage Theorem in partial metric spaces (Theorem 3.6) as an immediate corollary of a general inequality (Theorem 3.2) applied to the Hutchinson operator W acting on the hyperspace (Hp(X), hp), together with the contractivity of W (Theorem 2.13) and the condensation condition hp(C,C)=0 when present (Definition 3.5 and Theorem 3.4). This yields hp(L,A) ≤ (1−s)^{-1} hp(L,W(L)) exactly as claimed . The candidate model gives a standard one-step triangle-inequality proof tailored to the partial Hausdorff setting, also handling the condensation case by noting that adding a set C with hp(C,C)=0 does not increase hp; this is consistent with the paper’s condensation condition and yields the same estimate. Thus, the paper’s statement and justification are correct, and the model’s argument is also correct, though it uses a different but standard route.