Generalized Fiber Contraction Mapping Principle

The paper studies the non-stationary skew-product compositions F1 ∘ ··· ∘ Fn, explicitly in the left-to-right composition order where Fn acts first and F1 acts last, and proves universal convergence under hypotheses (0.1), (0.2), and (1)–(3) (see the statement of Theorem 4 and surrounding discussion). Under this composition order, the X-projection is handled by Theorem 3 via inequality (1.1), and the Y-projection convergence is established in Claims 1–2 culminating in (1.12), yielding a limit independent of the initial point and then combining to (x*, y*) for all (x, y) (Theorem 4) . The model’s counterexample uses the opposite (forward-iteration) order x_{n+1} = f_{n+1}(x_n), which is not the object of the paper’s theorem. For the model’s maps fn(x) = μx + (−1)^n and h_n^x(y)=λy with μ, λ ∈ (0,1), the paper’s composition F1 ∘ ··· ∘ Fn produces the X-component μ^n x − (1 − (−μ)^n)/(1+μ) → −1/(1+μ) and Y-component λ^n y → 0, so convergence does hold as claimed. Thus the candidate’s “counterexample” rests on a composition-order misinterpretation, and does not invalidate the paper’s result. The paper also illustrates the necessity of assumptions (2) and (3) with explicit examples, consistent with the main theorem .