Faculty Publications
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Item Extended Local Convergence for High Order Schemes Under ?-Continuity Conditions(Universal Wiser Publisher, 2020) Argyros, G.; Argyros, M.; Argyros, I.K.; George, S.There is a plethora of schemes of the same convergence order for generating a sequence approximating a solution of an equation involving Banach space operators. But the set of convergence criteria is not the same in general. This makes the comparison between them challenging and only numerically. Moreover, the convergence is established using Taylor series and by assuming the existence of high order derivatives that do not even appear on these schemes. Furthermore, no computable error estimates, uniqueness for the solution results or a ball of convergence is given. We address all these problems by using only the first derivative that actually appears on these schemes and under the same set of convergence conditions. Our technique is so general that it can be used to extend the applicability of other schemes along the same lines. © 2020 Ioannis K. Argyros, et al.Item Unified ball convergence of third and fourth convergence order algorithms under ??continuity conditions(University of Guilan, 2021) Argyros, G.; Argyros, M.; Argyros, I.K.; George, S.There is a plethora of third and fourth convergence order algorithms for solving Banach space valued equations. These orders are shown under conditions on higher than one derivatives not appearing on these algorithms. Moreover, error estimations on the distances involved or uniqueness of the solution results if given at all are also based on the existence of high order derivatives. But these problems limit the applicability of the algorithms. That is why we address all these problems under conditions only on the first derivative that appear in these algorithms. Our analysis includes computable error estimations as well as uniqueness results based on ?? continuity conditions on the Fréchet derivative of the operator involved. © 2021 University of Guilan.Item Semi-Local Convergence Of A Derivative-Free Method For Solving Equations(Petrozavodsk State University, 2021) Argyros, G.; Argyros, M.; Argyros, I.K.; George, S.We present the semi-local convergence analysis of a two-step derivative-free method for solving Banach space valued equations. The convergence criteria are based only on the first derivative and our idea of recurrent functions. © Petrozavodsk State University, 2021Item Extended newton-frank-wolfe-type algorithm for constrained systems(International Publications, 2021) Argyros, G.; Argyros, M.; Argyros, I.K.; George, S.The convergence region of algorithm is not large in general, limiting the choice of starters. Moreover, the error distances are pessimistic. Motivated by optimization concerns and these draw backs, we develop a technique that locates a smaller than before set which also contains the iterates of the algorithm. The majorant functions connected to this set are more precise resulting to a finer convergence analysis than before but without more conditions. In particular, our technique is used to solve constrained nonlinear systems using an inexact Newton-Frank-Wolfe-Type-Algorithm (INFWTA). This technique is so general that it can be used to extend the usage of other algorithms along the same lines. © 2021, International Publications. All rights reserved.Item A comparison between two competing sixth convergence order algorithms under the same set of conditions(SINUS Association, 2021) Argyros, G.; Argyros, M.; Argyros, I.K.; George, S.There is a plethora of algorithms of the same convergence order for generating a sequence approximating a solution of an equation involving Banach space operators. But the set of convergence criteria is not the same in general. This makes the comparison between them hard and only numerically. Moreover, the convergence is established using Taylor series and by assuming the existence of high order derivatives not even appearing on these algorithms. Furthermore, no computable error estimates, uniqueness for the solution results or a ball of convergence is given. We address all these problems by using only the first derivative that actually appears on these algorithms and under the same set of convergence conditions. Our technique is so general that it can be used to extend the applicability of other algorithms along the same lines. © 2021, SINUS Association. 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 Influence of Center-Lipschitz Conditions in the Convergence Analysis of Multi-point Iterative Methods(International Publications, 2022) Argyros, I.K.; George, S.; Argyros, M.The aim of this article is to extend the local as well as the semi-local convergence analysis of multi-point iterative methods using center Lipschitz conditions in combination with our idea, of the restricted convergence region. It turns out that this way a finer convergence analysis for these methods is obtained than in earlier works and without additional hypotheses. Numerical examples favoring our technique over earlier ones completes this article. © 2022, International Publications. All rights reserved.Item Extended Kantorovich theory for solving nonlinear equations with applications(Springer Nature, 2023) Regmi, S.; Argyros, I.K.; George, S.; Argyros, M.The Kantorovich theory plays an important role in the study of nonlinear equations. It is used to establish the existence of a solution for an equation defined in an abstract space. The solution is usually determined by using an iterative process such as Newton’s or its variants. A plethora of convergence results are available based mainly on Lipschitz-like conditions on the derivatives, and the celebrated Kantorovich convergence criterion. But there are even simple real equations for which this criterion is not satisfied. Consequently, the applicability of the theory is limited. The question there arises: is it possible to extend this theory without adding convergence conditions? The answer is, Yes! This is the novelty and motivation for this paper. Other extensions include the determination of better information about the solution, i.e. its uniqueness ball; the ratio of quadratic convergence as well as more precise error analysis. The numerical section contains a Hammerstein-type nonlinear equation and other examples as applications. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.