Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 11 1 2026 06 07 From Quadratic Convergence to Structural Invariance: A Selje Topological Framework for Newton-Type Methods 174 185 2093 10.22128/ansne.2026.3276.1203 EN Selva Nandhini N Assistant Professor Faculty in Mathematics, Dr. Mahalingam College of Engineering and Technology, Pollachi, India 0009-0002-1862-6784 Maryam Akhoundi Clinical Research Development Unit of Rouhani Hospital, Babol University of Medical Sciences, Babol, Iran 0000-0002-2419-1732 Senbaga Priya K Assistant Professor Faculty in Mathematics, Dr. Mahalingam College of Engineering and Technology, Pollachi, India Mostafa NouriJouybari Payame Noor University (PNU) Tehran, Iran 0000-0003-2295-1504 Farshid Mofidnakhaei Department of Physics, Sar.C., Islamic Azad University, Sari, Iran 0000-0002-2903-0428 Journal Article 2026 02 20 Newton-type methods are essential for solving nonlinear equations and systems, with classical metric-based analysis focusing on quadratic convergence and local error bounds. However, these results overlook the structural stability of iterations under perturbations. This paper introduces a Selje topological framework to analyze the stability of Newton-type methods beyond traditional numerical theory. We associate nonlinear operators with Selje topological structures and study the invariance and stability of iterative sequences via induced operators $\mathcal{T}_{\mathcal{R}}{(\mathbb{X})}$,$\mu_{\mathcal{R}}{(\mathbb{X})}$, $SJ_{\mathcal{R}}{(\mathbb{X})}$ . Sufficient conditions are established for preserving topological stability in Newton-type iterations, interpreting convergence as structural consistency in the Selje space. This framework yields a generalized stability characterization that complements classical convergence theory, advancing the analysis of nonlinear iterative solvers through topology. https://ansne.du.ac.ir/article_2093_4247dd41848ce59c482dc54eb75ba048.pdf