APPROXIMATE FIXED POINT THEOREMS OF CYCLICAL CONTRACTION MAPPING ON G-METRIC SPACES

The paper’s Theorem 4.11 claims that every G–Mohsenialhosseini cyclical operator (with α∈[0,1), β,γ∈[0,1/2)) has ε–fixed points, but its proof derives a recurrence D_{n} ≤ 3η D_{n-1} (with η = max{α, β/(1−β), γ/(1−γ)}) and then concludes D_n→0 without imposing the necessary 3η<1; this is a gap (indeed false if η≥1/3). See Definition 4.10 and Theorem 4.11, and the displayed inequalities (4.1)–(4.5) culminating in the (3η)^n bound and the erroneous “Therefore” limit claim . By contrast, the model proves existence of ε–fixed points under the correct contraction ranges (α<1/2 in case (i), β,γ<1/2 in (ii)/(iii)), giving D_{n+1}≤θD_n with θ∈[0,1), which does imply D_n→0. This aligns with the axioms for G-metrics (G1,G4,G5 used; and G3 is tacitly needed in case (iii)) .