Non-stationary φ-contractions and associated fractals

The paper’s Theorem 4.6 claims: (i) each Tk is a Matkowski contraction on (C*(I), d∞), and (ii) the backward trajectories Ψk(g) converge to a unique f* with f*(xi)=yi, invoking Proposition 3.11 and stating that “it is easy to check that all hypotheses” hold. However, Proposition 3.11 requires a summability hypothesis ∑k φ1∘⋯∘φk(t)<∞ that the theorem does not assume; only φn(t)→0 is assumed. This gap is not addressed in the paper’s proof, and the usual ‘join-up/anchoring’ conditions needed to guarantee f*(xi)=yi are also unstated. In contrast, the candidate solution proves convergence directly via d∞(Ψn(g),Ψm(g))≤φm(b−a), which only needs φn(t)→0, and it explicitly stipulates the standard join-up conditions to ensure interpolation. Thus the model’s solution is correct under the stated assumptions, while the paper’s proof is incomplete on the use of Proposition 3.11 and missing anchoring hypotheses for interpolation (cf. Theorem 4.6 statement and proof; Proposition 3.11).