Regmi, S.Argyros, I.K.George, S.Argyros, M.I.2026-02-042022Mathematics, 2022, 10, 11, pp. -https://doi.org/10.3390/math10111839https://idr.nitk.ac.in/handle/123456789/22540This article is an independently written continuation of an earlier study with the same title [Mathematics, 2022, 10, 1225] on the Newton Process (NP). This process is applied to solve nonlinear equations. The complementing features are: the smallness of the initial approximation is expressed explicitly in turns of the Lipschitz or Hölder constants and the convergence order 1 + p is shown for p ∈ (0, 1]. The first feature becomes attainable by further simplifying proofs of convergence criteria. The second feature is possible by choosing a bit larger upper bound on the smallness of the initial approximation. This way, the completed convergence analysis is finer and can replace the classical one by Kantorovich and others. A two-point boundary value problem (TPBVP) is solved to complement this article. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.Banach spaceiterative processessemi-local convergence