Faculty Publications
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Item Direct comparison between two third convergence order schemes for solving equations(MDPI AG, 2020) Regmi, S.; Argyros, I.K.; George, S.We provide a comparison between two schemes for solving equations on Banach space. A comparison between same convergence order schemes has been given using numerical examples which can go in favor of either scheme. However, we do not know in advance and under the same set of conditions which scheme has the largest ball of convergence, tighter error bounds or best information on the location of the solution. We present a technique that allows us to achieve this objective. Numerical examples are also given to further justify the theoretical results. Our technique can be used to compare other schemes of the same convergence order. © 2020 by the authors. Licensee MDPI, Basel, Switzerland.Item On the solution of equations by extended discretization(MDPI Multidisciplinary Digital Publishing Institute rasetti@mdpi.com, 2020) Argyros, G.I.; Argyros, M.I.; Regmi, S.; Argyros, I.K.; George, S.The method of discretization is used to solve nonlinear equations involving Banach space valued operators using Lipschitz or Hölder constants. But these constants cannot always be found. That is why we present results using ?- continuity conditions on the Fréchet derivative of the operator involved. This way, we extend the applicability of the discretization technique. It turns out that if we specialize ?- continuity our new results improve those in the literature too in the case of Lipschitz or Hölder continuity. Our analysis includes tighter upper error bounds on the distances involved. © 2020 by the authors.Item On the local convergence and comparison between two novel eighth convergence order schemes for solving nonlinear equations(Cambridge Scientific Publishers, 2021) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.We compare two eighth order schemes for solving nonlinear equations involving 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 ninth derivative (not on the scheme) are 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. © 2021. All Rights Reserved.Item Extended kung–traub methods for solving equations with applications(MDPI, 2021) Regmi, S.; Argyros, I.K.; George, S.; Magreñán Ruiz, Á.A.; Argyros, M.Kung and Traub (1974) proposed an iterative method for solving equations defined on the real line. The convergence order four was shown using Taylor expansions, requiring the existence of the fifth derivative not in this method. However, these hypotheses limit the utilization of it to functions that are at least five times differentiable, although the methods may converge. As far as we know, no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided difference, which appear in the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item On the semi-local convergence of an ostrowski-type method for solving equations(MDPI, 2021) Argyros, C.I.; Argyros, I.K.; Joshi, J.; Regmi, S.; George, S.Symmetries play a crucial role in the dynamics of physical systems. As an example, microworld and quantum physics problems are modeled on principles of symmetry. These problems are then formulated as equations defined on suitable abstract spaces. Then, these equations can be solved using iterative methods. In this article, an Ostrowski-type method for solving equations in Banach space is extended. This is achieved by finding a stricter set than before containing the iterates. The convergence analysis becomes finer. Due to the general nature of our technique, it can be utilized to enlarge the utilization of other methods. Examples finish the paper. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item Convergence criteria of three step schemes for solving equations(MDPI, 2021) Regmi, S.; Argyros, C.I.; Argyros, I.K.; George, S.We develop a unified convergence analysis of three-step iterative schemes for solving nonlinear Banach space valued equations. The local convergence order has been shown before to be five on the finite dimensional Euclidean space assuming Taylor expansions and the existence of the sixth derivative not on these schemes. So, the usage of them is restricted six or higher differentiable mappings. But in our paper only the first Frèchet derivative is utilized to show convergence. Consequently, the scheme is expanded. Numerical applications are also given to test convergence. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item Extended convergence of a sixth order scheme for solving equations under ω-continuity conditions(Sciendo, 2022) Regmi, S.; Argyros, C.I.; Argyros, I.K.; George, S.The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω-continuity conditions. Numerical examples complete this article. © 2021 Samundra Regmi et al., published by Sciendo.Item Kantorovich-type results for generalized equations with applications(Springer Science and Business Media B.V., 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.Kantorovich-type results for generalized equations are extended with no additional conditions using Newton procedures. Iterates are shown to belong in a smaller domain resulting to tighter Lipschitz constants and a finer convergence analysis than in earlier works. © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.Item Extending the Traub Theory for Solving Nonlinear Equations(Universal Wiser Publisher, 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.In this paper, we extend the Traub theory for solving nonlinear equation. The extension is based on recurrent functions, the center Lipschitz condition and the notion of the restricted convergence domain. Numerical examples indicate that the new results can be utilized to solve nonlinear equations, but not earlier ones. © 2022 Ioannis K. Argyros, et al.Item A Comparison Study of the Classical and Modern Results of Semi-Local Convergence of Newton-Kantorovich Iterations(MDPI, 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.I.There are a plethora of semi-local convergence results for Newton’s method (NM). These results rely on the Newton–Kantorovich criterion. However, this condition may not be satisfied even in the case of scalar equations. For this reason, we first present a comparative study of established classical and modern results. Moreover, using recurrent functions and at least as small constants or majorant functions, a finer convergence analysis for NM can be provided. The new constants and functions are specializations of earlier ones; hence, no new conditions are required to show convergence of NM. The technique is useful on other iterative methods as well. Numerical examples complement the theoretical results. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.