Argyros, I.K.George, S.2026-02-052017WSEAS Transactions on Mathematics, 2017, 16, , pp. 248-25611092769https://idr.nitk.ac.in/handle/123456789/25727The goal of this paper is to present a local convergence analysis of Newton's method for approximating a locally unique solution of an equation in a Banach space setting. Using the gauge function theory and our new idea of restricted convergence regions we present an extended and unified convergence theory.Banach spaceConvergence regionGauge functionNewton's methodNewton-Kantorovich theoremSemilocal convergence