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Item Numerical Processes for Approximating Solutions of Nonlinear Equations(MDPI, 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.In this article, we present generalized conditions of three-step iterative schemes for solving nonlinear equations. The convergence order is shown using Taylor series, but the existence of high-order derivatives is assumed. However, only the first derivative appears on these schemes. Therefore, the hypotheses limit the utilization of the schemes to operators that are at least nine times differentiable, although the schemes may converge. To the best of our knowledge, no semi-local convergence has been given in the setting of a Banach space. Our goal is to extend the applicability of these schemes in both the local and semi-local convergence cases. Moreover, we use our idea of recurrent functions and conditions only on the derivative or divided differences of order one that appear in these schemes. This idea can be applied to extend other high convergence multipoint and multistep schemes. Numerical applications where the convergence criteria are tested complement this article. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.Item Extended Convergence of Three Step Iterative Methods for Solving Equations in Banach Space with Applications(MDPI, 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.Symmetries are vital in the study of physical phenomena such as quantum physics and the micro-world, among others. Then, these phenomena reduce to solving nonlinear equations in abstract spaces. These equations in turn are mostly solved iteratively. That is why the objective of this paper was to obtain a uniform way to study three-step iterative methods to solve equations defined on Banach spaces. The convergence is established by using information appearing in these methods. This is in contrast to earlier works which relied on derivatives of the higher order to establish the convergence. The numerical example completes this paper. © 2022 by the authors.Item On the convergence of a novel seventh convergence order schemes for solving equations(Springer Science and Business Media B.V., 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.We study the local convergence of a seventh order scheme for solving nonlinear equations for Banach space valued equations. This is done by using assumptions only on the first derivative that does appear on the schemes, whereas in earlier works up to the eighth derivative (not on the scheme) are used to establish the convergence (not on the scheme). Our technique is so general that it can be used to extend the usage of other schemes along the same lines. © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.Item On a novel seventh convergence order method for solving nonlinear equations and its extensions(World Scientific, 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.We extend the applicability of a novel seventh-order method for solving nonlinear equations in the setting of Banach spaces. This is done by using assumptions only on the first derivative that does appear on the method, whereas in earlier works up to the eighth derivatives (not on the scheme) were used to establish the convergence. Our technique is so general that it can be used to extend the usage of other schemes along the same lines. © 2022 World Scientific Publishing Company.