Faculty Publications
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Item Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions(Springer-Verlag Italia s.r.l., 2016) Argyros, I.K.; George, S.We present a local convergence analysis of a sixth order iterative method for approximate a locally unique solution of an equation defined on the real line. Earlier studies such as Sharma et al. (Appl Math Comput 190:111–115, 2007) have shown convergence of these methods under hypotheses up to the fifth derivative of the function although only the first derivative appears in the method. In this study we expand the applicability of these methods using only hypotheses up to the first derivative of the function. Numerical examples are also presented in this study. © 2015, Springer-Verlag Italia.Item Ball convergence of some iterative methods for nonlinear equations in Banach space under weak conditions(Springer-Verlag Italia s.r.l., 2018) Argyros, I.K.; George, S.The aim of this paper is to expand the applicability of a fast iterative method in a Banach space setting. Moreover, we provide computable radius of convergence, error bounds on the distances involved and a uniqueness of the solution result based on Lipschitz-type functions not given before. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. The convegence order is determined using the computational order of convergence or the approximate order of convergence. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study. © 2017, Springer-Verlag Italia S.r.l.Item Expanding the applicability of an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems(Springer-Verlag Italia s.r.l., 2019) Argyros, I.K.; Cho, Y.J.; George, S.; Xiao, Y.We expand the applicability of an a posteriori parameter choice strategy for Tikhonov regularization of the nonlinear ill-posed problem presented in Jin and Hou (Numer Math 83:139–159, 1999) by weakening the conditions needed in Jin and Hou [13]. Using a center-type Lipschitz condition instead of a Lipschitz-type condition used in Jin and Hou [13], Scherzer et al. (SIAM J Numer Anal 30:1796–1838, 1993), we obtain a tighter convergence analysis. Numerical examples are presented to show that our results apply but earlier ones do not apply to solve equations. © 2019, The Royal Academy of Sciences, Madrid.Item On the complexity of convergence for high order iterative methods(Academic Press Inc., 2022) Argyros, I.K.; George, S.; Argyros, C.Lipschitz-type conditions on the second derivative or conditions on higher than two derivatives not appearing on these methods have been employed to prove convergence. But these restrictions limit the applicability of high convergence order iterative methods although they may converge. That is why a new semi-local analysis is presented using only information taken from the derivatives on these methods. The new results compare favorably to the earlier ones even if the earlier conditions are used, since the latter use tighter Lipschitz parameters. Special cases and applications test convergence criteria. © 2022 Elsevier Inc.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.Item On a unified convergence analysis for Newton-type methods solving generalized equations with the Aubin property(Academic Press Inc., 2024) Argyros, I.K.; George, S.A plethora of applications from diverse disciplines reduce to solving generalized equations involving Banach space valued operators. These equations are solved mostly iteratively, when a sequence is generated approximating a solution provided that certain conditions are valid on the starting point and the operators appearing on the method. Secant-type methods are developed whose specializations reduce to well known methods such as Newton, modified Newton, Secant, Kurchatov and Steffensen to mention a few. Unified local as well as semi-local analysis of these methods is presented using the celebrated contraction mapping principle in combination with the Aubin property of a set valued operator, and generalized continuity assumption on the operators on these methods. Numerical applications complement the theory. © 2023 Elsevier Inc.