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Item Convergence of Chun’s method in Banach spaces under weaker assumptions(Springer Science and Business Media B.V., 2025) George, S.; M, M.; Godavarma, C.In this paper, we give a modified convergence analysis for the fourth-order method studied in Cordero et al. (J Math Chem 53(1):430–449, 2015). Our analysis provides the convergence order using the derivative of the involved operator up to order two only, whereas their study needs it to be five times differentiable. Apart from this, this paper obtains the convergence ball radius and the number of iterations to reach the solution with the desired accuracy. Further, we use the general Banach space settings to get these results, while their work is done only for the space. At the end of the paper, we discuss a few numerical examples and compare them with other existing fourth-order methods. © The Author(s), under exclusive licence to The Forum D’Analystes 2025.Item Jarratt-type methods and their convergence analysis without using Taylor expansion(Elsevier Inc., 2025) Bate, I.; Senapati, K.; George, S.; M, M.; Godavarma, C.In this paper, we study the local convergence analysis of the Jarratt-type iterative methods for solving non-linear equations in the Banach space setting without using the Taylor expansion. Convergence analysis using Taylor series required the operator to be differentiable at least p+1 times, where p is the order of convergence. In our convergence analysis, we do not use the Taylor expansion, so we require only assumptions on the derivatives of the involved operator of order up to three only. Thus, we extended the applicability of the methods under study. Further, we obtained a six-order Jarratt-type method by utilising the method studied by Hueso et al. in 2015. Numerical examples and dynamics of the methods are presented to illustrate the theoretical results. © 2024 Elsevier Inc.Item A procedure for increasing the convergence order of iterative methods from p to 5p for solving nonlinear system(Academic Press Inc., 2025) George, S.; M, M.; Gopal, M.; Godavarma, C.; Argyros, I.K.In this paper, we propose a procedure to obtain an iterative method that increases its convergence order from p to 5p for solving nonlinear systems. Our analysis is given in more general Banach space settings and uses assumptions on the derivative of the involved operator only up to order max?{k,2}. Here, k is the order of the highest derivative used in the convergence analysis of the iterative method with convergence order p. A particular case of our analysis includes an existing fifth-order method and improves its applicability to more problems than the problems covered by the method's analysis in earlier study. © 2024Item Unified convergence analysis of a class of iterative methods(Springer, 2025) M, M.; George, S.; Godavarma, G.In this paper, unified convergence analyses for a class of iterative methods of order three, five, and six are studied to solve the nonlinear systems in Banach space settings. Our analysis gives the number of iterations needed to achieve the given accuracy and the radius of the convergence ball precisely using weaker conditions on the involved operator. Various numerical examples have been taken to illustrate the proposed method, and the theoretical convergence has been validated via these examples. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.Item Improved convergence analysis for an at least fourth and at least sixth order parametric family of iterative methods for nonlinear system(Springer-Verlag Italia s.r.l., 2025) George, S.; Gopal, M.; M, M.Hueso et al. (2015) introduced a new family of iterative methods for solving non-linear systems. However, the convergence analysis is based on Taylor series expansion, which requires the existence of derivatives of the involved operator up to the fifth and seventh orders, respectively, for the method with at least fourth-order convergence and the method with sixth-order convergence. In this paper, we obtain at least fourth- and sixth-order convergence for the respective methods by assuming derivatives only up to the third order. We also provide the semi-local convergence analysis (which is not given in Hueso et al. (2015)) in a more general Banach space form. Moreover, our semi-local and local convergence analyses are based on the same set of assumptions, unlike existing studies, where the authors typically use one set of assumptions for semi-local convergence analysis and another set of assumptions for local convergence analysis. Numerical examples and dynamics of the methods are also provided in this study. © The Author(s), under exclusive licence to Springer-Verlag Italia S.r.l., part of Springer Nature 2025.