Argyros, I.K.George, S.2026-02-032024Journal of Numerical Analysis and Approximation Theory, 2024, 53, 2, pp. 187-19824576794https://doi.org/10.33993/jnaat532-1178https://idr.nitk.ac.in/handle/123456789/20742The convergence of two-step iterative methods of third and fourth order of convergence are studied under weaker hypotheses than in earlier works using our new idea of the restricted convergence region. This way, we obtain a finer semilocal and local convergence analysis, and under the same or weaker hypotheses. Hence, we extend the applicability of these methods in cases not covered before. Numerical examples are used to compare our results favorably to earlier ones. © 2024, Publishing House of the Romanian Academy. All rights reserved.Banach spaceconvergence of iterative methodrestricted convergence region