A countable fractal interpolation scheme involving Rakotch contractions

The paper’s core construction (metric d_θ, Rakotch/Matkowski framework, graph transform T, existence/uniqueness of f*) is sound and matches the model. However, the paper defines the fractal operator for a countable IFS as FS(K)=⋃_n f_n(K) on P_cp(X), and later asserts FS(G)=G by claiming (b,f*(b)) belongs to the raw union ⋃_n f_n(G). This is incorrect: being a limit of points z_n∈f_n(G) does not place the limit in the countable union across varying indices; the point (b,f*(b)) is only in the closure of the union. Indeed, Definition 2.3 sets FS(K)=⋃_n f_n(K) (no closure) and Theorem 3.3’s proof attempts to put (b,f*(b)) in the union via a limit argument, which does not validate membership in the union . By contrast, the model explicitly uses the standard closure-of-union operator for countable IFS and proves cl(⋃_n f_n(G))=G, fixing the gap. The rest of the argument aligns with the paper’s Theorems 3.2–3.3 on d_θ-contractivity and the graph-transform contraction . Therefore, the paper’s proof as written is incomplete at the attractor/union step, while the model’s solution is correct and complete.